\(\int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [595]

Optimal result

Integrand size = 45, antiderivative size = 333 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (1304 A+1132 B+1015 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d}+\frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (680 A+628 B+545 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d} \]

[Out]

1/512*a^(5/2)*(1304*A+1132*B+1015*C)*arcsinh(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+1/60*a*(12*B+5*C)*se
c(d*x+c)^(7/2)*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/6*C*sec(d*x+c)^(7/2)*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d+
1/512*a^3*(1304*A+1132*B+1015*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/768*a^3*(1304*A+1132*B
+1015*C)*sec(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/960*a^3*(680*A+628*B+545*C)*sec(d*x+c)^(7/2)*s
in(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/480*a^2*(120*A+156*B+115*C)*sec(d*x+c)^(7/2)*sin(d*x+c)*(a+a*sec(d*x+c))^
(1/2)/d

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4173, 4103, 4101, 3888, 3886, 221} \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (1304 A+1132 B+1015 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{512 d}+\frac {a^3 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{960 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{768 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{512 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{480 d}+\frac {a (12 B+5 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac {C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d} \]

[In]

Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(
1304*A + 1132*B + 1015*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1
132*B + 1015*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(680*A + 628*B + 545*
C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Sec[c + d*
x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(12*B + 5*C)*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*
x])^(3/2)*Sin[c + d*x])/(60*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 3886

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(a/(b
*f))*Sqrt[a*(d/b)], Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]])], x] /; Free
Q[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*(d/b), 0]

Rule 3888

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*d*
Cot[e + f*x]*((d*Csc[e + f*x])^(n - 1)/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Dist[2*a*d*((n - 1)/(b*(2
*n - 1))), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a
^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]

Rule 4101

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(
B_.) + (A_)), x_Symbol] :> Simp[-2*b*B*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])
), x] + Dist[(A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1)), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n, x], x]
/; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n
, 0] &&  !LtQ[n, 0]

Rule 4103

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m +
n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*d*(m + n) + B*(b*d
*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] &&
NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1]

Rule 4173

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(
(d*Csc[e + f*x])^n/(f*(m + n + 1))), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^
n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b*B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A
, B, C, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)] &&  !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (12 A+5 C)+\frac {1}{2} a (12 B+5 C) \sec (c+d x)\right ) \, dx}{6 a} \\ & = \frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {15}{4} a^2 (8 A+4 B+5 C)+\frac {1}{4} a^2 (120 A+156 B+115 C) \sec (c+d x)\right ) \, dx}{30 a} \\ & = \frac {a^2 (120 A+156 B+115 C) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {5}{8} a^3 (312 A+252 B+235 C)+\frac {3}{8} a^3 (680 A+628 B+545 C) \sec (c+d x)\right ) \, dx}{120 a} \\ & = \frac {a^3 (680 A+628 B+545 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {1}{384} \left (a^2 (1304 A+1132 B+1015 C)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (680 A+628 B+545 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {1}{512} \left (a^2 (1304 A+1132 B+1015 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (680 A+628 B+545 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac {\left (a^2 (1304 A+1132 B+1015 C)\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx}{1024} \\ & = \frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (680 A+628 B+545 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}-\frac {\left (a^2 (1304 A+1132 B+1015 C)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d} \\ & = \frac {a^{5/2} (1304 A+1132 B+1015 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d}+\frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (680 A+628 B+545 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (12 B+5 C) \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {C \sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(757\) vs. \(2(333)=666\).

Time = 7.08 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.27 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 A \sqrt {a (1+\sec (c+d x))} \left (\frac {489 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {326 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {136 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {48 \sec ^{\frac {7}{2}}(c+d x) \sqrt {1+\sec (c+d x)} \sin (c+d x)}{d}+\frac {489 \arcsin \left (\sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\right )}{192 \sqrt {1+\sec (c+d x)}}+\frac {a^2 C \sqrt {a (1+\sec (c+d x))} \left (\frac {3045 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {2030 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {1624 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {1392 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {640 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {256 \sec ^{\frac {11}{2}}(c+d x) \sqrt {1+\sec (c+d x)} \sin (c+d x)}{d}+\frac {3045 \arcsin \left (\sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\right )}{1536 \sqrt {1+\sec (c+d x)}}+\frac {a^2 B \sqrt {a (1+\sec (c+d x))} \left (\frac {4245 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {2830 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {2264 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {1008 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {384 \sec ^{\frac {9}{2}}(c+d x) \sqrt {1+\sec (c+d x)} \sin (c+d x)}{d}+\frac {4245 \arcsin \left (\sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}\right )}{1920 \sqrt {1+\sec (c+d x)}} \]

[In]

Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^2*A*Sqrt[a*(1 + Sec[c + d*x])]*((489*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (326*Sec
[c + d*x]^(5/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (136*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*Sqrt[1 + S
ec[c + d*x]]) + (48*Sec[c + d*x]^(7/2)*Sqrt[1 + Sec[c + d*x]]*Sin[c + d*x])/d + (489*ArcSin[Sqrt[1 - Sec[c + d
*x]]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(192*Sqrt[1 + Sec[c + d*x]]) + (a^2*C*
Sqrt[a*(1 + Sec[c + d*x])]*((3045*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (2030*Sec[c +
d*x]^(5/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (1624*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c
+ d*x]]) + (1392*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (640*Sec[c + d*x]^(11/2)*Sin[c
+ d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (256*Sec[c + d*x]^(11/2)*Sqrt[1 + Sec[c + d*x]]*Sin[c + d*x])/d + (3045*
ArcSin[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(1536*Sqrt[1
+ Sec[c + d*x]]) + (a^2*B*Sqrt[a*(1 + Sec[c + d*x])]*((4245*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c
+ d*x]]) + (2830*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (2264*Sec[c + d*x]^(7/2)*Sin[c
+ d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (1008*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (384
*Sec[c + d*x]^(9/2)*Sqrt[1 + Sec[c + d*x]]*Sin[c + d*x])/d + (4245*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x]
)/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(1920*Sqrt[1 + Sec[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(778\) vs. \(2(289)=578\).

Time = 0.81 (sec) , antiderivative size = 779, normalized size of antiderivative = 2.34

\[-\frac {a^{2} \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (19560 A \arctan \left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{3}+19560 A \cos \left (d x +c \right )^{3} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )-39120 A \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}+16980 B \arctan \left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{3}+16980 B \cos \left (d x +c \right )^{3} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )-33960 B \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+15225 C \arctan \left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{3}+15225 C \cos \left (d x +c \right )^{3} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right )-30450 C \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}-26080 A \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}-22640 B \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )-20300 C \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}-14720 A \sin \left (d x +c \right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}-18112 B \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )-16240 C \sin \left (d x +c \right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}-3840 A \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \tan \left (d x +c \right )-11136 B \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \tan \left (d x +c \right )-13920 C \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \tan \left (d x +c \right )-3072 B \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )-8960 C \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )-2560 C \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}\right )}{15360 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\]

[In]

int(sec(d*x+c)^(5/2)*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

-1/15360*a^2/d*sec(d*x+c)^(5/2)*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2)*(19560*A*arc
tan(1/2*(-cos(d*x+c)+sin(d*x+c)-1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3+19560*A*cos(d*x+c)^3
*arctan(1/2*(cos(d*x+c)+sin(d*x+c)+1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2))-39120*A*cos(d*x+c)^2*sin(d*x+c
)*(-1/(cos(d*x+c)+1))^(1/2)+16980*B*arctan(1/2*(-cos(d*x+c)+sin(d*x+c)-1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(
1/2))*cos(d*x+c)^3+16980*B*cos(d*x+c)^3*arctan(1/2*(cos(d*x+c)+sin(d*x+c)+1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1)
)^(1/2))-33960*B*(-1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^2*sin(d*x+c)+15225*C*arctan(1/2*(-cos(d*x+c)+sin(d*x+c)-
1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3+15225*C*cos(d*x+c)^3*arctan(1/2*(cos(d*x+c)+sin(d*x+
c)+1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2))-30450*C*cos(d*x+c)^2*sin(d*x+c)*(-1/(cos(d*x+c)+1))^(1/2)-2608
0*A*cos(d*x+c)*sin(d*x+c)*(-1/(cos(d*x+c)+1))^(1/2)-22640*B*(-1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)-20
300*C*cos(d*x+c)*sin(d*x+c)*(-1/(cos(d*x+c)+1))^(1/2)-14720*A*sin(d*x+c)*(-1/(cos(d*x+c)+1))^(1/2)-18112*B*(-1
/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-16240*C*sin(d*x+c)*(-1/(cos(d*x+c)+1))^(1/2)-3840*A*(-1/(cos(d*x+c)+1))^(1/2
)*tan(d*x+c)-11136*B*(-1/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)-13920*C*(-1/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)-3072*B*
(-1/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)*sec(d*x+c)-8960*C*(-1/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)*sec(d*x+c)-2560*C*
(-1/(cos(d*x+c)+1))^(1/2)*tan(d*x+c)*sec(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.90 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {15 \, {\left ({\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - \frac {4 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac {4 \, {\left (15 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 10 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (920 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (40 \, A + 116 \, B + 145 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 128 \, {\left (12 \, B + 35 \, C\right )} a^{2} \cos \left (d x + c\right ) + 1280 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{30720 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}, \frac {15 \, {\left ({\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5}\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right ) + \frac {2 \, {\left (15 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 10 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (920 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (40 \, A + 116 \, B + 145 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 128 \, {\left (12 \, B + 35 \, C\right )} a^{2} \cos \left (d x + c\right ) + 1280 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{15360 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}\right ] \]

[In]

integrate(sec(d*x+c)^(5/2)*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/30720*(15*((1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^6 + (1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^5)*s
qrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*
x + c) + a)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)) + 8*a)/(cos(d*x + c)^3 + cos(d*x + c)^2)) + 4*(15*(1
304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^5 + 10*(1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^4 + 8*(920*A + 11
32*B + 1015*C)*a^2*cos(d*x + c)^3 + 48*(40*A + 116*B + 145*C)*a^2*cos(d*x + c)^2 + 128*(12*B + 35*C)*a^2*cos(d
*x + c) + 1280*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)
^6 + d*cos(d*x + c)^5), 1/15360*(15*((1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^6 + (1304*A + 1132*B + 1015*C
)*a^2*cos(d*x + c)^5)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*si
n(d*x + c)/(a*cos(d*x + c)^2 - a*cos(d*x + c) - 2*a)) + 2*(15*(1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^5 +
10*(1304*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^4 + 8*(920*A + 1132*B + 1015*C)*a^2*cos(d*x + c)^3 + 48*(40*A +
116*B + 145*C)*a^2*cos(d*x + c)^2 + 128*(12*B + 35*C)*a^2*cos(d*x + c) + 1280*C*a^2)*sqrt((a*cos(d*x + c) + a
)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)]

Sympy [F(-1)]

Timed out. \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**(5/2)*(a+a*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16461 vs. \(2 (289) = 578\).

Time = 2.74 (sec) , antiderivative size = 16461, normalized size of antiderivative = 49.43 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^(5/2)*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/30720*(40*(1956*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x +
4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(15/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 652*(sqrt(2)*a^
2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x
+ 2*c))*cos(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 6204*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2
)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(11/4*arctan2(sin
(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2060*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sq
rt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c
))) + 2060*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4
*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 6204*(sqrt(2)*a^2*sin(8*
d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))
*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 652*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(
6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*
c), cos(2*d*x + 2*c))) - 1956*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*s
in(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 489*(
a^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a
^2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*
d*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(
4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x +
2*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c)
+ 3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2
*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*
sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 489*(a^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6
*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*
d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^
2 + 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c)
+ a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*
a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(
2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*
cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))
)^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))) + 2) - 489*(a^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*
c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c
)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*
(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*c
os(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x
+ 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(
3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 489*(a
^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^
2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d
*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4
*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2
*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) +
3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*
d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2
(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*s
qrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) - 1956*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt
(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(
15/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 652*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*
x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(13/4*arctan2(sin
(2*d*x + 2*c), cos(2*d*x + 2*c))) - 6204*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sq
rt(2)*a^2*cos(4*d*x + 4*c) + 4*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(11/4*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c))) + 2060*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4
*d*x + 4*c) + 4*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)
)) - 2060*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*
sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 6204*(sqrt(
2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*sqrt(2)*a^2*cos(
2*d*x + 2*c) + sqrt(2)*a^2)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 652*(sqrt(2)*a^2*cos(8*d*x
+ 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*sqrt(2)*a^2*cos(2*d*x + 2*c) + sq
rt(2)*a^2)*sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1956*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(
2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(1
/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*A*sqrt(a)/(2*(4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos
(2*d*x + 2*c) + 1)*cos(8*d*x + 8*c) + cos(8*d*x + 8*c)^2 + 8*(6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos
(6*d*x + 6*c) + 16*cos(6*d*x + 6*c)^2 + 12*(4*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 36*cos(4*d*x + 4*c)^2 +
16*cos(2*d*x + 2*c)^2 + 4*(2*sin(6*d*x + 6*c) + 3*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + s
in(8*d*x + 8*c)^2 + 16*(3*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 16*sin(6*d*x + 6*c)^2 + 36
*sin(4*d*x + 4*c)^2 + 48*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*sin(2*d*x + 2*c)^2 + 8*cos(2*d*x + 2*c) + 1) +
4*(16980*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) +
10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(19/4*arctan2(sin(2*d*x + 2*c), cos(2*d*
x + 2*c))) + 5660*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x
+ 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(17/4*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c))) + 81504*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*s
in(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(15/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) + 8320*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(
2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(13/4*arctan2(s
in(2*d*x + 2*c), cos(2*d*x + 2*c))) + 86440*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) +
10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(11/4*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 86440*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x
+ 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*
cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 8320*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*si
n(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x +
2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 81504*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2
)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin
(2*d*x + 2*c))*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 5660*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5
*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*
a^2*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 16980*(sqrt(2)*a^2*sin(10*d*x + 1
0*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*
sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 4245*(a^2*cos(10*d*x + 10
*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2*cos(2*d*x
+ 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*a^2*sin(4*
d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*d*x + 2*c)
+ a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c
) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) +
a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 20*(5*
a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4
*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) +
a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)
)*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 4245*(a^2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*co
s(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*
sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*
d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*co
s(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6
*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x
+ 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) +
10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x
+ 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*
(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*
d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) - 4245*(a
^2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2
+ 25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^
2 + 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^
2*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a
^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*
cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d
*x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x +
6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*
sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c)
)*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin
(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 4245*(a^2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8
*c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x +
10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4
*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x
+ 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c)
+ 10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*
(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*
cos(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x
+ 2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8
*d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(
1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c
))) + 2) - 16980*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x +
6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(19/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) - 5660*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*s
qrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*
sin(17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 81504*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*
cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x
+ 2*c) + sqrt(2)*a^2)*sin(15/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 8320*(sqrt(2)*a^2*cos(10*d*x +
10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5
*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 86440*(sq
rt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a
^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d
*x + 2*c))) + 86440*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*
x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(9/4*arctan2(sin
(2*d*x + 2*c), cos(2*d*x + 2*c))) + 8320*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10
*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2
)*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 81504*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2
*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*
x + 2*c) + sqrt(2)*a^2)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 5660*(sqrt(2)*a^2*cos(10*d*x +
10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5
*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 16980*(sqr
t(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^
2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))))*B*sqrt(a)/(2*(5*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 10*cos(4*d*x + 4*c) + 5*cos(2*d*x + 2*c) +
1)*cos(10*d*x + 10*c) + cos(10*d*x + 10*c)^2 + 10*(10*cos(6*d*x + 6*c) + 10*cos(4*d*x + 4*c) + 5*cos(2*d*x +
2*c) + 1)*cos(8*d*x + 8*c) + 25*cos(8*d*x + 8*c)^2 + 20*(10*cos(4*d*x + 4*c) + 5*cos(2*d*x + 2*c) + 1)*cos(6*d
*x + 6*c) + 100*cos(6*d*x + 6*c)^2 + 20*(5*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 100*cos(4*d*x + 4*c)^2 + 2
5*cos(2*d*x + 2*c)^2 + 10*(sin(8*d*x + 8*c) + 2*sin(6*d*x + 6*c) + 2*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(
10*d*x + 10*c) + sin(10*d*x + 10*c)^2 + 50*(2*sin(6*d*x + 6*c) + 2*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*
d*x + 8*c) + 25*sin(8*d*x + 8*c)^2 + 100*(2*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 100*sin(6*
d*x + 6*c)^2 + 100*sin(4*d*x + 4*c)^2 + 100*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*sin(2*d*x + 2*c)^2 + 10*cos
(2*d*x + 2*c) + 1) + 5*(12180*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*
a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2
*d*x + 2*c))*cos(23/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 4060*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*
sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2
)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(21/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))
+ 70644*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c)
+ 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(19/4
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 22620*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d
*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*
c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 147592*(sqrt(2)*a
^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*si
n(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(15/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) - 37800*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqr
t(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*
sin(2*d*x + 2*c))*cos(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 37800*(sqrt(2)*a^2*sin(12*d*x + 12*c
) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*
sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) - 147592*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x
+ 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*c
os(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 22620*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*si
n(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*
x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 70644*(sqrt(
2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^
2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(5/4*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))) - 4060*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*s
qrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^
2*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 12180*(sqrt(2)*a^2*sin(12*d*x + 12*
c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15
*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) - 3045*(a^2*cos(12*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 + 225*a^2*cos(8*d*x + 8*c)^2 + 400*a^2*c
os(6*d*x + 6*c)^2 + 225*a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(12*d*x + 12*c)^2 + 36*a^2
*sin(10*d*x + 10*c)^2 + 225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x + 6*c)^2 + 225*a^2*sin(4*d*x + 4*c)^2 +
180*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2 + 12*a^2*cos(2*d*x + 2*c) + a^2 + 2*(6*
a^2*cos(10*d*x + 10*c) + 15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*c
os(2*d*x + 2*c) + a^2)*cos(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos
(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*
d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x +
2*c) + a^2)*cos(6*d*x + 6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 2*(6*a^2*sin(10*d*x + 10*c
) + 15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(
12*d*x + 12*c) + 12*(15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2
*d*x + 2*c))*sin(10*d*x + 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*
c))*sin(8*d*x + 8*c) + 120*(5*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*s
qrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c))) + 2) + 3045*(a^2*cos(12*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 + 225*a^2*cos(8*d*x + 8*c)^
2 + 400*a^2*cos(6*d*x + 6*c)^2 + 225*a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(12*d*x + 12*
c)^2 + 36*a^2*sin(10*d*x + 10*c)^2 + 225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x + 6*c)^2 + 225*a^2*sin(4*d
*x + 4*c)^2 + 180*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2 + 12*a^2*cos(2*d*x + 2*c)
+ a^2 + 2*(6*a^2*cos(10*d*x + 10*c) + 15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4
*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c)
+ 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(20*a^2*cos(6*d*x + 6*c) +
15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^2*cos(4*d*x + 4*c) + 6*a^2
*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 2*(6*a^2*sin(
10*d*x + 10*c) + 15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x
+ 2*c))*sin(12*d*x + 12*c) + 12*(15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c)
+ 6*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*s
in(2*d*x + 2*c))*sin(8*d*x + 8*c) + 120*(5*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*lo
g(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2
*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) + 2) - 3045*(a^2*cos(12*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 + 225*a^2*cos(
8*d*x + 8*c)^2 + 400*a^2*cos(6*d*x + 6*c)^2 + 225*a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c)^2 + a^2*sin
(12*d*x + 12*c)^2 + 36*a^2*sin(10*d*x + 10*c)^2 + 225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x + 6*c)^2 + 22
5*a^2*sin(4*d*x + 4*c)^2 + 180*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2 + 12*a^2*cos(
2*d*x + 2*c) + a^2 + 2*(6*a^2*cos(10*d*x + 10*c) + 15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*
cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos
(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(20*a^2*cos(6*
d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^2*cos(4*d*x +
4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) +
2*(6*a^2*sin(10*d*x + 10*c) + 15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*
a^2*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 12*(15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(
4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4
*c) + 6*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 120*(5*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d
*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 3045*(a^2*cos(12*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 +
225*a^2*cos(8*d*x + 8*c)^2 + 400*a^2*cos(6*d*x + 6*c)^2 + 225*a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c
)^2 + a^2*sin(12*d*x + 12*c)^2 + 36*a^2*sin(10*d*x + 10*c)^2 + 225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x
+ 6*c)^2 + 225*a^2*sin(4*d*x + 4*c)^2 + 180*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2
+ 12*a^2*cos(2*d*x + 2*c) + a^2 + 2*(6*a^2*cos(10*d*x + 10*c) + 15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6
*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c)
+ 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(
20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^
2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*
d*x + 4*c) + 2*(6*a^2*sin(10*d*x + 10*c) + 15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*
x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 12*(15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c)
+ 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*
sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 120*(5*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x +
2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)
*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) - 12180*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)
*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*c
os(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(23/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) - 4060*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x +
8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + s
qrt(2)*a^2)*sin(21/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 70644*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*
sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2
)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(19/4*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c))) - 22620*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos
(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x +
2*c) + sqrt(2)*a^2)*sin(17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 147592*(sqrt(2)*a^2*cos(12*d*x + 1
2*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) +
15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(15/4*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))) + 37800*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2
)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos
(2*d*x + 2*c) + sqrt(2)*a^2)*sin(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 37800*(sqrt(2)*a^2*cos(12
*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x +
6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(11/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) + 147592*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) +
15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2
)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 22620*(sqrt(2)*a^
2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos
(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(7/4*arctan
2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 70644*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10
*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*
sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 4060*(sqrt(
2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^
2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(3/4*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 12180*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x
+ 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c)
+ 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*C*sqrt(
a)/(2*(6*cos(10*d*x + 10*c) + 15*cos(8*d*x + 8*c) + 20*cos(6*d*x + 6*c) + 15*cos(4*d*x + 4*c) + 6*cos(2*d*x +
2*c) + 1)*cos(12*d*x + 12*c) + cos(12*d*x + 12*c)^2 + 12*(15*cos(8*d*x + 8*c) + 20*cos(6*d*x + 6*c) + 15*cos(4
*d*x + 4*c) + 6*cos(2*d*x + 2*c) + 1)*cos(10*d*x + 10*c) + 36*cos(10*d*x + 10*c)^2 + 30*(20*cos(6*d*x + 6*c) +
15*cos(4*d*x + 4*c) + 6*cos(2*d*x + 2*c) + 1)*cos(8*d*x + 8*c) + 225*cos(8*d*x + 8*c)^2 + 40*(15*cos(4*d*x +
4*c) + 6*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + 400*cos(6*d*x + 6*c)^2 + 30*(6*cos(2*d*x + 2*c) + 1)*cos(4*d
*x + 4*c) + 225*cos(4*d*x + 4*c)^2 + 36*cos(2*d*x + 2*c)^2 + 2*(6*sin(10*d*x + 10*c) + 15*sin(8*d*x + 8*c) + 2
0*sin(6*d*x + 6*c) + 15*sin(4*d*x + 4*c) + 6*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + sin(12*d*x + 12*c)^2 + 12*
(15*sin(8*d*x + 8*c) + 20*sin(6*d*x + 6*c) + 15*sin(4*d*x + 4*c) + 6*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 36
*sin(10*d*x + 10*c)^2 + 30*(20*sin(6*d*x + 6*c) + 15*sin(4*d*x + 4*c) + 6*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) +
225*sin(8*d*x + 8*c)^2 + 120*(5*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 400*sin(6*d*x + 6*c
)^2 + 225*sin(4*d*x + 4*c)^2 + 180*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*sin(2*d*x + 2*c)^2 + 12*cos(2*d*x +
2*c) + 1))/d

Giac [F]

\[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(sec(d*x+c)^(5/2)*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]

[In]

int((a + a/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

int((a + a/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2), x)