3.1.81 \(\int (a+i a \tan (c+d x))^8 \, dx\) [81]

Optimal. Leaf size=200 \[ 128 a^8 x-\frac {128 i a^8 \log (\cos (c+d x))}{d}-\frac {64 a^8 \tan (c+d x)}{d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {16 i a^2 \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d} \]

[Out]

128*a^8*x-128*I*a^8*ln(cos(d*x+c))/d-64*a^8*tan(d*x+c)/d+4/5*I*a^3*(a+I*a*tan(d*x+c))^5/d+1/3*I*a^2*(a+I*a*tan
(d*x+c))^6/d+1/7*I*a*(a+I*a*tan(d*x+c))^7/d+16/3*I*a^2*(a^2+I*a^2*tan(d*x+c))^3/d+2*I*(a^2+I*a^2*tan(d*x+c))^4
/d+16*I*(a^4+I*a^4*tan(d*x+c))^2/d

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Rubi [A]
time = 0.12, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3559, 3558, 3556} \begin {gather*} -\frac {64 a^8 \tan (c+d x)}{d}-\frac {128 i a^8 \log (\cos (c+d x))}{d}+128 a^8 x+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {16 i a^2 \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[c + d*x])^8,x]

[Out]

128*a^8*x - ((128*I)*a^8*Log[Cos[c + d*x]])/d - (64*a^8*Tan[c + d*x])/d + (((4*I)/5)*a^3*(a + I*a*Tan[c + d*x]
)^5)/d + ((I/3)*a^2*(a + I*a*Tan[c + d*x])^6)/d + ((I/7)*a*(a + I*a*Tan[c + d*x])^7)/d + (((16*I)/3)*a^2*(a^2
+ I*a^2*Tan[c + d*x])^3)/d + ((2*I)*(a^2 + I*a^2*Tan[c + d*x])^4)/d + ((16*I)*(a^4 + I*a^4*Tan[c + d*x])^2)/d

Rule 3556

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

Rule 3558

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2)*x, x] + (Dist[2*a*b, Int[Tan[c + d
*x], x], x] + Simp[b^2*(Tan[c + d*x]/d), x]) /; FreeQ[{a, b, c, d}, x]

Rule 3559

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && G
tQ[n, 1]

Rubi steps

\begin {align*} \int (a+i a \tan (c+d x))^8 \, dx &=\frac {i a (a+i a \tan (c+d x))^7}{7 d}+(2 a) \int (a+i a \tan (c+d x))^7 \, dx\\ &=\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\left (4 a^2\right ) \int (a+i a \tan (c+d x))^6 \, dx\\ &=\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\left (8 a^3\right ) \int (a+i a \tan (c+d x))^5 \, dx\\ &=\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\left (16 a^4\right ) \int (a+i a \tan (c+d x))^4 \, dx\\ &=\frac {16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\left (32 a^5\right ) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac {16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}+\left (64 a^6\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=128 a^8 x-\frac {64 a^8 \tan (c+d x)}{d}+\frac {16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}+\left (128 i a^8\right ) \int \tan (c+d x) \, dx\\ &=128 a^8 x-\frac {128 i a^8 \log (\cos (c+d x))}{d}-\frac {64 a^8 \tan (c+d x)}{d}+\frac {16 i a^5 (a+i a \tan (c+d x))^3}{3 d}+\frac {4 i a^3 (a+i a \tan (c+d x))^5}{5 d}+\frac {i a^2 (a+i a \tan (c+d x))^6}{3 d}+\frac {i a (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right )^4}{d}+\frac {16 i \left (a^4+i a^4 \tan (c+d x)\right )^2}{d}\\ \end {align*}

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Mathematica [A]
time = 2.51, size = 383, normalized size = 1.92 \begin {gather*} \frac {a^8 \sec (c) \sec ^7(c+d x) \left (70 \cos (d x) \left (-139 i+210 d x-105 i \log \left (\cos ^2(c+d x)\right )\right )+70 \cos (2 c+d x) \left (-139 i+210 d x-105 i \log \left (\cos ^2(c+d x)\right )\right )+3 \left (-420 i \cos (4 c+5 d x)+980 d x \cos (4 c+5 d x)-420 i \cos (6 c+5 d x)+980 d x \cos (6 c+5 d x)+140 d x \cos (6 c+7 d x)+140 d x \cos (8 c+7 d x)+70 \cos (2 c+3 d x) \left (-25 i+42 d x-21 i \log \left (\cos ^2(c+d x)\right )\right )+70 \cos (4 c+3 d x) \left (-25 i+42 d x-21 i \log \left (\cos ^2(c+d x)\right )\right )-490 i \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-490 i \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )-70 i \cos (6 c+7 d x) \log \left (\cos ^2(c+d x)\right )-70 i \cos (8 c+7 d x) \log \left (\cos ^2(c+d x)\right )-6965 \sin (d x)+5740 \sin (2 c+d x)-4963 \sin (2 c+3 d x)+2660 \sin (4 c+3 d x)-1981 \sin (4 c+5 d x)+560 \sin (6 c+5 d x)-363 \sin (6 c+7 d x)\right )\right )}{420 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*Sec[c]*Sec[c + d*x]^7*(70*Cos[d*x]*(-139*I + 210*d*x - (105*I)*Log[Cos[c + d*x]^2]) + 70*Cos[2*c + d*x]*(
-139*I + 210*d*x - (105*I)*Log[Cos[c + d*x]^2]) + 3*((-420*I)*Cos[4*c + 5*d*x] + 980*d*x*Cos[4*c + 5*d*x] - (4
20*I)*Cos[6*c + 5*d*x] + 980*d*x*Cos[6*c + 5*d*x] + 140*d*x*Cos[6*c + 7*d*x] + 140*d*x*Cos[8*c + 7*d*x] + 70*C
os[2*c + 3*d*x]*(-25*I + 42*d*x - (21*I)*Log[Cos[c + d*x]^2]) + 70*Cos[4*c + 3*d*x]*(-25*I + 42*d*x - (21*I)*L
og[Cos[c + d*x]^2]) - (490*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - (490*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]
^2] - (70*I)*Cos[6*c + 7*d*x]*Log[Cos[c + d*x]^2] - (70*I)*Cos[8*c + 7*d*x]*Log[Cos[c + d*x]^2] - 6965*Sin[d*x
] + 5740*Sin[2*c + d*x] - 4963*Sin[2*c + 3*d*x] + 2660*Sin[4*c + 3*d*x] - 1981*Sin[4*c + 5*d*x] + 560*Sin[6*c
+ 5*d*x] - 363*Sin[6*c + 7*d*x])))/(420*d)

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Maple [A]
time = 0.06, size = 103, normalized size = 0.52

method	result	size
derivativedivides	\(\frac {a^{8} \left (-127 \tan \left (d x +c \right )+\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {4 i \left (\tan ^{6}\left (d x +c \right )\right )}{3}-\frac {29 \left (\tan ^{5}\left (d x +c \right )\right )}{5}+16 i \left (\tan ^{4}\left (d x +c \right )\right )+33 \left (\tan ^{3}\left (d x +c \right )\right )-60 i \left (\tan ^{2}\left (d x +c \right )\right )+64 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+128 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(103\)
default	\(\frac {a^{8} \left (-127 \tan \left (d x +c \right )+\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {4 i \left (\tan ^{6}\left (d x +c \right )\right )}{3}-\frac {29 \left (\tan ^{5}\left (d x +c \right )\right )}{5}+16 i \left (\tan ^{4}\left (d x +c \right )\right )+33 \left (\tan ^{3}\left (d x +c \right )\right )-60 i \left (\tan ^{2}\left (d x +c \right )\right )+64 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+128 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(103\)
risch	\(-\frac {256 a^{8} c}{d}-\frac {32 i a^{8} \left (2940 \,{\mathrm e}^{12 i \left (d x +c \right )}+13230 \,{\mathrm e}^{10 i \left (d x +c \right )}+26950 \,{\mathrm e}^{8 i \left (d x +c \right )}+30625 \,{\mathrm e}^{6 i \left (d x +c \right )}+20139 \,{\mathrm e}^{4 i \left (d x +c \right )}+7203 \,{\mathrm e}^{2 i \left (d x +c \right )}+1089\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {128 i a^{8} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(122\)
norman	\(128 a^{8} x -\frac {127 a^{8} \tan \left (d x +c \right )}{d}+\frac {33 a^{8} \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {29 a^{8} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{8} \left (\tan ^{7}\left (d x +c \right )\right )}{7 d}-\frac {60 i a^{8} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {16 i a^{8} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {4 i a^{8} \left (\tan ^{6}\left (d x +c \right )\right )}{3 d}+\frac {64 i a^{8} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d*a^8*(-127*tan(d*x+c)+1/7*tan(d*x+c)^7-4/3*I*tan(d*x+c)^6-29/5*tan(d*x+c)^5+16*I*tan(d*x+c)^4+33*tan(d*x+c)
^3-60*I*tan(d*x+c)^2+64*I*ln(1+tan(d*x+c)^2)+128*arctan(tan(d*x+c)))

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Maxima [A]
time = 0.51, size = 121, normalized size = 0.60 \begin {gather*} \frac {15 \, a^{8} \tan \left (d x + c\right )^{7} - 140 i \, a^{8} \tan \left (d x + c\right )^{6} - 609 \, a^{8} \tan \left (d x + c\right )^{5} + 1680 i \, a^{8} \tan \left (d x + c\right )^{4} + 3465 \, a^{8} \tan \left (d x + c\right )^{3} - 6300 i \, a^{8} \tan \left (d x + c\right )^{2} + 13440 \, {\left (d x + c\right )} a^{8} + 6720 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 13335 \, a^{8} \tan \left (d x + c\right )}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*a^8*tan(d*x + c)^7 - 140*I*a^8*tan(d*x + c)^6 - 609*a^8*tan(d*x + c)^5 + 1680*I*a^8*tan(d*x + c)^4 +
3465*a^8*tan(d*x + c)^3 - 6300*I*a^8*tan(d*x + c)^2 + 13440*(d*x + c)*a^8 + 6720*I*a^8*log(tan(d*x + c)^2 + 1
) - 13335*a^8*tan(d*x + c))/d

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Fricas [A]
time = 0.35, size = 297, normalized size = 1.48 \begin {gather*} -\frac {32 \, {\left (2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 13230 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 26950 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 30625 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 20139 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7203 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 1089 i \, a^{8} + 420 \, {\left (i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 7 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 21 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 35 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 35 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 21 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{8}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{105 \, {\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/105*(2940*I*a^8*e^(12*I*d*x + 12*I*c) + 13230*I*a^8*e^(10*I*d*x + 10*I*c) + 26950*I*a^8*e^(8*I*d*x + 8*I*c
) + 30625*I*a^8*e^(6*I*d*x + 6*I*c) + 20139*I*a^8*e^(4*I*d*x + 4*I*c) + 7203*I*a^8*e^(2*I*d*x + 2*I*c) + 1089*
I*a^8 + 420*(I*a^8*e^(14*I*d*x + 14*I*c) + 7*I*a^8*e^(12*I*d*x + 12*I*c) + 21*I*a^8*e^(10*I*d*x + 10*I*c) + 35
*I*a^8*e^(8*I*d*x + 8*I*c) + 35*I*a^8*e^(6*I*d*x + 6*I*c) + 21*I*a^8*e^(4*I*d*x + 4*I*c) + 7*I*a^8*e^(2*I*d*x
+ 2*I*c) + I*a^8)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(14*I*d*x + 14*I*c) + 7*d*e^(12*I*d*x + 12*I*c) + 21*d*e^
(10*I*d*x + 10*I*c) + 35*d*e^(8*I*d*x + 8*I*c) + 35*d*e^(6*I*d*x + 6*I*c) + 21*d*e^(4*I*d*x + 4*I*c) + 7*d*e^(
2*I*d*x + 2*I*c) + d)

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Sympy [A]
time = 0.41, size = 301, normalized size = 1.50 \begin {gather*} - \frac {128 i a^{8} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 94080 i a^{8} e^{12 i c} e^{12 i d x} - 423360 i a^{8} e^{10 i c} e^{10 i d x} - 862400 i a^{8} e^{8 i c} e^{8 i d x} - 980000 i a^{8} e^{6 i c} e^{6 i d x} - 644448 i a^{8} e^{4 i c} e^{4 i d x} - 230496 i a^{8} e^{2 i c} e^{2 i d x} - 34848 i a^{8}}{105 d e^{14 i c} e^{14 i d x} + 735 d e^{12 i c} e^{12 i d x} + 2205 d e^{10 i c} e^{10 i d x} + 3675 d e^{8 i c} e^{8 i d x} + 3675 d e^{6 i c} e^{6 i d x} + 2205 d e^{4 i c} e^{4 i d x} + 735 d e^{2 i c} e^{2 i d x} + 105 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128*I*a**8*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-94080*I*a**8*exp(12*I*c)*exp(12*I*d*x) - 423360*I*a**8*exp(1
0*I*c)*exp(10*I*d*x) - 862400*I*a**8*exp(8*I*c)*exp(8*I*d*x) - 980000*I*a**8*exp(6*I*c)*exp(6*I*d*x) - 644448*
I*a**8*exp(4*I*c)*exp(4*I*d*x) - 230496*I*a**8*exp(2*I*c)*exp(2*I*d*x) - 34848*I*a**8)/(105*d*exp(14*I*c)*exp(
14*I*d*x) + 735*d*exp(12*I*c)*exp(12*I*d*x) + 2205*d*exp(10*I*c)*exp(10*I*d*x) + 3675*d*exp(8*I*c)*exp(8*I*d*x
) + 3675*d*exp(6*I*c)*exp(6*I*d*x) + 2205*d*exp(4*I*c)*exp(4*I*d*x) + 735*d*exp(2*I*c)*exp(2*I*d*x) + 105*d)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (166) = 332\).
time = 0.70, size = 378, normalized size = 1.89 \begin {gather*} -\frac {32 \, {\left (420 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8820 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 14700 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 14700 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8820 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2940 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 13230 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 26950 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 30625 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 20139 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 7203 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 420 i \, a^{8} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1089 i \, a^{8}\right )}}{105 \, {\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/105*(420*I*a^8*e^(14*I*d*x + 14*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 2940*I*a^8*e^(12*I*d*x + 12*I*c)*log(e
^(2*I*d*x + 2*I*c) + 1) + 8820*I*a^8*e^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 14700*I*a^8*e^(8*I*d
*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 14700*I*a^8*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 8820
*I*a^8*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 2940*I*a^8*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*
c) + 1) + 2940*I*a^8*e^(12*I*d*x + 12*I*c) + 13230*I*a^8*e^(10*I*d*x + 10*I*c) + 26950*I*a^8*e^(8*I*d*x + 8*I*
c) + 30625*I*a^8*e^(6*I*d*x + 6*I*c) + 20139*I*a^8*e^(4*I*d*x + 4*I*c) + 7203*I*a^8*e^(2*I*d*x + 2*I*c) + 420*
I*a^8*log(e^(2*I*d*x + 2*I*c) + 1) + 1089*I*a^8)/(d*e^(14*I*d*x + 14*I*c) + 7*d*e^(12*I*d*x + 12*I*c) + 21*d*e
^(10*I*d*x + 10*I*c) + 35*d*e^(8*I*d*x + 8*I*c) + 35*d*e^(6*I*d*x + 6*I*c) + 21*d*e^(4*I*d*x + 4*I*c) + 7*d*e^
(2*I*d*x + 2*I*c) + d)

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Mupad [B]
time = 3.40, size = 113, normalized size = 0.56 \begin {gather*} \frac {33\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^3-127\,a^8\,\mathrm {tan}\left (c+d\,x\right )-\frac {29\,a^8\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+a^8\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,128{}\mathrm {i}-a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2\,60{}\mathrm {i}+a^8\,{\mathrm {tan}\left (c+d\,x\right )}^4\,16{}\mathrm {i}-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^6\,4{}\mathrm {i}}{3}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(c + d*x)*1i)^8,x)

[Out]

(a^8*log(tan(c + d*x) + 1i)*128i - 127*a^8*tan(c + d*x) - a^8*tan(c + d*x)^2*60i + 33*a^8*tan(c + d*x)^3 + a^8
*tan(c + d*x)^4*16i - (29*a^8*tan(c + d*x)^5)/5 - (a^8*tan(c + d*x)^6*4i)/3 + (a^8*tan(c + d*x)^7)/7)/d

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