Optimal. Leaf size=143 \[ -\frac {25 i x}{8 a^3}+\frac {3 \log (\cos (c+d x))}{a^3 d}+\frac {25 i \tan (c+d x)}{8 a^3 d}-\frac {\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.19, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3639, 3676,
3606, 3556} \begin {gather*} \frac {3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {25 i \tan (c+d x)}{8 a^3 d}+\frac {3 \log (\cos (c+d x))}{a^3 d}-\frac {25 i x}{8 a^3}-\frac {\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3606
Rule 3639
Rule 3676
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac {\int \frac {\tan ^3(c+d x) (-4 a+7 i a \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=-\frac {\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\tan ^2(c+d x) \left (-33 i a^2-39 a^2 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=-\frac {\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {\int \tan (c+d x) \left (144 a^3-150 i a^3 \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac {25 i x}{8 a^3}+\frac {25 i \tan (c+d x)}{8 a^3 d}-\frac {\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {3 \int \tan (c+d x) \, dx}{a^3}\\ &=-\frac {25 i x}{8 a^3}+\frac {3 \log (\cos (c+d x))}{a^3 d}+\frac {25 i \tan (c+d x)}{8 a^3 d}-\frac {\tan ^4(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {11 i \tan ^3(c+d x)}{24 a d (a+i a \tan (c+d x))^2}+\frac {3 \tan ^2(c+d x)}{2 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 2.70, size = 239, normalized size = 1.67 \begin {gather*} \frac {\sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 (-138 i \cos (2 d x) \sin (c)-21 i \cos (4 d x) \sin (c)+300 d x \sin (3 c)+2 i \cos (6 d x) \sin (3 c)+288 i \log (\cos (c+d x)) \sin (3 c)-96 \sec (c) \sec (c+d x) \sin (3 c) \sin (d x)-138 \sin (c) \sin (2 d x)+\cos (c) (39 \cos (d x)+53 i \sin (d x)) (-3 \cos (3 d x)+3 i \sin (3 d x))-21 \sin (c) \sin (4 d x)+\cos (3 c) (-300 i d x-2 \cos (6 d x)+288 \log (\cos (c+d x))+96 i \sec (c) \sec (c+d x) \sin (d x)+2 i \sin (6 d x))+2 \sin (3 c) \sin (6 d x))}{96 d (a+i a \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 83, normalized size = 0.58
method | result | size |
derivativedivides | \(\frac {i \tan \left (d x +c \right )+\frac {31 i}{8 \left (\tan \left (d x +c \right )-i\right )}-\frac {i}{6 \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {9}{8 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {49 \ln \left (\tan \left (d x +c \right )-i\right )}{16}+\frac {\ln \left (\tan \left (d x +c \right )+i\right )}{16}}{d \,a^{3}}\) | \(83\) |
default | \(\frac {i \tan \left (d x +c \right )+\frac {31 i}{8 \left (\tan \left (d x +c \right )-i\right )}-\frac {i}{6 \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {9}{8 \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {49 \ln \left (\tan \left (d x +c \right )-i\right )}{16}+\frac {\ln \left (\tan \left (d x +c \right )+i\right )}{16}}{d \,a^{3}}\) | \(83\) |
risch | \(-\frac {49 i x}{8 a^{3}}-\frac {23 \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 a^{3} d}+\frac {7 \,{\mathrm e}^{-4 i \left (d x +c \right )}}{32 a^{3} d}-\frac {{\mathrm e}^{-6 i \left (d x +c \right )}}{48 a^{3} d}-\frac {6 i c}{a^{3} d}-\frac {2}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{3} d}\) | \(111\) |
norman | \(\frac {-\frac {5 \left (\tan ^{4}\left (d x +c \right )\right )}{d a}+\frac {i \left (\tan ^{7}\left (d x +c \right )\right )}{d a}-\frac {35}{12 d a}-\frac {25 i x}{8 a}-\frac {29 \left (\tan ^{2}\left (d x +c \right )\right )}{4 d a}-\frac {75 i x \left (\tan ^{2}\left (d x +c \right )\right )}{8 a}-\frac {75 i x \left (\tan ^{4}\left (d x +c \right )\right )}{8 a}-\frac {25 i x \left (\tan ^{6}\left (d x +c \right )\right )}{8 a}+\frac {25 i \tan \left (d x +c \right )}{8 d a}+\frac {25 i \left (\tan ^{3}\left (d x +c \right )\right )}{3 d a}+\frac {55 i \left (\tan ^{5}\left (d x +c \right )\right )}{8 d a}}{a^{2} \left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 a^{3} d}\) | \(196\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 120, normalized size = 0.84 \begin {gather*} \frac {-588 i \, d x e^{\left (8 i \, d x + 8 i \, c\right )} - 6 \, {\left (98 i \, d x + 55\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 288 \, {\left (e^{\left (8 i \, d x + 8 i \, c\right )} + e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 117 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2}{96 \, {\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 214, normalized size = 1.50 \begin {gather*} \begin {cases} \frac {\left (- 35328 a^{6} d^{2} e^{10 i c} e^{- 2 i d x} + 5376 a^{6} d^{2} e^{8 i c} e^{- 4 i d x} - 512 a^{6} d^{2} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (\frac {\left (- 49 i e^{6 i c} + 23 i e^{4 i c} - 7 i e^{2 i c} + i\right ) e^{- 6 i c}}{8 a^{3}} + \frac {49 i}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {2}{a^{3} d e^{2 i c} e^{2 i d x} + a^{3} d} - \frac {49 i x}{8 a^{3}} + \frac {3 \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.22, size = 91, normalized size = 0.64 \begin {gather*} \frac {\frac {6 \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{3}} - \frac {294 \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {96 i \, \tan \left (d x + c\right )}{a^{3}} + \frac {539 \, \tan \left (d x + c\right )^{3} - 1245 i \, \tan \left (d x + c\right )^{2} - 981 \, \tan \left (d x + c\right ) + 259 i}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.81, size = 122, normalized size = 0.85 \begin {gather*} -\frac {\frac {35}{12\,a^3}-\frac {31\,{\mathrm {tan}\left (c+d\,x\right )}^2}{8\,a^3}+\frac {\mathrm {tan}\left (c+d\,x\right )\,53{}\mathrm {i}}{8\,a^3}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1\right )}-\frac {49\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{16\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{16\,a^3\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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