Optimal. Leaf size=124 \[ 80 a^8 x-\frac {80 i a^8 \log (\cos (c+d x))}{d}-\frac {31 a^8 \tan (c+d x)}{d}-\frac {4 i a^8 \tan ^2(c+d x)}{d}+\frac {a^8 \tan ^3(c+d x)}{3 d}-\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}+\frac {80 i a^9}{d (a-i a \tan (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45}
\begin {gather*} -\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}+\frac {80 i a^9}{d (a-i a \tan (c+d x))}+\frac {a^8 \tan ^3(c+d x)}{3 d}-\frac {4 i a^8 \tan ^2(c+d x)}{d}-\frac {31 a^8 \tan (c+d x)}{d}-\frac {80 i a^8 \log (\cos (c+d x))}{d}+80 a^8 x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {\left (i a^5\right ) \text {Subst}\left (\int \frac {(a+x)^5}{(a-x)^3} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^5\right ) \text {Subst}\left (\int \left (-31 a^2+\frac {32 a^5}{(a-x)^3}-\frac {80 a^4}{(a-x)^2}+\frac {80 a^3}{a-x}-8 a x-x^2\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=80 a^8 x-\frac {80 i a^8 \log (\cos (c+d x))}{d}-\frac {31 a^8 \tan (c+d x)}{d}-\frac {4 i a^8 \tan ^2(c+d x)}{d}+\frac {a^8 \tan ^3(c+d x)}{3 d}-\frac {16 i a^{10}}{d (a-i a \tan (c+d x))^2}+\frac {80 i a^9}{d (a-i a \tan (c+d x))}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(566\) vs. \(2(124)=248\).
time = 1.97, size = 566, normalized size = 4.56 \begin {gather*} \frac {a^8 \sec (c) \sec ^3(c+d x) (\cos (2 (c+5 d x))+i \sin (2 (c+5 d x))) \left (-66 i \cos (2 c+3 d x)+180 d x \cos (2 c+3 d x)+75 i \cos (4 c+3 d x)+180 d x \cos (4 c+3 d x)-50 i \cos (4 c+5 d x)+60 d x \cos (4 c+5 d x)-3 i \cos (6 c+5 d x)+60 d x \cos (6 c+5 d x)+3 \cos (2 c+d x) \left (71 i+80 d x-40 i \log \left (\cos ^2(c+d x)\right )\right )+\cos (d x) \left (119 i+240 d x-120 i \log \left (\cos ^2(c+d x)\right )\right )-90 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-90 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-30 i \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-30 i \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )-101 \sin (d x)-120 i d x \sin (d x)-60 \log \left (\cos ^2(c+d x)\right ) \sin (d x)+87 \sin (2 c+d x)-120 i d x \sin (2 c+d x)-60 \log \left (\cos ^2(c+d x)\right ) \sin (2 c+d x)-96 \sin (2 c+3 d x)-180 i d x \sin (2 c+3 d x)-90 \log \left (\cos ^2(c+d x)\right ) \sin (2 c+3 d x)+45 \sin (4 c+3 d x)-180 i d x \sin (4 c+3 d x)-90 \log \left (\cos ^2(c+d x)\right ) \sin (4 c+3 d x)-44 \sin (4 c+5 d x)-60 i d x \sin (4 c+5 d x)-30 \log \left (\cos ^2(c+d x)\right ) \sin (4 c+5 d x)+3 \sin (6 c+5 d x)-60 i d x \sin (6 c+5 d x)-30 \log \left (\cos ^2(c+d x)\right ) \sin (6 c+5 d x)\right )}{12 d (\cos (d x)+i \sin (d x))^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 408 vs. \(2 (116 ) = 232\).
time = 0.22, size = 409, normalized size = 3.30
method | result | size |
risch | \(-\frac {4 i a^{8} {\mathrm e}^{4 i \left (d x +c \right )}}{d}+\frac {32 i a^{8} {\mathrm e}^{2 i \left (d x +c \right )}}{d}-\frac {160 a^{8} c}{d}-\frac {4 i a^{8} \left (60 \,{\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{2 i \left (d x +c \right )}+47\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {80 i a^{8} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(114\) |
derivativedivides | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )-14 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )-2 i a^{8} \left (\cos ^{4}\left (d x +c \right )\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+56 i a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+a^{8} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(409\) |
default | \(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )-14 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )-2 i a^{8} \left (\cos ^{4}\left (d x +c \right )\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+56 i a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-28 a^{8} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+a^{8} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(409\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 135, normalized size = 1.09 \begin {gather*} \frac {a^{8} \tan \left (d x + c\right )^{3} - 12 i \, a^{8} \tan \left (d x + c\right )^{2} + 240 \, {\left (d x + c\right )} a^{8} + 120 i \, a^{8} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 93 \, a^{8} \tan \left (d x + c\right ) - \frac {48 \, {\left (5 \, a^{8} \tan \left (d x + c\right )^{3} - 6 i \, a^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{8} \tan \left (d x + c\right ) - 4 i \, a^{8}\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 179, normalized size = 1.44 \begin {gather*} -\frac {4 \, {\left (3 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 15 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 63 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 9 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 81 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 47 i \, a^{8} + 60 \, {\left (i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 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 )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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.
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Sympy [A]
time = 0.45, size = 216, normalized size = 1.74 \begin {gather*} - \frac {80 i a^{8} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 240 i a^{8} e^{4 i c} e^{4 i d x} - 420 i a^{8} e^{2 i c} e^{2 i d x} - 188 i a^{8}}{3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} + 3 d} + \begin {cases} \frac {- 4 i a^{8} d e^{4 i c} e^{4 i d x} + 32 i a^{8} d e^{2 i c} e^{2 i d x}}{d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (16 a^{8} e^{4 i c} - 64 a^{8} e^{2 i c}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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. 785 vs. \(2 (110) = 220\).
time = 1.12, size = 785, normalized size = 6.33 \begin {gather*} -\frac {4 \, {\left (60 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 840 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 5460 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 21840 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 60060 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 120120 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 180180 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 180180 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 120120 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 60060 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 21840 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 5460 i \, a^{8} e^{\left (4 i \, d x - 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 840 i \, a^{8} e^{\left (2 i \, d x - 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 205920 i \, a^{8} e^{\left (14 i \, d x\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 60 i \, a^{8} e^{\left (-14 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 i \, a^{8} e^{\left (32 i \, d x + 18 i \, c\right )} + 18 i \, a^{8} e^{\left (30 i \, d x + 16 i \, c\right )} - 63 i \, a^{8} e^{\left (28 i \, d x + 14 i \, c\right )} - 1032 i \, a^{8} e^{\left (26 i \, d x + 12 i \, c\right )} - 4968 i \, a^{8} e^{\left (24 i \, d x + 10 i \, c\right )} - 13516 i \, a^{8} e^{\left (22 i \, d x + 8 i \, c\right )} - 22847 i \, a^{8} e^{\left (20 i \, d x + 6 i \, c\right )} - 22066 i \, a^{8} e^{\left (18 i \, d x + 4 i \, c\right )} - 3234 i \, a^{8} e^{\left (16 i \, d x + 2 i \, c\right )} + 44979 i \, a^{8} e^{\left (12 i \, d x - 2 i \, c\right )} + 43332 i \, a^{8} e^{\left (10 i \, d x - 4 i \, c\right )} + 27672 i \, a^{8} e^{\left (8 i \, d x - 6 i \, c\right )} + 12048 i \, a^{8} e^{\left (6 i \, d x - 8 i \, c\right )} + 3467 i \, a^{8} e^{\left (4 i \, d x - 10 i \, c\right )} + 598 i \, a^{8} e^{\left (2 i \, d x - 12 i \, c\right )} + 25674 i \, a^{8} e^{\left (14 i \, d x\right )} + 47 i \, a^{8} e^{\left (-14 i \, c\right )}\right )}}{3 \, {\left (d e^{\left (28 i \, d x + 14 i \, c\right )} + 14 \, d e^{\left (26 i \, d x + 12 i \, c\right )} + 91 \, d e^{\left (24 i \, d x + 10 i \, c\right )} + 364 \, d e^{\left (22 i \, d x + 8 i \, c\right )} + 1001 \, d e^{\left (20 i \, d x + 6 i \, c\right )} + 2002 \, d e^{\left (18 i \, d x + 4 i \, c\right )} + 3003 \, d e^{\left (16 i \, d x + 2 i \, c\right )} + 3003 \, d e^{\left (12 i \, d x - 2 i \, c\right )} + 2002 \, d e^{\left (10 i \, d x - 4 i \, c\right )} + 1001 \, d e^{\left (8 i \, d x - 6 i \, c\right )} + 364 \, d e^{\left (6 i \, d x - 8 i \, c\right )} + 91 \, d e^{\left (4 i \, d x - 10 i \, c\right )} + 14 \, d e^{\left (2 i \, d x - 12 i \, c\right )} + 3432 \, d e^{\left (14 i \, d x\right )} + d e^{\left (-14 i \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.37, size = 111, normalized size = 0.90 \begin {gather*} \frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d}-\frac {80\,a^8\,\mathrm {tan}\left (c+d\,x\right )+a^8\,64{}\mathrm {i}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,2{}\mathrm {i}-1\right )}-\frac {31\,a^8\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {a^8\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,80{}\mathrm {i}}{d}-\frac {a^8\,{\mathrm {tan}\left (c+d\,x\right )}^2\,4{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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