Optimal. Leaf size=112 \[ -\frac {a^2 \tan ^5(c+d x)}{5 d}+\frac {i a^2 \tan ^4(c+d x)}{2 d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}-\frac {i a^2 \tan ^2(c+d x)}{d}-\frac {2 a^2 \tan (c+d x)}{d}-\frac {2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x \]
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Rubi [A] time = 0.14, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3543, 3528, 3525, 3475} \[ -\frac {a^2 \tan ^5(c+d x)}{5 d}+\frac {i a^2 \tan ^4(c+d x)}{2 d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}-\frac {i a^2 \tan ^2(c+d x)}{d}-\frac {2 a^2 \tan (c+d x)}{d}-\frac {2 i a^2 \log (\cos (c+d x))}{d}+2 a^2 x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rule 3543
Rubi steps
\begin {align*} \int \tan ^4(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {a^2 \tan ^5(c+d x)}{5 d}+\int \tan ^4(c+d x) \left (2 a^2+2 i a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {i a^2 \tan ^4(c+d x)}{2 d}-\frac {a^2 \tan ^5(c+d x)}{5 d}+\int \tan ^3(c+d x) \left (-2 i a^2+2 a^2 \tan (c+d x)\right ) \, dx\\ &=\frac {2 a^2 \tan ^3(c+d x)}{3 d}+\frac {i a^2 \tan ^4(c+d x)}{2 d}-\frac {a^2 \tan ^5(c+d x)}{5 d}+\int \tan ^2(c+d x) \left (-2 a^2-2 i a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac {i a^2 \tan ^2(c+d x)}{d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}+\frac {i a^2 \tan ^4(c+d x)}{2 d}-\frac {a^2 \tan ^5(c+d x)}{5 d}+\int \tan (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=2 a^2 x-\frac {2 a^2 \tan (c+d x)}{d}-\frac {i a^2 \tan ^2(c+d x)}{d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}+\frac {i a^2 \tan ^4(c+d x)}{2 d}-\frac {a^2 \tan ^5(c+d x)}{5 d}+\left (2 i a^2\right ) \int \tan (c+d x) \, dx\\ &=2 a^2 x-\frac {2 i a^2 \log (\cos (c+d x))}{d}-\frac {2 a^2 \tan (c+d x)}{d}-\frac {i a^2 \tan ^2(c+d x)}{d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}+\frac {i a^2 \tan ^4(c+d x)}{2 d}-\frac {a^2 \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 108, normalized size = 0.96 \[ \frac {2 a^2 \tan ^{-1}(\tan (c+d x))}{d}-\frac {a^2 \tan ^5(c+d x)}{5 d}+\frac {2 a^2 \tan ^3(c+d x)}{3 d}-\frac {2 a^2 \tan (c+d x)}{d}-\frac {i a^2 \left (-\tan ^4(c+d x)+2 \tan ^2(c+d x)+4 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 216, normalized size = 1.93 \[ \frac {-270 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 600 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 740 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 400 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 86 i \, a^{2} + {\left (-30 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 150 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 300 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 300 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 150 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 30 i \, a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.94, size = 274, normalized size = 2.45 \[ \frac {-30 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 150 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 300 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 300 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 150 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 270 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 600 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 740 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 400 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 30 i \, a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 86 i \, a^{2}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 117, normalized size = 1.04 \[ -\frac {2 a^{2} \tan \left (d x +c \right )}{d}-\frac {a^{2} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {i a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{2 d}+\frac {2 a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {i a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {i a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {2 a^{2} \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 95, normalized size = 0.85 \[ -\frac {6 \, a^{2} \tan \left (d x + c\right )^{5} - 15 i \, a^{2} \tan \left (d x + c\right )^{4} - 20 \, a^{2} \tan \left (d x + c\right )^{3} + 30 i \, a^{2} \tan \left (d x + c\right )^{2} - 60 \, {\left (d x + c\right )} a^{2} - 30 i \, a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, a^{2} \tan \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.70, size = 86, normalized size = 0.77 \[ \frac {\frac {2\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-2\,a^2\,\mathrm {tan}\left (c+d\,x\right )-\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+a^2\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}-a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.60, size = 219, normalized size = 1.96 \[ - \frac {2 i a^{2} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {270 i a^{2} e^{8 i c} e^{8 i d x} + 600 i a^{2} e^{6 i c} e^{6 i d x} + 740 i a^{2} e^{4 i c} e^{4 i d x} + 400 i a^{2} e^{2 i c} e^{2 i d x} + 86 i a^{2}}{- 15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} - 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} - 75 d e^{2 i c} e^{2 i d x} - 15 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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