Optimal. Leaf size=89 \[ -\frac {4 a^4 \tan (c+d x)}{d}-\frac {8 i a^4 \log (\cos (c+d x))}{d}+8 a^4 x+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3478, 3477, 3475} \[ -\frac {4 a^4 \tan (c+d x)}{d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac {8 i a^4 \log (\cos (c+d x))}{d}+8 a^4 x+\frac {i a (a+i a \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rule 3478
Rubi steps
\begin {align*} \int (a+i a \tan (c+d x))^4 \, dx &=\frac {i a (a+i a \tan (c+d x))^3}{3 d}+(2 a) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (4 a^2\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=8 a^4 x-\frac {4 a^4 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (8 i a^4\right ) \int \tan (c+d x) \, dx\\ &=8 a^4 x-\frac {8 i a^4 \log (\cos (c+d x))}{d}-\frac {4 a^4 \tan (c+d x)}{d}+\frac {i a (a+i a \tan (c+d x))^3}{3 d}+\frac {i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \end {align*}
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Mathematica [A] time = 1.17, size = 176, normalized size = 1.98 \[ \frac {a^4 \sec (c) \sec ^3(c+d x) \left (12 \sin (2 c+d x)-11 \sin (2 c+3 d x)+6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)-3 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+3 \cos (d x) \left (-3 i \log \left (\cos ^2(c+d x)\right )+6 d x-2 i\right )+3 \cos (2 c+d x) \left (-3 i \log \left (\cos ^2(c+d x)\right )+6 d x-2 i\right )-3 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-21 \sin (d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 136, normalized size = 1.53 \[ \frac {-72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 108 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 44 i \, a^{4} + {\left (-24 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 72 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 24 i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.78, size = 170, normalized size = 1.91 \[ \frac {-24 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 72 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 108 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 24 i \, a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 44 i \, a^{4}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 0.94 \[ -\frac {7 a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {2 i a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {4 i a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {8 a^{4} \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 108, normalized size = 1.21 \[ a^{4} x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac {6 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{4}}{d} + \frac {2 i \, a^{4} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} + \frac {4 i \, a^{4} \log \left (\sec \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 59, normalized size = 0.66 \[ \frac {\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-7\,a^4\,\mathrm {tan}\left (c+d\,x\right )+a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,8{}\mathrm {i}-a^4\,{\mathrm {tan}\left (c+d\,x\right )}^2\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 138, normalized size = 1.55 \[ - \frac {8 i a^{4} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {72 i a^{4} e^{4 i c} e^{4 i d x} + 108 i a^{4} e^{2 i c} e^{2 i d x} + 44 i a^{4}}{- 3 d e^{6 i c} e^{6 i d x} - 9 d e^{4 i c} e^{4 i d x} - 9 d e^{2 i c} e^{2 i d x} - 3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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