\(\int \frac {(a+a \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x))}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) [1153]

Optimal result

Integrand size = 37, antiderivative size = 332 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {a^{5/2} (1304 A+1015 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{512 d}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)} \]

[Out]

1/12*a*C*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(7/2)+1/6*C*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d/cos(d*
x+c)^(7/2)+1/512*a^(5/2)*(1304*A+1015*C)*arcsinh(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*s
ec(d*x+c)^(1/2)/d+1/192*a^3*(136*A+109*C)*sin(d*x+c)/d/cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^(1/2)+1/768*a^3*(1304
*A+1015*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(1/2)+1/512*a^3*(1304*A+1015*C)*sin(d*x+c)/d/cos(d*x
+c)^(3/2)/(a+a*sec(d*x+c))^(1/2)+1/96*a^2*(24*A+23*C)*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d/cos(d*x+c)^(7/2)

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {4350, 4174, 4103, 4101, 3888, 3886, 221} \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {a^{5/2} (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{512 d}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (24 A+23 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d \cos ^{\frac {7}{2}}(c+d x)} \]

[In]

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

[Out]

(a^(5/2)*(1304*A + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Se
c[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]])
+ (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 10
15*C)*Sin[c + d*x])/(512*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 23*C)*Sqrt[a + a*Sec[c
+ d*x]]*Sin[c + d*x])/(96*d*Cos[c + d*x]^(7/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Cos[c +
d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*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 4174

Int[((A_.) + 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*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, 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))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 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))^{5/2} \left (\frac {1}{2} a (12 A+5 C)+\frac {5}{2} a C \sec (c+d x)\right ) \, dx}{6 a} \\ & = \frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 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 {15}{4} a^2 (8 A+5 C)+\frac {5}{4} a^2 (24 A+23 C) \sec (c+d x)\right ) \, dx}{30 a} \\ & = \frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 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}{8} a^3 (312 A+235 C)+\frac {15}{8} a^3 (136 A+109 C) \sec (c+d x)\right ) \, dx}{120 a} \\ & = \frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{384} \left (a^2 (1304 A+1015 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^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{512} \left (a^2 (1304 A+1015 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^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (a^2 (1304 A+1015 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}{1024} \\ & = \frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\left (a^2 (1304 A+1015 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 )}{512 d} \\ & = \frac {a^{5/2} (1304 A+1015 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)}}{512 d}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 13.02 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.50 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {4 \sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )} \left (\frac {163 A \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{512 \sqrt {2}}+\frac {1015 C \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{4096 \sqrt {2}}+\frac {C \sin \left (\frac {1}{2} (c+d x)\right )}{48 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {7 C \sin \left (\frac {1}{2} (c+d x)\right )}{96 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {A \sin \left (\frac {1}{2} (c+d x)\right )}{32 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {29 C \sin \left (\frac {1}{2} (c+d x)\right )}{256 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {23 A \sin \left (\frac {1}{2} (c+d x)\right )}{192 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {203 C \sin \left (\frac {1}{2} (c+d x)\right )}{1536 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {163 A \sin \left (\frac {1}{2} (c+d x)\right )}{768 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {1015 C \sin \left (\frac {1}{2} (c+d x)\right )}{6144 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {163 A \sin \left (\frac {1}{2} (c+d x)\right )}{512 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {1015 C \sin \left (\frac {1}{2} (c+d x)\right )}{4096 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}\right )}{d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]

[In]

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

[Out]

(4*Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*(A + 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]*((163*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])/(512*Sqrt[2]) + (1015*C*ArcTanh[Sq
rt[2]*Sin[(c + d*x)/2]])/(4096*Sqrt[2]) + (C*Sin[(c + d*x)/2])/(48*(1 - 2*Sin[(c + d*x)/2]^2)^6) + (7*C*Sin[(c
+ d*x)/2])/(96*(1 - 2*Sin[(c + d*x)/2]^2)^5) + (A*Sin[(c + d*x)/2])/(32*(1 - 2*Sin[(c + d*x)/2]^2)^4) + (29*C
*Sin[(c + d*x)/2])/(256*(1 - 2*Sin[(c + d*x)/2]^2)^4) + (23*A*Sin[(c + d*x)/2])/(192*(1 - 2*Sin[(c + d*x)/2]^2
)^3) + (203*C*Sin[(c + d*x)/2])/(1536*(1 - 2*Sin[(c + d*x)/2]^2)^3) + (163*A*Sin[(c + d*x)/2])/(768*(1 - 2*Sin
[(c + d*x)/2]^2)^2) + (1015*C*Sin[(c + d*x)/2])/(6144*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (163*A*Sin[(c + d*x)/2])
/(512*(1 - 2*Sin[(c + d*x)/2]^2)) + (1015*C*Sin[(c + d*x)/2])/(4096*(1 - 2*Sin[(c + d*x)/2]^2))))/(d*(A + 2*C
+ A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.70

method	result	size
default	\(\frac {a^{2} \left (7824 A \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}-3912 A \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 ) \cos \left (d x +c \right )^{6}-3912 A \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 ) \cos \left (d x +c \right )^{6}+6090 C \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}-3045 C \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 ) \cos \left (d x +c \right )^{6}-3045 C \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 ) \cos \left (d x +c \right )^{6}+5216 A \cos \left (d x +c \right )^{4} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+4060 C \cos \left (d x +c \right )^{4} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+2944 A \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+3248 C \cos \left (d x +c \right )^{3} \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+768 A \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+2784 C \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2} \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+1792 C \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-\frac {1}{1+\cos \left (d x +c \right )}}+512 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 )}}{3072 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 {11}{2}}}\)	\(566\)

[In]

int((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3072*a^2/d*(7824*A*(-1/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^5-3912*A*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))*cos(d*x+c)^6-3912*A*arctan(1/2*(-cos(d*x+c)+sin(d*x+c)-1)/(1+co
s(d*x+c))/(-1/(1+cos(d*x+c)))^(1/2))*cos(d*x+c)^6+6090*C*(-1/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)^5-304
5*C*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))*cos(d*x+c)^6-3045*C*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))*cos(d*x+c)^6+5216*A*cos(d*x+c)^4*sin(
d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+4060*C*cos(d*x+c)^4*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+2944*A*cos(d*x+c)^3*
sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+3248*C*cos(d*x+c)^3*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+768*A*sin(d*x+c)
*cos(d*x+c)^2*(-1/(1+cos(d*x+c)))^(1/2)+2784*C*sin(d*x+c)*cos(d*x+c)^2*(-1/(1+cos(d*x+c)))^(1/2)+1792*C*cos(d*
x+c)*sin(d*x+c)*(-1/(1+cos(d*x+c)))^(1/2)+512*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)^(11/2)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.74 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\left [\frac {4 \, {\left (3 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 2 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 896 \, C a^{2} \cos \left (d x + c\right ) + 256 \, C a^{2}\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 ) + 3 \, {\left ({\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\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 )}{6144 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}, \frac {2 \, {\left (3 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 2 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 896 \, C a^{2} \cos \left (d x + c\right ) + 256 \, C a^{2}\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 ) + 3 \, {\left ({\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\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 )}{3072 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}\right ] \]

[In]

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

[Out]

[1/6144*(4*(3*(1304*A + 1015*C)*a^2*cos(d*x + c)^5 + 2*(1304*A + 1015*C)*a^2*cos(d*x + c)^4 + 8*(184*A + 203*C
)*a^2*cos(d*x + c)^3 + 48*(8*A + 29*C)*a^2*cos(d*x + c)^2 + 896*C*a^2*cos(d*x + c) + 256*C*a^2)*sqrt((a*cos(d*
x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 3*((1304*A + 1015*C)*a^2*cos(d*x + c)^7 + (1304*A
+ 1015*C)*a^2*cos(d*x + c)^6)*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)^7 + d*cos(d*x + c)^6), 1/3072*(2*(3*(1304*A + 1015*C)*a^2*cos(d*x + c)^5 + 2*(1304*A +
1015*C)*a^2*cos(d*x + c)^4 + 8*(184*A + 203*C)*a^2*cos(d*x + c)^3 + 48*(8*A + 29*C)*a^2*cos(d*x + c)^2 + 896*C
*a^2*cos(d*x + c) + 256*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 3*((1
304*A + 1015*C)*a^2*cos(d*x + c)^7 + (1304*A + 1015*C)*a^2*cos(d*x + c)^6)*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)^7 + d*cos(d*x + c)^6)]

Sympy [F(-1)]

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

[In]

integrate((a+a*sec(d*x+c))**(5/2)*(A+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. 11081 vs. \(2 (284) = 568\).

Time = 2.10 (sec) , antiderivative size = 11081, normalized size of antiderivative = 33.38 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6144*(8*(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*sqrt
(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*s
qrt(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))*c
os(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*sin
(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*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*sq
rt(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*co
s(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*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(s
in(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*sqr
t(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*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(11/4*arctan2(sin(2*d*x + 2*c), co
s(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*sq
rt(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) + sqrt
(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 + 1
6*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*s
in(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) + (
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) + 2
0*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*ar
ctan2(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*si
n(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*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(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*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(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))*cos(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*sin(10*d*x + 10*c) + 15*sq
rt(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*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(
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(1
0*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(1
2*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)*c
os(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*co
s(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) + 1
20*(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(s
in(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) + 30
45*(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*si
n(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) - 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*c
os(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))*si
n(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), co
s(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 + 1
2*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*si
n(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*cos(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*c
os(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(21/4*arc
tan2(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 + 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(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*c
os(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*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(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*arctan2(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 + 1
2*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) + 20*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) + 2
0*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]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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