Optimal. Leaf size=67 \[ i a x+\frac {a \log (\cos (c+d x))}{d}-\frac {i a \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {i a \tan ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3609, 3606,
3556} \begin {gather*} \frac {i a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^2(c+d x)}{2 d}-\frac {i a \tan (c+d x)}{d}+\frac {a \log (\cos (c+d x))}{d}+i a x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3606
Rule 3609
Rubi steps
\begin {align*} \int \tan ^3(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \tan ^3(c+d x)}{3 d}+\int \tan ^2(c+d x) (-i a+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^2(c+d x)}{2 d}+\frac {i a \tan ^3(c+d x)}{3 d}+\int \tan (c+d x) (-a-i a \tan (c+d x)) \, dx\\ &=i a x-\frac {i a \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {i a \tan ^3(c+d x)}{3 d}-a \int \tan (c+d x) \, dx\\ &=i a x+\frac {a \log (\cos (c+d x))}{d}-\frac {i a \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {i a \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 74, normalized size = 1.10 \begin {gather*} \frac {i a \text {ArcTan}(\tan (c+d x))}{d}-\frac {i a \tan (c+d x)}{d}+\frac {i a \tan ^3(c+d x)}{3 d}+\frac {a \left (2 \log (\cos (c+d x))+\tan ^2(c+d x)\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 60, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {a \left (-i \tan \left (d x +c \right )+\frac {i \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(60\) |
default | \(\frac {a \left (-i \tan \left (d x +c \right )+\frac {i \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+i \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(60\) |
norman | \(i a x +\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {i a \tan \left (d x +c \right )}{d}+\frac {i a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(66\) |
risch | \(-\frac {2 i a c}{d}+\frac {2 a \left (9 \,{\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}+4\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 59, normalized size = 0.88 \begin {gather*} -\frac {-2 i \, a \tan \left (d x + c\right )^{3} - 3 \, a \tan \left (d x + c\right )^{2} - 6 i \, {\left (d x + c\right )} a + 3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 i \, a \tan \left (d x + c\right )}{6 \, 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. 120 vs. \(2 (57) = 114\).
time = 0.50, size = 120, normalized size = 1.79 \begin {gather*} \frac {18 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, {\left (a e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 \, a}{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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 121 vs. \(2 (56) = 112\).
time = 0.20, size = 121, normalized size = 1.81 \begin {gather*} \frac {a \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {18 a e^{4 i c} e^{4 i d x} + 18 a e^{2 i c} e^{2 i d x} + 8 a}{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} \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. 156 vs. \(2 (57) = 114\).
time = 0.90, size = 156, normalized size = 2.33 \begin {gather*} \frac {3 \, a e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9 \, a e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9 \, a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 \, a}{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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Mupad [B]
time = 3.69, size = 51, normalized size = 0.76 \begin {gather*} -\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}}{3}+a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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