Optimal. Leaf size=83 \[ \frac {i a \tan ^4(c+d x)}{4 d}+\frac {a \tan ^3(c+d x)}{3 d}-\frac {i a \tan ^2(c+d x)}{2 d}-\frac {a \tan (c+d x)}{d}-\frac {i a \log (\cos (c+d x))}{d}+a x \]
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Rubi [A] time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3528, 3525, 3475} \[ \frac {i a \tan ^4(c+d x)}{4 d}+\frac {a \tan ^3(c+d x)}{3 d}-\frac {i a \tan ^2(c+d x)}{2 d}-\frac {a \tan (c+d x)}{d}-\frac {i a \log (\cos (c+d x))}{d}+a x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rubi steps
\begin {align*} \int \tan ^4(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \tan ^4(c+d x)}{4 d}+\int \tan ^3(c+d x) (-i a+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^3(c+d x)}{3 d}+\frac {i a \tan ^4(c+d x)}{4 d}+\int \tan ^2(c+d x) (-a-i a \tan (c+d x)) \, dx\\ &=-\frac {i a \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {i a \tan ^4(c+d x)}{4 d}+\int \tan (c+d x) (i a-a \tan (c+d x)) \, dx\\ &=a x-\frac {a \tan (c+d x)}{d}-\frac {i a \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {i a \tan ^4(c+d x)}{4 d}+(i a) \int \tan (c+d x) \, dx\\ &=a x-\frac {i a \log (\cos (c+d x))}{d}-\frac {a \tan (c+d x)}{d}-\frac {i a \tan ^2(c+d x)}{2 d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {i a \tan ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 81, normalized size = 0.98 \[ \frac {a \tan ^{-1}(\tan (c+d x))}{d}+\frac {a \tan ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}-\frac {i a \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 158, normalized size = 1.90 \[ \frac {-24 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 36 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 32 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-3 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 12 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 18 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 12 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 8 i \, a}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.61, size = 204, normalized size = 2.46 \[ \frac {-3 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 24 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 36 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 32 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 8 i \, a}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 88, normalized size = 1.06 \[ -\frac {a \tan \left (d x +c \right )}{d}+\frac {i a \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {i a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 70, normalized size = 0.84 \[ -\frac {-3 i \, a \tan \left (d x + c\right )^{4} - 4 \, a \tan \left (d x + c\right )^{3} + 6 i \, a \tan \left (d x + c\right )^{2} - 12 \, {\left (d x + c\right )} a - 6 i \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.71, size = 63, normalized size = 0.76 \[ \frac {\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2}-a\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}}{4}+a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.45, size = 168, normalized size = 2.02 \[ - \frac {i a \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {24 i a e^{6 i c} e^{6 i d x} + 36 i a e^{4 i c} e^{4 i d x} + 32 i a e^{2 i c} e^{2 i d x} + 8 i a}{- 3 d e^{8 i c} e^{8 i d x} - 12 d e^{6 i c} e^{6 i d x} - 18 d e^{4 i c} e^{4 i d x} - 12 d e^{2 i c} e^{2 i d x} - 3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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