Optimal. Leaf size=67 \[ \frac {i a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^2(c+d x)}{2 d}-\frac {i a \tan (c+d x)}{d}+\frac {a \log (\cos (c+d x))}{d}+i a x \]
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Rubi [A] time = 0.06, 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 = {3528, 3525, 3475} \[ \frac {i a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^2(c+d x)}{2 d}-\frac {i a \tan (c+d x)}{d}+\frac {a \log (\cos (c+d x))}{d}+i a x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
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 = 0.15, size = 74, normalized size = 1.10 \[ \frac {i a \tan ^{-1}(\tan (c+d x))}{d}+\frac {i a \tan ^3(c+d x)}{3 d}-\frac {i a \tan (c+d x)}{d}+\frac {a \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 120, normalized size = 1.79 \[ \frac {18 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, {\left (a e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 \, a}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.38, size = 156, normalized size = 2.33 \[ \frac {3 \, a e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9 \, a e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 9 \, a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 \, a}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 75, normalized size = 1.12 \[ -\frac {i a \tan \left (d x +c \right )}{d}+\frac {i a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {i a \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 59, normalized size = 0.88 \[ -\frac {-2 i \, a \tan \left (d x + c\right )^{3} - 3 \, a \tan \left (d x + c\right )^{2} - 6 i \, {\left (d x + c\right )} a + 3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 i \, a \tan \left (d x + c\right )}{6 \, d} \]
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 \[ -\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}}{3}+a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.41, size = 136, normalized size = 2.03 \[ \frac {a \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 18 i a e^{4 i c} e^{4 i d x} - 18 i a e^{2 i c} e^{2 i d x} - 8 i a}{- 3 i d e^{6 i c} e^{6 i d x} - 9 i d e^{4 i c} e^{4 i d x} - 9 i d e^{2 i c} e^{2 i d x} - 3 i d} \]
Verification of antiderivative is not currently implemented for this CAS.
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