3.176 \(\int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=205 \[ \frac{1155 \tanh ^{-1}(\sin (c+d x))}{8 a^8 d}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}-\frac{154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{385 \tan (c+d x) \sec ^3(c+d x)}{4 a^8 d}+\frac{1155 \tan (c+d x) \sec (c+d x)}{8 a^8 d}+\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7} \]

[Out]

(1155*ArcTanh[Sin[c + d*x]])/(8*a^8*d) + (1155*Sec[c + d*x]*Tan[c + d*x])/(8*a^8*d) + (385*Sec[c + d*x]^3*Tan[
c + d*x])/(4*a^8*d) + (((2*I)/3)*Sec[c + d*x]^11)/(a*d*(a + I*a*Tan[c + d*x])^7) - (((22*I)/3)*Sec[c + d*x]^9)
/(a^3*d*(a + I*a*Tan[c + d*x])^5) - ((66*I)*Sec[c + d*x]^7)/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) - ((154*I)*Se
c[c + d*x]^5)/(d*(a^8 + I*a^8*Tan[c + d*x]))

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Rubi [A]  time = 0.216992, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3500, 3768, 3770} \[ \frac{1155 \tanh ^{-1}(\sin (c+d x))}{8 a^8 d}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}-\frac{154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{385 \tan (c+d x) \sec ^3(c+d x)}{4 a^8 d}+\frac{1155 \tan (c+d x) \sec (c+d x)}{8 a^8 d}+\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(1155*ArcTanh[Sin[c + d*x]])/(8*a^8*d) + (1155*Sec[c + d*x]*Tan[c + d*x])/(8*a^8*d) + (385*Sec[c + d*x]^3*Tan[
c + d*x])/(4*a^8*d) + (((2*I)/3)*Sec[c + d*x]^11)/(a*d*(a + I*a*Tan[c + d*x])^7) - (((22*I)/3)*Sec[c + d*x]^9)
/(a^3*d*(a + I*a*Tan[c + d*x])^5) - ((66*I)*Sec[c + d*x]^7)/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) - ((154*I)*Se
c[c + d*x]^5)/(d*(a^8 + I*a^8*Tan[c + d*x]))

Rule 3500

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*d^2
*(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + 2*n)), x] - Dist[(d^2*(m - 2))/(b^2*(m + 2*n
)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a
^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (Integers
Q[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac{11 \int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{3 a^2}\\ &=\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}+\frac{33 \int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{a^4}\\ &=\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}+\frac{231 \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^6}\\ &=\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac{154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{385 \int \sec ^5(c+d x) \, dx}{a^8}\\ &=\frac{385 \sec ^3(c+d x) \tan (c+d x)}{4 a^8 d}+\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac{154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{1155 \int \sec ^3(c+d x) \, dx}{4 a^8}\\ &=\frac{1155 \sec (c+d x) \tan (c+d x)}{8 a^8 d}+\frac{385 \sec ^3(c+d x) \tan (c+d x)}{4 a^8 d}+\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac{154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{1155 \int \sec (c+d x) \, dx}{8 a^8}\\ &=\frac{1155 \tanh ^{-1}(\sin (c+d x))}{8 a^8 d}+\frac{1155 \sec (c+d x) \tan (c+d x)}{8 a^8 d}+\frac{385 \sec ^3(c+d x) \tan (c+d x)}{4 a^8 d}+\frac{2 i \sec ^{11}(c+d x)}{3 a d (a+i a \tan (c+d x))^7}-\frac{22 i \sec ^9(c+d x)}{3 a^3 d (a+i a \tan (c+d x))^5}-\frac{66 i \sec ^7(c+d x)}{a^5 d (a+i a \tan (c+d x))^3}-\frac{154 i \sec ^5(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 6.2131, size = 1704, normalized size = 8.31 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(-1155*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8)/(8*d*(a
+ I*a*Tan[c + d*x])^8) + (1155*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x]
+ I*Sin[d*x])^8)/(8*d*(a + I*a*Tan[c + d*x])^8) + (Cos[3*d*x]*Sec[c + d*x]^8*(((32*I)/3)*Cos[5*c] - (32*Sin[5
*c])/3)*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Cos[d*x]*Sec[c + d*x]^8*((-160*I)*Cos[7*c]
+ 160*Sin[7*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) - (((1155*I)/8)*Log[Cos[c/2 + (d*x)/2]
- Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (((11
55*I)/8)*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a
+ I*a*Tan[c + d*x])^8) + (Sec[c]*Sec[c + d*x]^8*(((-236*I)/3)*Cos[8*c] + (236*Sin[8*c])/3)*(Cos[d*x] + I*Sin[
d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(-160*Cos[7*c] - (160*I)*Sin[7*c])*(Cos[d*x] + I*Sin[d
*x])^8*Sin[d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*((32*Cos[5*c])/3 + ((32*I)/3)*Sin[5*c])*(Cos[d
*x] + I*Sin[d*x])^8*Sin[3*d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(Cos[8*c]/16 + (I/16)*Sin[8*c])
*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4*(a + I*a*Tan[c + d*x])^8) - ((1/96
+ I/96)*Sec[c + d*x]^8*((-407*I)*Cos[(15*c)/2] + 343*Cos[(17*c)/2] + 407*Sin[(15*c)/2] + (343*I)*Sin[(17*c)/2]
)*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan
[c + d*x])^8) + (Sec[c + d*x]^8*(-Cos[8*c]/16 - (I/16)*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2 + (d*x
)/2] + Sin[c/2 + (d*x)/2])^4*(a + I*a*Tan[c + d*x])^8) + ((1/96 + I/96)*Sec[c + d*x]^8*(407*Cos[(15*c)/2] - (3
43*I)*Cos[(17*c)/2] + (407*I)*Sin[(15*c)/2] + 343*Sin[(17*c)/2])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2] + Sin
[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^8) + (236*Sec[c + d*x]^8*(Cos[d*x] +
I*Sin[d*x])^8*(Cos[8*c - (d*x)/2]/2 - Cos[8*c + (d*x)/2]/2 + (I/2)*Sin[8*c - (d*x)/2] - (I/2)*Sin[8*c + (d*x)
/2]))/(3*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^8) + (4*Sec[
c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(Cos[8*c - (d*x)/2]/2 - Cos[8*c + (d*x)/2]/2 + (I/2)*Sin[8*c - (d*x)/2] -
(I/2)*Sin[8*c + (d*x)/2]))/(3*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3*(a + I*a*Ta
n[c + d*x])^8) + (4*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(-Cos[8*c - (d*x)/2]/2 + Cos[8*c + (d*x)/2]/2 - (
I/2)*Sin[8*c - (d*x)/2] + (I/2)*Sin[8*c + (d*x)/2]))/(3*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2
+ (d*x)/2])^3*(a + I*a*Tan[c + d*x])^8) + (236*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(-Cos[8*c - (d*x)/2]/2
+ Cos[8*c + (d*x)/2]/2 - (I/2)*Sin[8*c - (d*x)/2] + (I/2)*Sin[8*c + (d*x)/2]))/(3*d*(Cos[c/2] + Sin[c/2])*(Co
s[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^8)

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Maple [B]  time = 0.146, size = 409, normalized size = 2. \begin{align*}{\frac{121}{8\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{76\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{2\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{{\frac{8\,i}{3}}}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{123}{8\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{128\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-2}}-{\frac{1}{4\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-4}}+{\frac{1155}{8\,d{a}^{8}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{4\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{256}{3\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-3}}-256\,{\frac{1}{d{a}^{8} \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) }}+{\frac{1}{2\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}+{\frac{76\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{121}{8\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{{\frac{8\,i}{3}}}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{123}{8\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{4\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{4\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-4}}-{\frac{1155}{8\,d{a}^{8}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^8,x)

[Out]

121/8/d/a^8/(tan(1/2*d*x+1/2*c)+1)^2-76*I/d/a^8/(tan(1/2*d*x+1/2*c)+1)+1/2/d/a^8/(tan(1/2*d*x+1/2*c)+1)^3+8/3*
I/d/a^8/(tan(1/2*d*x+1/2*c)+1)^3-123/8/d/a^8/(tan(1/2*d*x+1/2*c)+1)+128*I/d/a^8/(tan(1/2*d*x+1/2*c)-I)^2-1/4/d
/a^8/(tan(1/2*d*x+1/2*c)+1)^4+1155/8/d/a^8*ln(tan(1/2*d*x+1/2*c)+1)-4*I/d/a^8/(tan(1/2*d*x+1/2*c)+1)^2-256/3/d
/a^8/(tan(1/2*d*x+1/2*c)-I)^3-256/d/a^8/(tan(1/2*d*x+1/2*c)-I)+1/2/d/a^8/(tan(1/2*d*x+1/2*c)-1)^3+76*I/d/a^8/(
tan(1/2*d*x+1/2*c)-1)-121/8/d/a^8/(tan(1/2*d*x+1/2*c)-1)^2-8/3*I/d/a^8/(tan(1/2*d*x+1/2*c)-1)^3-123/8/d/a^8/(t
an(1/2*d*x+1/2*c)-1)-4*I/d/a^8/(tan(1/2*d*x+1/2*c)-1)^2+1/4/d/a^8/(tan(1/2*d*x+1/2*c)-1)^4-1155/8/d/a^8*ln(tan
(1/2*d*x+1/2*c)-1)

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Maxima [B]  time = 2.09246, size = 1075, normalized size = 5.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-((6930*cos(11*d*x + 11*c) + 27720*cos(9*d*x + 9*c) + 41580*cos(7*d*x + 7*c) + 27720*cos(5*d*x + 5*c) + 6930*c
os(3*d*x + 3*c) + 6930*I*sin(11*d*x + 11*c) + 27720*I*sin(9*d*x + 9*c) + 41580*I*sin(7*d*x + 7*c) + 27720*I*si
n(5*d*x + 5*c) + 6930*I*sin(3*d*x + 3*c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) + (6930*cos(11*d*x + 11*c) +
27720*cos(9*d*x + 9*c) + 41580*cos(7*d*x + 7*c) + 27720*cos(5*d*x + 5*c) + 6930*cos(3*d*x + 3*c) + 6930*I*sin
(11*d*x + 11*c) + 27720*I*sin(9*d*x + 9*c) + 41580*I*sin(7*d*x + 7*c) + 27720*I*sin(5*d*x + 5*c) + 6930*I*sin(
3*d*x + 3*c))*arctan2(cos(d*x + c), -sin(d*x + c) + 1) - (-3465*I*cos(11*d*x + 11*c) - 13860*I*cos(9*d*x + 9*c
) - 20790*I*cos(7*d*x + 7*c) - 13860*I*cos(5*d*x + 5*c) - 3465*I*cos(3*d*x + 3*c) + 3465*sin(11*d*x + 11*c) +
13860*sin(9*d*x + 9*c) + 20790*sin(7*d*x + 7*c) + 13860*sin(5*d*x + 5*c) + 3465*sin(3*d*x + 3*c))*log(cos(d*x
+ c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) - (3465*I*cos(11*d*x + 11*c) + 13860*I*cos(9*d*x + 9*c) + 20790*
I*cos(7*d*x + 7*c) + 13860*I*cos(5*d*x + 5*c) + 3465*I*cos(3*d*x + 3*c) - 3465*sin(11*d*x + 11*c) - 13860*sin(
9*d*x + 9*c) - 20790*sin(7*d*x + 7*c) - 13860*sin(5*d*x + 5*c) - 3465*sin(3*d*x + 3*c))*log(cos(d*x + c)^2 + s
in(d*x + c)^2 - 2*sin(d*x + c) + 1) + 13860*cos(10*d*x + 10*c) + 50820*cos(8*d*x + 8*c) + 67452*cos(6*d*x + 6*
c) + 36828*cos(4*d*x + 4*c) + 5632*cos(2*d*x + 2*c) + 13860*I*sin(10*d*x + 10*c) + 50820*I*sin(8*d*x + 8*c) +
67452*I*sin(6*d*x + 6*c) + 36828*I*sin(4*d*x + 4*c) + 5632*I*sin(2*d*x + 2*c) - 512)/((-48*I*a^8*cos(11*d*x +
11*c) - 192*I*a^8*cos(9*d*x + 9*c) - 288*I*a^8*cos(7*d*x + 7*c) - 192*I*a^8*cos(5*d*x + 5*c) - 48*I*a^8*cos(3*
d*x + 3*c) + 48*a^8*sin(11*d*x + 11*c) + 192*a^8*sin(9*d*x + 9*c) + 288*a^8*sin(7*d*x + 7*c) + 192*a^8*sin(5*d
*x + 5*c) + 48*a^8*sin(3*d*x + 3*c))*d)

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Fricas [A]  time = 2.95583, size = 826, normalized size = 4.03 \begin{align*} \frac{3465 \,{\left (e^{\left (11 i \, d x + 11 i \, c\right )} + 4 \, e^{\left (9 i \, d x + 9 i \, c\right )} + 6 \, e^{\left (7 i \, d x + 7 i \, c\right )} + 4 \, e^{\left (5 i \, d x + 5 i \, c\right )} + e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3465 \,{\left (e^{\left (11 i \, d x + 11 i \, c\right )} + 4 \, e^{\left (9 i \, d x + 9 i \, c\right )} + 6 \, e^{\left (7 i \, d x + 7 i \, c\right )} + 4 \, e^{\left (5 i \, d x + 5 i \, c\right )} + e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 6930 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 25410 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 33726 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 18414 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 2816 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 256 i}{24 \,{\left (a^{8} d e^{\left (11 i \, d x + 11 i \, c\right )} + 4 \, a^{8} d e^{\left (9 i \, d x + 9 i \, c\right )} + 6 \, a^{8} d e^{\left (7 i \, d x + 7 i \, c\right )} + 4 \, a^{8} d e^{\left (5 i \, d x + 5 i \, c\right )} + a^{8} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/24*(3465*(e^(11*I*d*x + 11*I*c) + 4*e^(9*I*d*x + 9*I*c) + 6*e^(7*I*d*x + 7*I*c) + 4*e^(5*I*d*x + 5*I*c) + e^
(3*I*d*x + 3*I*c))*log(e^(I*d*x + I*c) + I) - 3465*(e^(11*I*d*x + 11*I*c) + 4*e^(9*I*d*x + 9*I*c) + 6*e^(7*I*d
*x + 7*I*c) + 4*e^(5*I*d*x + 5*I*c) + e^(3*I*d*x + 3*I*c))*log(e^(I*d*x + I*c) - I) - 6930*I*e^(10*I*d*x + 10*
I*c) - 25410*I*e^(8*I*d*x + 8*I*c) - 33726*I*e^(6*I*d*x + 6*I*c) - 18414*I*e^(4*I*d*x + 4*I*c) - 2816*I*e^(2*I
*d*x + 2*I*c) + 256*I)/(a^8*d*e^(11*I*d*x + 11*I*c) + 4*a^8*d*e^(9*I*d*x + 9*I*c) + 6*a^8*d*e^(7*I*d*x + 7*I*c
) + 4*a^8*d*e^(5*I*d*x + 5*I*c) + a^8*d*e^(3*I*d*x + 3*I*c))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**13/(a+I*a*tan(d*x+c))**8,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.24793, size = 266, normalized size = 1.3 \begin{align*} \frac{\frac{3465 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{8}} - \frac{3465 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{8}} - \frac{1024 \,{\left (6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 7\right )}}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{3}} - \frac{2 \,{\left (369 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1728 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 393 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 5568 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 393 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5696 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 369 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1856 i\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{4} a^{8}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^13/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/24*(3465*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^8 - 3465*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^8 - 1024*(6*tan(
1/2*d*x + 1/2*c)^2 - 15*I*tan(1/2*d*x + 1/2*c) - 7)/(a^8*(tan(1/2*d*x + 1/2*c) - I)^3) - 2*(369*tan(1/2*d*x +
1/2*c)^7 - 1728*I*tan(1/2*d*x + 1/2*c)^6 - 393*tan(1/2*d*x + 1/2*c)^5 + 5568*I*tan(1/2*d*x + 1/2*c)^4 - 393*ta
n(1/2*d*x + 1/2*c)^3 - 5696*I*tan(1/2*d*x + 1/2*c)^2 + 369*tan(1/2*d*x + 1/2*c) + 1856*I)/((tan(1/2*d*x + 1/2*
c)^2 - 1)^4*a^8))/d