3.92 \(\int \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=205 \[ \frac{1155 i a^8 \sec (c+d x)}{8 d}+\frac{1155 a^8 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}+\frac{33 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{4 d}+\frac{77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac{385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d} \]

[Out]

(1155*a^8*ArcTanh[Sin[c + d*x]])/(8*d) + (((1155*I)/8)*a^8*Sec[c + d*x])/d + (((22*I)/3)*a^3*Cos[c + d*x]*(a +
I*a*Tan[c + d*x])^5)/d - (((2*I)/3)*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^7)/d + (((33*I)/4)*a^2*Sec[c + d*
x]*(a^2 + I*a^2*Tan[c + d*x])^3)/d + (((77*I)/4)*Sec[c + d*x]*(a^4 + I*a^4*Tan[c + d*x])^2)/d + (((385*I)/8)*S
ec[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/d

________________________________________________________________________________________

Rubi [A]  time = 0.193307, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3496, 3498, 3486, 3770} \[ \frac{1155 i a^8 \sec (c+d x)}{8 d}+\frac{1155 a^8 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}+\frac{33 i a^2 \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{4 d}+\frac{77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac{385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(1155*a^8*ArcTanh[Sin[c + d*x]])/(8*d) + (((1155*I)/8)*a^8*Sec[c + d*x])/d + (((22*I)/3)*a^3*Cos[c + d*x]*(a +
I*a*Tan[c + d*x])^5)/d - (((2*I)/3)*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^7)/d + (((33*I)/4)*a^2*Sec[c + d*
x]*(a^2 + I*a^2*Tan[c + d*x])^3)/d + (((77*I)/4)*Sec[c + d*x]*(a^4 + I*a^4*Tan[c + d*x])^2)/d + (((385*I)/8)*S
ec[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/d

Rule 3496

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*b*(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*m), x] - Dist[(b^2*(m + 2*n - 2))/(d^2*m), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]

Rule 3498

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

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

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 \cos ^3(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}-\frac{1}{3} \left (11 a^2\right ) \int \cos (c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\left (33 a^4\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=\frac{33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac{1}{4} \left (231 a^5\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=\frac{33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac{77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac{1}{4} \left (385 a^6\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=\frac{33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac{77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac{385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}+\frac{1}{8} \left (1155 a^7\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx\\ &=\frac{1155 i a^8 \sec (c+d x)}{8 d}+\frac{33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac{77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac{385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}+\frac{1}{8} \left (1155 a^8\right ) \int \sec (c+d x) \, dx\\ &=\frac{1155 a^8 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{1155 i a^8 \sec (c+d x)}{8 d}+\frac{33 i a^5 \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac{22 i a^3 \cos (c+d x) (a+i a \tan (c+d x))^5}{3 d}-\frac{2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^7}{3 d}+\frac{77 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )^2}{4 d}+\frac{385 i \sec (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{8 d}\\ \end{align*}

Mathematica [B]  time = 6.94833, size = 1540, normalized size = 7.51 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(-1155*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(a + I*a*Tan[c + d*x])^8)/(8*d*(Co
s[d*x] + I*Sin[d*x])^8) + (1155*Cos[8*c]*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(a + I*a*
Tan[c + d*x])^8)/(8*d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[3*d*x]*Cos[c + d*x]^8*(((-32*I)/3)*Cos[5*c] - (32*Sin[
5*c])/3)*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[d*x]*Cos[c + d*x]^8*((160*I)*Cos[7*c]
+ 160*Sin[7*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (((1155*I)/8)*Cos[c + d*x]^8*Log[Cos
[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) - (((11
55*I)/8)*Cos[c + d*x]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sin[8*c]*(a + I*a*Tan[c + d*x])^8)/(d*(Co
s[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*Sec[c]*(((236*I)/3)*Cos[8*c] + (236*Sin[8*c])/3)*(a + I*a*Tan[c + d*
x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(-160*Cos[7*c] + (160*I)*Sin[7*c])*Sin[d*x]*(a + I*a*Ta
n[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*((32*Cos[5*c])/3 - ((32*I)/3)*Sin[5*c])*Sin[3*d
*x]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8) + (Cos[c + d*x]^8*(Cos[8*c]/16 - (I/16)*Sin[8*c])*
(a + I*a*Tan[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) - (I*Cos[c
+ d*x]^8*((4*Cos[8*c])/3 - ((4*I)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2]
)*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^8*((-375 - 32*I)*Cos[
c/2] + (375 - 32*I)*Sin[c/2])*(Cos[8*c]/48 - (I/48)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2
])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (I*Cos[c + d*x]^8*((236*Cos[8*c])/
3 - ((236*I)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] - Sin[c/2])*(Cos[d*x] + I*Sin[d*
x])^8*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c + d*x]^8*(-Cos[8*c]/16 + (I/16)*Sin[8*c])*(a + I*a*T
an[c + d*x])^8)/(d*(Cos[d*x] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (I*Cos[c + d*x]^8*
((4*Cos[8*c])/3 - ((4*I)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x
] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^8*((375 - 32*I)*Cos[c/2] + (375
+ 32*I)*Sin[c/2])*(Cos[8*c]/48 - (I/48)*Sin[8*c])*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x
] + I*Sin[d*x])^8*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) - (I*Cos[c + d*x]^8*((236*Cos[8*c])/3 - ((236*I
)/3)*Sin[8*c])*Sin[(d*x)/2]*(a + I*a*Tan[c + d*x])^8)/(d*(Cos[c/2] + Sin[c/2])*(Cos[d*x] + I*Sin[d*x])^8*(Cos[
c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

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Maple [A]  time = 0.076, size = 356, normalized size = 1.7 \begin{align*} -{\frac{3449\,{a}^{8}\sin \left ( dx+c \right ) }{24\,d}}+{\frac{{\frac{688\,i}{3}}{a}^{8}\cos \left ( dx+c \right ) }{d}}-14\,{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{\frac{8\,i}{3}}{a}^{8} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{8}}{3\,d}}+{\frac{{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1379\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{24\,d}}-{\frac{5\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{8\,d}}-{\frac{119\,{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d}}+{\frac{1155\,{a}^{8}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{\frac{8\,i}{3}}{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{56\,i{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d\cos \left ( dx+c \right ) }}+{\frac{{\frac{344\,i}{3}}{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{72\,i{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{{\frac{40\,i}{3}}{a}^{8}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d}}+{\frac{{\frac{40\,i}{3}}{a}^{8} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{d\cos \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a+I*a*tan(d*x+c))^8,x)

[Out]

-3449/24*a^8*sin(d*x+c)/d+688/3*I/d*a^8*cos(d*x+c)-14/d*a^8*sin(d*x+c)^7/cos(d*x+c)^2-8/3*I/d*a^8*cos(d*x+c)^3
+1/3/d*sin(d*x+c)*cos(d*x+c)^2*a^8+1/4/d*a^8*sin(d*x+c)^9/cos(d*x+c)^4-5/8/d*a^8*sin(d*x+c)^9/cos(d*x+c)^2-137
9/24*a^8*sin(d*x+c)^3/d-5/8*a^8*sin(d*x+c)^7/d-119/8*a^8*sin(d*x+c)^5/d+1155/8/d*a^8*ln(sec(d*x+c)+tan(d*x+c))
-8/3*I/d*a^8*sin(d*x+c)^8/cos(d*x+c)^3+56*I/d*a^8*sin(d*x+c)^6/cos(d*x+c)+344/3*I/d*a^8*cos(d*x+c)*sin(d*x+c)^
2+72*I/d*a^8*cos(d*x+c)*sin(d*x+c)^4+40/3*I/d*a^8*cos(d*x+c)*sin(d*x+c)^6+40/3*I/d*a^8*sin(d*x+c)^8/cos(d*x+c)

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Maxima [B]  time = 1.19282, size = 475, normalized size = 2.32 \begin{align*} -\frac{128 i \, a^{8} \cos \left (d x + c\right )^{3} + 448 \, a^{8} \sin \left (d x + c\right )^{3} + 896 i \,{\left (\cos \left (d x + c\right )^{3} - \frac{3}{\cos \left (d x + c\right )} - 6 \, \cos \left (d x + c\right )\right )} a^{8} + 128 i \,{\left (\cos \left (d x + c\right )^{3} - \frac{9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{8} + 896 i \,{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{8} +{\left (16 \, \sin \left (d x + c\right )^{3} - \frac{6 \,{\left (13 \, \sin \left (d x + c\right )^{3} - 11 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, \sin \left (d x + c\right )\right )} a^{8} + 112 \,{\left (4 \, \sin \left (d x + c\right )^{3} - \frac{6 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, \sin \left (d x + c\right )\right )} a^{8} + 560 \,{\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{8} + 16 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{8}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/48*(128*I*a^8*cos(d*x + c)^3 + 448*a^8*sin(d*x + c)^3 + 896*I*(cos(d*x + c)^3 - 3/cos(d*x + c) - 6*cos(d*x
+ c))*a^8 + 128*I*(cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 - 9*cos(d*x + c))*a^8 + 896*I*(cos(d
*x + c)^3 - 3*cos(d*x + c))*a^8 + (16*sin(d*x + c)^3 - 6*(13*sin(d*x + c)^3 - 11*sin(d*x + c))/(sin(d*x + c)^4
- 2*sin(d*x + c)^2 + 1) - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 144*sin(d*x + c))*a^8 + 112
*(4*sin(d*x + c)^3 - 6*sin(d*x + c)/(sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)
+ 24*sin(d*x + c))*a^8 + 560*(2*sin(d*x + c)^3 - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1) + 6*sin(d*
x + c))*a^8 + 16*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^8)/d

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Fricas [A]  time = 1.86591, size = 819, normalized size = 4. \begin{align*} \frac{-256 i \, a^{8} e^{\left (11 i \, d x + 11 i \, c\right )} + 2816 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} + 18414 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} + 33726 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} + 25410 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} + 6930 i \, a^{8} e^{\left (i \, d x + i \, c\right )} + 3465 \,{\left (a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3465 \,{\left (a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{8}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{24 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/24*(-256*I*a^8*e^(11*I*d*x + 11*I*c) + 2816*I*a^8*e^(9*I*d*x + 9*I*c) + 18414*I*a^8*e^(7*I*d*x + 7*I*c) + 33
726*I*a^8*e^(5*I*d*x + 5*I*c) + 25410*I*a^8*e^(3*I*d*x + 3*I*c) + 6930*I*a^8*e^(I*d*x + I*c) + 3465*(a^8*e^(8*
I*d*x + 8*I*c) + 4*a^8*e^(6*I*d*x + 6*I*c) + 6*a^8*e^(4*I*d*x + 4*I*c) + 4*a^8*e^(2*I*d*x + 2*I*c) + a^8)*log(
e^(I*d*x + I*c) + I) - 3465*(a^8*e^(8*I*d*x + 8*I*c) + 4*a^8*e^(6*I*d*x + 6*I*c) + 6*a^8*e^(4*I*d*x + 4*I*c) +
4*a^8*e^(2*I*d*x + 2*I*c) + a^8)*log(e^(I*d*x + I*c) - I))/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) +
6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 4.80562, size = 282, normalized size = 1.38 \begin{align*} \frac{1155 a^{8} \left (- \frac{\log{\left (e^{i d x} - i e^{- i c} \right )}}{8} + \frac{\log{\left (e^{i d x} + i e^{- i c} \right )}}{8}\right )}{d} + \frac{\frac{765 i a^{8} e^{- i c} e^{7 i d x}}{4 d} + \frac{5855 i a^{8} e^{- 3 i c} e^{5 i d x}}{12 d} + \frac{5153 i a^{8} e^{- 5 i c} e^{3 i d x}}{12 d} + \frac{515 i a^{8} e^{- 7 i c} e^{i d x}}{4 d}}{e^{8 i d x} + 4 e^{- 2 i c} e^{6 i d x} + 6 e^{- 4 i c} e^{4 i d x} + 4 e^{- 6 i c} e^{2 i d x} + e^{- 8 i c}} + \begin{cases} \frac{- 32 i a^{8} d e^{3 i c} e^{3 i d x} + 480 i a^{8} d e^{i c} e^{i d x}}{3 d^{2}} & \text{for}\: 3 d^{2} \neq 0 \\x \left (32 a^{8} e^{3 i c} - 160 a^{8} e^{i c}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(a+I*a*tan(d*x+c))**8,x)

[Out]

1155*a**8*(-log(exp(I*d*x) - I*exp(-I*c))/8 + log(exp(I*d*x) + I*exp(-I*c))/8)/d + (765*I*a**8*exp(-I*c)*exp(7
*I*d*x)/(4*d) + 5855*I*a**8*exp(-3*I*c)*exp(5*I*d*x)/(12*d) + 5153*I*a**8*exp(-5*I*c)*exp(3*I*d*x)/(12*d) + 51
5*I*a**8*exp(-7*I*c)*exp(I*d*x)/(4*d))/(exp(8*I*d*x) + 4*exp(-2*I*c)*exp(6*I*d*x) + 6*exp(-4*I*c)*exp(4*I*d*x)
+ 4*exp(-6*I*c)*exp(2*I*d*x) + exp(-8*I*c)) + Piecewise(((-32*I*a**8*d*exp(3*I*c)*exp(3*I*d*x) + 480*I*a**8*d
*exp(I*c)*exp(I*d*x))/(3*d**2), Ne(3*d**2, 0)), (x*(32*a**8*exp(3*I*c) - 160*a**8*exp(I*c)), True))

________________________________________________________________________________________

Giac [B]  time = 2.99755, size = 3827, normalized size = 18.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/3440640*(26725545*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 374157630*a^8*e^(26*I*d*x + 12*I*c)
*log(I*e^(I*d*x + I*c) + 1) + 2432024595*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 9728098380*a^8
*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 26752270545*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c)
+ 1) + 53504541090*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 80256811635*a^8*e^(16*I*d*x + 2*I*c)*
log(I*e^(I*d*x + I*c) + 1) + 80256811635*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 53504541090*a^8
*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 26752270545*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) +
1) + 9728098380*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2432024595*a^8*e^(4*I*d*x - 10*I*c)*log(
I*e^(I*d*x + I*c) + 1) + 374157630*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 91722070440*a^8*e^(14
*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 26725545*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 523464480*a^8*e^(28
*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 7328502720*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1)
+ 47635267680*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 190541070720*a^8*e^(22*I*d*x + 8*I*c)*log
(I*e^(I*d*x + I*c) - 1) + 523987944480*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1047975888960*a^8
*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1571963833440*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c
) - 1) + 1571963833440*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1047975888960*a^8*e^(10*I*d*x - 4
*I*c)*log(I*e^(I*d*x + I*c) - 1) + 523987944480*a^8*e^(8*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 190541070
720*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 47635267680*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x +
I*c) - 1) + 7328502720*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1796530095360*a^8*e^(14*I*d*x)*l
og(I*e^(I*d*x + I*c) - 1) + 523464480*a^8*e^(-14*I*c)*log(I*e^(I*d*x + I*c) - 1) - 26725545*a^8*e^(28*I*d*x +
14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 374157630*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 243202
4595*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9728098380*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d
*x + I*c) + 1) - 26752270545*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 53504541090*a^8*e^(18*I*d*
x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 80256811635*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 80
256811635*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 53504541090*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e
^(I*d*x + I*c) + 1) - 26752270545*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 9728098380*a^8*e^(6*I*
d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2432024595*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 3
74157630*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 91722070440*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x +
I*c) + 1) - 26725545*a^8*e^(-14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 523464480*a^8*e^(28*I*d*x + 14*I*c)*log(-I
*e^(I*d*x + I*c) - 1) - 7328502720*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 47635267680*a^8*e^(
24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 190541070720*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c)
- 1) - 523987944480*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1047975888960*a^8*e^(18*I*d*x + 4*I
*c)*log(-I*e^(I*d*x + I*c) - 1) - 1571963833440*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1571963
833440*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1047975888960*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^
(I*d*x + I*c) - 1) - 523987944480*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 190541070720*a^8*e^(6*
I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 47635267680*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1)
- 7328502720*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1796530095360*a^8*e^(14*I*d*x)*log(-I*e^(I
*d*x + I*c) - 1) - 523464480*a^8*e^(-14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3465*a^8*e^(28*I*d*x + 14*I*c)*log(
I*e^(I*d*x) + e^(-I*c)) - 48510*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 315315*a^8*e^(24*I*d*x
+ 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 1261260*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 346846
5*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 6936930*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^
(-I*c)) - 10405395*a^8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 10405395*a^8*e^(12*I*d*x - 2*I*c)*lo
g(I*e^(I*d*x) + e^(-I*c)) - 6936930*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 3468465*a^8*e^(8*I*
d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 1261260*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 31531
5*a^8*e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 48510*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e^(-
I*c)) - 11891880*a^8*e^(14*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) - 3465*a^8*e^(-14*I*c)*log(I*e^(I*d*x) + e^(-I*c
)) + 3465*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 48510*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I
*d*x) + e^(-I*c)) + 315315*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 1261260*a^8*e^(22*I*d*x +
8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 3468465*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 6936930*
a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 10405395*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e
^(-I*c)) + 10405395*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 6936930*a^8*e^(10*I*d*x - 4*I*c)*l
og(-I*e^(I*d*x) + e^(-I*c)) + 3468465*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 1261260*a^8*e^(6*
I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 315315*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 4
8510*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 11891880*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x) + e^(-
I*c)) + 3465*a^8*e^(-14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 36700160*I*a^8*e^(31*I*d*x + 17*I*c) + 36700160*I*
a^8*e^(29*I*d*x + 15*I*c) + 5025341440*I*a^8*e^(27*I*d*x + 13*I*c) + 44995829760*I*a^8*e^(25*I*d*x + 11*I*c) +
211521945600*I*a^8*e^(23*I*d*x + 9*I*c) + 647303086080*I*a^8*e^(21*I*d*x + 7*I*c) + 1402445291520*I*a^8*e^(19
*I*d*x + 5*I*c) + 2242792366080*I*a^8*e^(17*I*d*x + 3*I*c) + 2703768453120*I*a^8*e^(15*I*d*x + I*c) + 24765325
31200*I*a^8*e^(13*I*d*x - I*c) + 1718329303040*I*a^8*e^(11*I*d*x - 3*I*c) + 890140303360*I*a^8*e^(9*I*d*x - 5*
I*c) + 334132592640*I*a^8*e^(7*I*d*x - 7*I*c) + 85969551360*I*a^8*e^(5*I*d*x - 9*I*c) + 13577625600*I*a^8*e^(3
*I*d*x - 11*I*c) + 993484800*I*a^8*e^(I*d*x - 13*I*c))/(d*e^(28*I*d*x + 14*I*c) + 14*d*e^(26*I*d*x + 12*I*c) +
91*d*e^(24*I*d*x + 10*I*c) + 364*d*e^(22*I*d*x + 8*I*c) + 1001*d*e^(20*I*d*x + 6*I*c) + 2002*d*e^(18*I*d*x +
4*I*c) + 3003*d*e^(16*I*d*x + 2*I*c) + 3003*d*e^(12*I*d*x - 2*I*c) + 2002*d*e^(10*I*d*x - 4*I*c) + 1001*d*e^(8
*I*d*x - 6*I*c) + 364*d*e^(6*I*d*x - 8*I*c) + 91*d*e^(4*I*d*x - 10*I*c) + 14*d*e^(2*I*d*x - 12*I*c) + 3432*d*e
^(14*I*d*x) + d*e^(-14*I*c))