\(\int \frac {(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) [1261]
Optimal result
Integrand size = 45, antiderivative size = 303 \[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {a^{3/2} (176 A+150 B+133 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{128 d}+\frac {a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a (10 B+3 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)}
\]
[Out]
1/5*C*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(7/2)+1/128*a^(3/2)*(176*A+150*B+133*C)*arcsinh(a^(1/2)*t
an(d*x+c)/(a+a*sec(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+1/240*a^2*(80*A+90*B+67*C)*sin(d*x+c)/d/
cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^(1/2)+1/192*a^2*(176*A+150*B+133*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*sec(d
*x+c))^(1/2)+1/128*a^2*(176*A+150*B+133*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/(a+a*sec(d*x+c))^(1/2)+1/40*a*(10*B+3
*C)*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d/cos(d*x+c)^(7/2)
Rubi [A] (verified)
Time = 1.05 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4350, 4173, 4103, 4101, 3888,
3886, 221} \[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {a^{3/2} (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{128 d}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a (10 B+3 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac {7}{2}}(c+d x)}
\]
[In]
Int[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]
[Out]
(a^(3/2)*(176*A + 150*B + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*S
qrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 90*B + 67*C)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[
c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) +
(a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(10*B + 3*
C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*
x])/(5*d*Cos[c + d*x]^(7/2))
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]
Rule 4350
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]
Rubi steps \begin{align*}
\text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {5}{2} a (2 A+C)+\frac {1}{2} a (10 B+3 C) \sec (c+d x)\right ) \, dx}{5 a} \\ & = \frac {a (10 B+3 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {5}{4} a^2 (16 A+10 B+11 C)+\frac {1}{4} a^2 (80 A+90 B+67 C) \sec (c+d x)\right ) \, dx}{20 a} \\ & = \frac {a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a (10 B+3 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{96} \left (a (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a (10 B+3 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{128} \left (a (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a (10 B+3 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{256} \left (a (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a (10 B+3 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\left (a (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\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 )}{128 d} \\ & = \frac {a^{3/2} (176 A+150 B+133 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{128 d}+\frac {a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a (10 B+3 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x)} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(608\) vs. \(2(303)=606\).
Time = 12.83 (sec) , antiderivative size = 608, normalized size of antiderivative = 2.01
\[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {4 \sec ^3\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )} \left (\frac {11 A \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32 \sqrt {2}}+\frac {75 B \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{256 \sqrt {2}}+\frac {133 C \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{512 \sqrt {2}}+\frac {C \sin \left (\frac {1}{2} (c+d x)\right )}{20 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {B \sin \left (\frac {1}{2} (c+d x)\right )}{16 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {19 C \sin \left (\frac {1}{2} (c+d x)\right )}{160 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {A \sin \left (\frac {1}{2} (c+d x)\right )}{12 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {5 B \sin \left (\frac {1}{2} (c+d x)\right )}{32 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {133 C \sin \left (\frac {1}{2} (c+d x)\right )}{960 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {11 A \sin \left (\frac {1}{2} (c+d x)\right )}{48 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {25 B \sin \left (\frac {1}{2} (c+d x)\right )}{128 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {133 C \sin \left (\frac {1}{2} (c+d x)\right )}{768 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {11 A \sin \left (\frac {1}{2} (c+d x)\right )}{32 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {75 B \sin \left (\frac {1}{2} (c+d x)\right )}{256 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {133 C \sin \left (\frac {1}{2} (c+d x)\right )}{512 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {7}{2}}(c+d x)}
\]
[In]
Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]
[Out]
(4*Sec[(c + d*x)/2]^3*(a*(1 + Sec[c + d*x]))^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c
+ d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*((11*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])/(32*Sqrt[2]) + (75
*B*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])/(256*Sqrt[2]) + (133*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])/(512*Sqrt[2])
+ (C*Sin[(c + d*x)/2])/(20*(1 - 2*Sin[(c + d*x)/2]^2)^5) + (B*Sin[(c + d*x)/2])/(16*(1 - 2*Sin[(c + d*x)/2]^2)
^4) + (19*C*Sin[(c + d*x)/2])/(160*(1 - 2*Sin[(c + d*x)/2]^2)^4) + (A*Sin[(c + d*x)/2])/(12*(1 - 2*Sin[(c + d*
x)/2]^2)^3) + (5*B*Sin[(c + d*x)/2])/(32*(1 - 2*Sin[(c + d*x)/2]^2)^3) + (133*C*Sin[(c + d*x)/2])/(960*(1 - 2*
Sin[(c + d*x)/2]^2)^3) + (11*A*Sin[(c + d*x)/2])/(48*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (25*B*Sin[(c + d*x)/2])/(
128*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (133*C*Sin[(c + d*x)/2])/(768*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (11*A*Sin[(c
+ d*x)/2])/(32*(1 - 2*Sin[(c + d*x)/2]^2)) + (75*B*Sin[(c + d*x)/2])/(256*(1 - 2*Sin[(c + d*x)/2]^2)) + (133*
C*Sin[(c + d*x)/2])/(512*(1 - 2*Sin[(c + d*x)/2]^2))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Se
c[c + d*x]^(7/2))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(729\) vs. \(2(261)=522\).
Time = 1.08 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.41
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\(\frac {a \left (5280 A \cos \left (d x +c \right )^{4} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}-2640 A \cos \left (d x +c \right )^{5} \arctan \left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}}\right )-2640 A \cos \left (d x +c \right )^{5} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}}\right )+4500 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4} \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}-2250 B \cos \left (d x +c \right )^{5} \arctan \left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}}\right )-2250 B \cos \left (d x +c \right )^{5} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}}\right )+3990 C \cos \left (d x +c \right )^{4} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}-1995 C \cos \left (d x +c \right )^{5} \arctan \left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}}\right )-1995 C \cos \left (d x +c \right )^{5} \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}}\right )+3520 A \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+3000 B \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3} \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+2660 C \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+1280 A \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+2400 B \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+2128 C \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+960 B \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+1824 C \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+768 C \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{3840 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )^{\frac {9}{2}}}\) |
\(730\) |
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[In]
int((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
[Out]
1/3840*a/d*(5280*A*cos(d*x+c)^4*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)-2640*A*cos(d*x+c)^5*arctan(1/2*(-cos(d*x+
c)+sin(d*x+c)-1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))-2640*A*cos(d*x+c)^5*arctan(1/2*(cos(d*x+c)+sin(d*x+
c)+1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))+4500*B*sin(d*x+c)*cos(d*x+c)^4*(-1/(1+cos(d*x+c)))^(1/2)-2250*
B*cos(d*x+c)^5*arctan(1/2*(-cos(d*x+c)+sin(d*x+c)-1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))-2250*B*cos(d*x+
c)^5*arctan(1/2*(cos(d*x+c)+sin(d*x+c)+1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))+3990*C*cos(d*x+c)^4*sin(d*
x+c)*(-1/(1+cos(d*x+c)))^(1/2)-1995*C*cos(d*x+c)^5*arctan(1/2*(-cos(d*x+c)+sin(d*x+c)-1)/(1+cos(d*x+c))/(-1/(1
+cos(d*x+c)))^(1/2))-1995*C*cos(d*x+c)^5*arctan(1/2*(cos(d*x+c)+sin(d*x+c)+1)/(1+cos(d*x+c))/(-1/(1+cos(d*x+c)
))^(1/2))+3520*A*cos(d*x+c)^3*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+3000*B*sin(d*x+c)*cos(d*x+c)^3*(-1/(1+cos(d
*x+c)))^(1/2)+2660*C*cos(d*x+c)^3*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+1280*A*sin(d*x+c)*cos(d*x+c)^2*(-1/(1+c
os(d*x+c)))^(1/2)+2400*B*cos(d*x+c)^2*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+2128*C*sin(d*x+c)*cos(d*x+c)^2*(-1/
(1+cos(d*x+c)))^(1/2)+960*B*cos(d*x+c)*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+1824*C*cos(d*x+c)*sin(d*x+c)*(-1/(
1+cos(d*x+c)))^(1/2)+768*C*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2))*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(-1/(
1+cos(d*x+c)))^(1/2)/cos(d*x+c)^(9/2)
Fricas [A] (verification not implemented)
none
Time = 0.53 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.82
\[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\left [\frac {4 \, {\left (15 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{4} + 10 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{3} + 8 \, {\left (80 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{2} + 48 \, {\left (10 \, B + 19 \, C\right )} a \cos \left (d x + c\right ) + 384 \, C a\right )} \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 ) + 15 \, {\left ({\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{6} + {\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{5}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 4 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{7680 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}, \frac {2 \, {\left (15 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{4} + 10 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{3} + 8 \, {\left (80 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{2} + 48 \, {\left (10 \, B + 19 \, C\right )} a \cos \left (d x + c\right ) + 384 \, C a\right )} \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 ) + 15 \, {\left ({\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{6} + {\left (176 \, A + 150 \, B + 133 \, C\right )} a \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 )}{3840 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}\right ]
\]
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x, algorithm="fricas")
[Out]
[1/7680*(4*(15*(176*A + 150*B + 133*C)*a*cos(d*x + c)^4 + 10*(176*A + 150*B + 133*C)*a*cos(d*x + c)^3 + 8*(80*
A + 150*B + 133*C)*a*cos(d*x + c)^2 + 48*(10*B + 19*C)*a*cos(d*x + c) + 384*C*a)*sqrt((a*cos(d*x + c) + a)/cos
(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 15*((176*A + 150*B + 133*C)*a*cos(d*x + c)^6 + (176*A + 150*B + 1
33*C)*a*cos(d*x + c)^5)*sqrt(a)*log((a*cos(d*x + c)^3 - 4*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(cos
(d*x + c) - 2)*sqrt(cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c)^2 + 8*a)/(cos(d*x + c)^3 + cos(d*x + c)^2)))
/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5), 1/3840*(2*(15*(176*A + 150*B + 133*C)*a*cos(d*x + c)^4 + 10*(176*A + 1
50*B + 133*C)*a*cos(d*x + c)^3 + 8*(80*A + 150*B + 133*C)*a*cos(d*x + c)^2 + 48*(10*B + 19*C)*a*cos(d*x + c) +
384*C*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 15*((176*A + 150*B + 133*C
)*a*cos(d*x + c)^6 + (176*A + 150*B + 133*C)*a*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))*sin(d*x + c)/(a*cos(d*x + c)^2 - a*cos(d*x + c) - 2*a)))/(d*cos(d*x + c
)^6 + d*cos(d*x + c)^5)]
Sympy [F(-1)]
Timed out. \[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out}
\]
[In]
integrate((a+a*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/cos(d*x+c)**(5/2),x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 10749 vs. \(2 (261) = 522\).
Time = 1.41 (sec) , antiderivative size = 10749, normalized size of antiderivative = 35.48
\[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x, algorithm="maxima")
[Out]
-1/7680*(80*(132*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*co
s(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 44*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x +
4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 216*(sqrt(2)*a*si
n(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c))) - 216*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x
+ 2*c))*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*si
n(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 132*(sqr
t(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))) - 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a
*sin(6*d*x + 6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2
+ 2*(3*a*cos(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d
*x + 4*c) + 6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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) + 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a*sin
(6*d*x + 6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2 + 2
*(3*a*cos(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d*x +
4*c) + 6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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), co
s(2*d*x + 2*c))) + 2) - 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a*sin(6*d
*x + 6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2 + 2*(3*
a*cos(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c
) + 6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arc
tan2(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*sqr
t(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) + 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a*sin(6*d*x +
6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2 + 2*(3*a*co
s(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) +
6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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) - 132*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c)
+ sqrt(2)*a)*sin(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2
)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) - 216*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt
(2)*a)*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 216*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*co
s(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))
) + 44*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*
sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 132*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x
+ 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*A*sq
rt(a)/(2*(3*cos(4*d*x + 4*c) + 3*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + cos(6*d*x + 6*c)^2 + 6*(3*cos(2*d*x
+ 2*c) + 1)*cos(4*d*x + 4*c) + 9*cos(4*d*x + 4*c)^2 + 9*cos(2*d*x + 2*c)^2 + 6*(sin(4*d*x + 4*c) + sin(2*d*x +
2*c))*sin(6*d*x + 6*c) + sin(6*d*x + 6*c)^2 + 9*sin(4*d*x + 4*c)^2 + 18*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9
*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c) + 1) + 30*(300*(sqrt(2)*a*sin(8*d*x + 8*c) + 4*sqrt(2)*a*sin(6*d*x +
6*c) + 6*sqrt(2)*a*sin(4*d*x + 4*c) + 4*sqrt(2)*a*sin(2*d*x + 2*c))*cos(15/4*arctan2(sin(2*d*x + 2*c), cos(2*d
*x + 2*c))) + 100*(sqrt(2)*a*sin(8*d*x + 8*c) + 4*sqrt(2)*a*sin(6*d*x + 6*c) + 6*sqrt(2)*a*sin(4*d*x + 4*c) +
4*sqrt(2)*a*sin(2*d*x + 2*c))*cos(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1140*(sqrt(2)*a*sin(8*d*
x + 8*c) + 4*sqrt(2)*a*sin(6*d*x + 6*c) + 6*sqrt(2)*a*sin(4*d*x + 4*c) + 4*sqrt(2)*a*sin(2*d*x + 2*c))*cos(11/
4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 228*(sqrt(2)*a*sin(8*d*x + 8*c) + 4*sqrt(2)*a*sin(6*d*x + 6*c
) + 6*sqrt(2)*a*sin(4*d*x + 4*c) + 4*sqrt(2)*a*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) + 228*(sqrt(2)*a*sin(8*d*x + 8*c) + 4*sqrt(2)*a*sin(6*d*x + 6*c) + 6*sqrt(2)*a*sin(4*d*x + 4*c) + 4*sq
rt(2)*a*sin(2*d*x + 2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1140*(sqrt(2)*a*sin(8*d*x + 8
*c) + 4*sqrt(2)*a*sin(6*d*x + 6*c) + 6*sqrt(2)*a*sin(4*d*x + 4*c) + 4*sqrt(2)*a*sin(2*d*x + 2*c))*cos(5/4*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 100*(sqrt(2)*a*sin(8*d*x + 8*c) + 4*sqrt(2)*a*sin(6*d*x + 6*c) + 6*
sqrt(2)*a*sin(4*d*x + 4*c) + 4*sqrt(2)*a*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))
) - 300*(sqrt(2)*a*sin(8*d*x + 8*c) + 4*sqrt(2)*a*sin(6*d*x + 6*c) + 6*sqrt(2)*a*sin(4*d*x + 4*c) + 4*sqrt(2)*
a*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 75*(a*cos(8*d*x + 8*c)^2 + 16*a*cos
(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 16*a*cos(2*d*x + 2*c)^2 + a*sin(8*d*x + 8*c)^2 + 16*a*sin(6*d*x +
6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 48*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a*sin(2*d*x + 2*c)^2 + 2*(4*a*c
os(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 8*(6*a*cos(4*d*x + 4*c)
+ 4*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 12*(4*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 8*a*cos(2*d*x
+ 2*c) + 4*(2*a*sin(6*d*x + 6*c) + 3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a*sin
(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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) + 75*(a*cos(8*d
*x + 8*c)^2 + 16*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 16*a*cos(2*d*x + 2*c)^2 + a*sin(8*d*x + 8*c)
^2 + 16*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 48*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a*sin(2*d
*x + 2*c)^2 + 2*(4*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 8*
(6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 12*(4*a*cos(2*d*x + 2*c) + a)*cos(4*d*x +
4*c) + 8*a*cos(2*d*x + 2*c) + 4*(2*a*sin(6*d*x + 6*c) + 3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(8*d*
x + 8*c) + 16*(3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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) - 75*(a*cos(8*d*x + 8*c)^2 + 16*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 + 16*a*cos(2*d*x + 2*c)^
2 + a*sin(8*d*x + 8*c)^2 + 16*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 48*a*sin(4*d*x + 4*c)*sin(2*d*x
+ 2*c) + 16*a*sin(2*d*x + 2*c)^2 + 2*(4*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)
*cos(8*d*x + 8*c) + 8*(6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 12*(4*a*cos(2*d*x +
2*c) + a)*cos(4*d*x + 4*c) + 8*a*cos(2*d*x + 2*c) + 4*(2*a*sin(6*d*x + 6*c) + 3*a*sin(4*d*x + 4*c) + 2*a*sin(
2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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) + 75*(a*cos(8*d*x + 8*c)^2 + 16*a*cos(6*d*x + 6*c)^2 + 36*a*cos(4*d*x + 4*c)^2 +
16*a*cos(2*d*x + 2*c)^2 + a*sin(8*d*x + 8*c)^2 + 16*a*sin(6*d*x + 6*c)^2 + 36*a*sin(4*d*x + 4*c)^2 + 48*a*sin(
4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a*sin(2*d*x + 2*c)^2 + 2*(4*a*cos(6*d*x + 6*c) + 6*a*cos(4*d*x + 4*c) + 4*a
*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 8*(6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c)
+ 12*(4*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 8*a*cos(2*d*x + 2*c) + 4*(2*a*sin(6*d*x + 6*c) + 3*a*sin(4
*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a*sin(4*d*x + 4*c) + 2*a*sin(2*d*x + 2*c))*sin(6*
d*x + 6*c) + a)*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) - 300*(sqrt(2)*a*cos(8*d*x + 8*c) + 4*sqrt(2)*a*cos(6*d*x +
6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(15/4*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c))) - 100*(sqrt(2)*a*cos(8*d*x + 8*c) + 4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*
d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) -
1140*(sqrt(2)*a*cos(8*d*x + 8*c) + 4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*
cos(2*d*x + 2*c) + sqrt(2)*a)*sin(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 228*(sqrt(2)*a*cos(8*d*x
+ 8*c) + 4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)
*a)*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 228*(sqrt(2)*a*cos(8*d*x + 8*c) + 4*sqrt(2)*a*cos(6
*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(7/4*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))) + 1140*(sqrt(2)*a*cos(8*d*x + 8*c) + 4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*
cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c
))) + 100*(sqrt(2)*a*cos(8*d*x + 8*c) + 4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2
)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 300*(sqrt(2)*a*cos(8*
d*x + 8*c) + 4*sqrt(2)*a*cos(6*d*x + 6*c) + 6*sqrt(2)*a*cos(4*d*x + 4*c) + 4*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt
(2)*a)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*B*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) + sin(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) + (7980*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c
) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(19/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))) + 2660*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c)
+ 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) + 38304*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) +
10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(15/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) + 12160*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) +
10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2
*c))) + 45400*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 1
0*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*
c))) - 45400*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10
*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)
)) - 12160*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*s
qrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))
- 38304*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqr
t(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) -
2660*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2
)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 79
80*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a
*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1995*
(a*cos(10*d*x + 10*c)^2 + 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^2 + 100*a*cos(4*d*x + 4*c)^2 + 25*a
*cos(2*d*x + 2*c)^2 + a*sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 100*a*sin(6*d*x + 6*c)^2 + 100*a*sin(
4*d*x + 4*c)^2 + 100*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x + 2*c)^2 + 2*(5*a*cos(8*d*x + 8*c) +
10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(10*d*x + 10*c) + 10*(10*a*cos(6
*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 20*(10*a*cos(4*d*x + 4*c) +
5*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 10*a*cos(2*d*x
+ 2*c) + 10*(a*sin(8*d*x + 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x
+ 10*c) + 50*(2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a*s
in(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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) + 1995*(a*cos(1
0*d*x + 10*c)^2 + 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^2 + 100*a*cos(4*d*x + 4*c)^2 + 25*a*cos(2*d
*x + 2*c)^2 + a*sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 100*a*sin(6*d*x + 6*c)^2 + 100*a*sin(4*d*x +
4*c)^2 + 100*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x + 2*c)^2 + 2*(5*a*cos(8*d*x + 8*c) + 10*a*co
s(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(10*d*x + 10*c) + 10*(10*a*cos(6*d*x + 6
*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 20*(10*a*cos(4*d*x + 4*c) + 5*a*cos
(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 10*a*cos(2*d*x + 2*c) +
10*(a*sin(8*d*x + 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x + 10*c)
+ 50*(2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a*sin(4*d*x
+ 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*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), co
s(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) - 1995*(a*cos(10*d*x +
10*c)^2 + 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^2 + 100*a*cos(4*d*x + 4*c)^2 + 25*a*cos(2*d*x + 2*c
)^2 + a*sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 100*a*sin(6*d*x + 6*c)^2 + 100*a*sin(4*d*x + 4*c)^2 +
100*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x + 2*c)^2 + 2*(5*a*cos(8*d*x + 8*c) + 10*a*cos(6*d*x
+ 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(10*d*x + 10*c) + 10*(10*a*cos(6*d*x + 6*c) + 10
*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 20*(10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x +
2*c) + a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 10*a*cos(2*d*x + 2*c) + 10*(a*s
in(8*d*x + 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2
*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a*sin(4*d*x + 4*c)
+ a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*s
in(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) + 1995*(a*cos(10*d*x + 10*c)^2
+ 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^2 + 100*a*cos(4*d*x + 4*c)^2 + 25*a*cos(2*d*x + 2*c)^2 + a*
sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 100*a*sin(6*d*x + 6*c)^2 + 100*a*sin(4*d*x + 4*c)^2 + 100*a*s
in(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x + 2*c)^2 + 2*(5*a*cos(8*d*x + 8*c) + 10*a*cos(6*d*x + 6*c) +
10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(10*d*x + 10*c) + 10*(10*a*cos(6*d*x + 6*c) + 10*a*cos(4
*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 20*(10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) +
a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 10*a*cos(2*d*x + 2*c) + 10*(a*sin(8*d*x
+ 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a*sin(6
*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a*sin(4*d*x + 4*c) + a*sin(
2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*a
rctan2(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) - 7980*(sqrt(2)*a*cos(10*d*x + 10*c) +
5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos
(2*d*x + 2*c) + sqrt(2)*a)*sin(19/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2660*(sqrt(2)*a*cos(10*d*x
+ 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqr
t(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 38304*(sqrt(2)*a*
cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4
*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(15/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 12160*
(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*co
s(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)
)) - 45400*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*s
qrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(11/4*arctan2(sin(2*d*x + 2*c), cos(2
*d*x + 2*c))) + 45400*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x +
6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(9/4*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))) + 12160*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*co
s(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(7/4*arctan2(sin
(2*d*x + 2*c), cos(2*d*x + 2*c))) + 38304*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sq
rt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(5/4*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2660*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c
) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*
sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 7980*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*
d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sq
rt(2)*a)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*C*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*(1
0*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 + 25*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))/d
Giac [F]
\[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x }
\]
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^{5/2}} \,d x
\]
[In]
int(((a + a/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^(5/2),x)
[Out]
int(((a + a/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^(5/2), x)