Optimal. Leaf size=93 \[ 2 i a^2 x+\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {2 i a^2 \tan (c+d x)}{d}+\frac {a^2 \tan ^2(c+d x)}{d}+\frac {2 i a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan ^4(c+d x)}{4 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 93, 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 = {3624, 3609,
3606, 3556} \begin {gather*} -\frac {a^2 \tan ^4(c+d x)}{4 d}+\frac {2 i a^2 \tan ^3(c+d x)}{3 d}+\frac {a^2 \tan ^2(c+d x)}{d}-\frac {2 i a^2 \tan (c+d x)}{d}+\frac {2 a^2 \log (\cos (c+d x))}{d}+2 i a^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3606
Rule 3609
Rule 3624
Rubi steps
\begin {align*} \int \tan ^3(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {a^2 \tan ^4(c+d x)}{4 d}+\int \tan ^3(c+d x) \left (2 a^2+2 i a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {2 i a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan ^4(c+d x)}{4 d}+\int \tan ^2(c+d x) \left (-2 i a^2+2 a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {a^2 \tan ^2(c+d x)}{d}+\frac {2 i a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan ^4(c+d x)}{4 d}+\int \tan (c+d x) \left (-2 a^2-2 i a^2 \tan (c+d x)\right ) \, dx\\ &=2 i a^2 x-\frac {2 i a^2 \tan (c+d x)}{d}+\frac {a^2 \tan ^2(c+d x)}{d}+\frac {2 i a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan ^4(c+d x)}{4 d}-\left (2 a^2\right ) \int \tan (c+d x) \, dx\\ &=2 i a^2 x+\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {2 i a^2 \tan (c+d x)}{d}+\frac {a^2 \tan ^2(c+d x)}{d}+\frac {2 i a^2 \tan ^3(c+d x)}{3 d}-\frac {a^2 \tan ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 73, normalized size = 0.78 \begin {gather*} \frac {a^2 \left (24 i \text {ArcTan}(\tan (c+d x))+24 \log (\cos (c+d x))-24 i \tan (c+d x)+12 \tan ^2(c+d x)+8 i \tan ^3(c+d x)-3 \tan ^4(c+d x)\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 70, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-2 i \tan \left (d x +c \right )-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {2 i \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan ^{2}\left (d x +c \right )-\ln \left (1+\tan ^{2}\left (d x +c \right )\right )+2 i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(70\) |
default | \(\frac {a^{2} \left (-2 i \tan \left (d x +c \right )-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {2 i \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan ^{2}\left (d x +c \right )-\ln \left (1+\tan ^{2}\left (d x +c \right )\right )+2 i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(70\) |
risch | \(-\frac {4 i a^{2} c}{d}+\frac {2 a^{2} \left (21 \,{\mathrm e}^{6 i \left (d x +c \right )}+36 \,{\mathrm e}^{4 i \left (d x +c \right )}+29 \,{\mathrm e}^{2 i \left (d x +c \right )}+8\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(88\) |
norman | \(\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+2 i a^{2} x -\frac {2 i a^{2} \tan \left (d x +c \right )}{d}+\frac {2 i a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 82, normalized size = 0.88 \begin {gather*} -\frac {3 \, a^{2} \tan \left (d x + c\right )^{4} - 8 i \, a^{2} \tan \left (d x + c\right )^{3} - 12 \, a^{2} \tan \left (d x + c\right )^{2} - 24 i \, {\left (d x + c\right )} a^{2} + 12 \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 24 i \, a^{2} \tan \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 174 vs. \(2 (83) = 166\).
time = 0.42, size = 174, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (21 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 29 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 8 \, a^{2} + 3 \, {\left (a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \, {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 168, normalized size = 1.81 \begin {gather*} \frac {2 a^{2} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {42 a^{2} e^{6 i c} e^{6 i d x} + 72 a^{2} e^{4 i c} e^{4 i d x} + 58 a^{2} e^{2 i c} e^{2 i d x} + 16 a^{2}}{3 d e^{8 i c} e^{8 i d x} + 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} + 12 d e^{2 i c} e^{2 i d x} + 3 d} \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. 222 vs. \(2 (83) = 166\).
time = 0.89, size = 222, normalized size = 2.39 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 12 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 12 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 21 \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 29 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 \, a^{2}\right )}}{3 \, {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.80, size = 73, normalized size = 0.78 \begin {gather*} -\frac {2\,a^2\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}+a^2\,\mathrm {tan}\left (c+d\,x\right )\,2{}\mathrm {i}-\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\,2{}\mathrm {i}}{3}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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