Optimal. Leaf size=63 \[ 4 a^3 x-\frac {4 i a^3 \log (\cos (c+d x))}{d}-\frac {2 a^3 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^2}{2 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {2 a^3 \tan (c+d x)}{d}-\frac {4 i a^3 \log (\cos (c+d x))}{d}+4 a^3 x+\frac {i a (a+i a \tan (c+d x))^2}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3558
Rule 3559
Rubi steps
\begin {align*} \int (a+i a \tan (c+d x))^3 \, dx &=\frac {i a (a+i a \tan (c+d x))^2}{2 d}+(2 a) \int (a+i a \tan (c+d x))^2 \, dx\\ &=4 a^3 x-\frac {2 a^3 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^2}{2 d}+\left (4 i a^3\right ) \int \tan (c+d x) \, dx\\ &=4 a^3 x-\frac {4 i a^3 \log (\cos (c+d x))}{d}-\frac {2 a^3 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 119, normalized size = 1.89 \begin {gather*} \frac {a^3 \sec (c) \sec ^2(c+d x) \left (2 d x \cos (3 c+2 d x)+\cos (c+2 d x) \left (2 d x-i \log \left (\cos ^2(c+d x)\right )\right )+\cos (c) \left (-i+4 d x-2 i \log \left (\cos ^2(c+d x)\right )\right )-i \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )+3 \sin (c)-3 \sin (c+2 d x)\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 51, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-3 \tan \left (d x +c \right )-\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+4 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(51\) |
default | \(\frac {a^{3} \left (-3 \tan \left (d x +c \right )-\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 i \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+4 \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(51\) |
norman | \(4 a^{3} x -\frac {3 a^{3} \tan \left (d x +c \right )}{d}-\frac {i a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(59\) |
risch | \(-\frac {8 a^{3} c}{d}-\frac {2 i a^{3} \left (4 \,{\mathrm e}^{2 i \left (d x +c \right )}+3\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {4 i a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 76, normalized size = 1.21 \begin {gather*} a^{3} x + \frac {3 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{3}}{d} + \frac {i \, a^{3} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{2 \, d} + \frac {3 i \, a^{3} \log \left (\sec \left (d x + c\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 97, normalized size = 1.54 \begin {gather*} -\frac {2 \, {\left (4 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, a^{3} + 2 \, {\left (i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 94, normalized size = 1.49 \begin {gather*} - \frac {4 i a^{3} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 8 i a^{3} e^{2 i c} e^{2 i d x} - 6 i a^{3}}{d e^{4 i c} e^{4 i d x} + 2 d e^{2 i c} e^{2 i d x} + 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. 118 vs. \(2 (55) = 110\).
time = 0.50, size = 118, normalized size = 1.87 \begin {gather*} -\frac {2 \, {\left (2 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 4 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 4 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, a^{3} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 i \, a^{3}\right )}}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.68, size = 41, normalized size = 0.65 \begin {gather*} -\frac {a^3\,\left (6\,\mathrm {tan}\left (c+d\,x\right )-\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}+{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}\right )}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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