Optimal. Leaf size=119 \[ -\frac {7 x}{8 a^3}-\frac {i \log (\cos (c+d x))}{a^3 d}-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}+\frac {7 i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.16, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3639, 3676,
3670, 3556, 12, 3607, 8} \begin {gather*} \frac {7 i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {i \log (\cos (c+d x))}{a^3 d}-\frac {7 x}{8 a^3}-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 3556
Rule 3607
Rule 3639
Rule 3670
Rule 3676
Rubi steps
\begin {align*} \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}-\frac {\int \frac {\tan ^2(c+d x) (-3 a+6 i a \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\tan (c+d x) \left (-18 i a^2-24 a^2 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}-\frac {i \int \frac {42 a^3 \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{24 a^5}+\frac {i \int \tan (c+d x) \, dx}{a^3}\\ &=-\frac {i \log (\cos (c+d x))}{a^3 d}-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}-\frac {(7 i) \int \frac {\tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=-\frac {i \log (\cos (c+d x))}{a^3 d}-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}+\frac {7 i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {7 \int 1 \, dx}{8 a^3}\\ &=-\frac {7 x}{8 a^3}-\frac {i \log (\cos (c+d x))}{a^3 d}-\frac {\tan ^3(c+d x)}{6 d (a+i a \tan (c+d x))^3}+\frac {3 i \tan ^2(c+d x)}{8 a d (a+i a \tan (c+d x))^2}+\frac {7 i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 118, normalized size = 0.99 \begin {gather*} \frac {\sec ^3(c+d x) (-51 \cos (c+d x)+\cos (3 (c+d x)) (-2-84 i d x+96 \log (\cos (c+d x)))-81 i \sin (c+d x)+2 i \sin (3 (c+d x))+84 d x \sin (3 (c+d x))+96 i \log (\cos (c+d x)) \sin (3 (c+d x)))}{96 a^3 d (-i+\tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 75, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {\frac {7 i}{8 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {15 i \ln \left (\tan \left (d x +c \right )-i\right )}{16}-\frac {1}{6 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {17}{8 \left (\tan \left (d x +c \right )-i\right )}+\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{16}}{d \,a^{3}}\) | \(75\) |
default | \(\frac {\frac {7 i}{8 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {15 i \ln \left (\tan \left (d x +c \right )-i\right )}{16}-\frac {1}{6 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {17}{8 \left (\tan \left (d x +c \right )-i\right )}+\frac {i \ln \left (\tan \left (d x +c \right )+i\right )}{16}}{d \,a^{3}}\) | \(75\) |
risch | \(-\frac {15 x}{8 a^{3}}+\frac {11 i {\mathrm e}^{-2 i \left (d x +c \right )}}{16 a^{3} d}-\frac {5 i {\mathrm e}^{-4 i \left (d x +c \right )}}{32 a^{3} d}+\frac {i {\mathrm e}^{-6 i \left (d x +c \right )}}{48 a^{3} d}-\frac {2 c}{a^{3} d}-\frac {i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{3} d}\) | \(92\) |
norman | \(\frac {-\frac {7 x}{8 a}+\frac {17 \left (\tan ^{5}\left (d x +c \right )\right )}{8 d a}-\frac {21 x \left (\tan ^{2}\left (d x +c \right )\right )}{8 a}-\frac {21 x \left (\tan ^{4}\left (d x +c \right )\right )}{8 a}-\frac {7 x \left (\tan ^{6}\left (d x +c \right )\right )}{8 a}+\frac {17 i}{12 d a}+\frac {7 \tan \left (d x +c \right )}{8 d a}+\frac {7 \left (\tan ^{3}\left (d x +c \right )\right )}{3 d a}+\frac {3 i \left (\tan ^{4}\left (d x +c \right )\right )}{d a}+\frac {15 i \left (\tan ^{2}\left (d x +c \right )\right )}{4 d a}}{a^{2} \left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 a^{3} d}\) | \(176\) |
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 = 0.36, size = 77, normalized size = 0.65 \begin {gather*} -\frac {{\left (180 \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 96 i \, e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 66 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 184, normalized size = 1.55 \begin {gather*} \begin {cases} \frac {\left (16896 i a^{6} d^{2} e^{10 i c} e^{- 2 i d x} - 3840 i a^{6} d^{2} e^{8 i c} e^{- 4 i d x} + 512 i 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 (- 15 e^{6 i c} + 11 e^{4 i c} - 5 e^{2 i c} + 1\right ) e^{- 6 i c}}{8 a^{3}} + \frac {15}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {15 x}{8 a^{3}} - \frac {i \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 = 1.39, size = 80, normalized size = 0.67 \begin {gather*} -\frac {-\frac {90 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} - \frac {6 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {165 i \, \tan \left (d x + c\right )^{3} + 291 \, \tan \left (d x + c\right )^{2} - 171 i \, \tan \left (d x + c\right ) - 29}{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 = 3.85, size = 110, normalized size = 0.92 \begin {gather*} -\frac {\frac {27\,\mathrm {tan}\left (c+d\,x\right )}{8\,a^3}-\frac {17{}\mathrm {i}}{12\,a^3}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,17{}\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 {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,15{}\mathrm {i}}{16\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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