Optimal. Leaf size=117 \[ -\frac {8 a^5 \tan (c+d x)}{d}-\frac {16 i a^5 \log (\cos (c+d x))}{d}+16 a^5 x+\frac {2 i a^2 (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac {i a (a+i a \tan (c+d x))^4}{4 d} \]
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Rubi [A] time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3478, 3477, 3475} \[ -\frac {8 a^5 \tan (c+d x)}{d}+\frac {2 i a^2 (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac {16 i a^5 \log (\cos (c+d x))}{d}+16 a^5 x+\frac {i a (a+i a \tan (c+d x))^4}{4 d} \]
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))^5 \, dx &=\frac {i a (a+i a \tan (c+d x))^4}{4 d}+(2 a) \int (a+i a \tan (c+d x))^4 \, dx\\ &=\frac {2 i a^2 (a+i a \tan (c+d x))^3}{3 d}+\frac {i a (a+i a \tan (c+d x))^4}{4 d}+\left (4 a^2\right ) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac {2 i a^3 (a+i a \tan (c+d x))^2}{d}+\frac {2 i a^2 (a+i a \tan (c+d x))^3}{3 d}+\frac {i a (a+i a \tan (c+d x))^4}{4 d}+\left (8 a^3\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=16 a^5 x-\frac {8 a^5 \tan (c+d x)}{d}+\frac {2 i a^3 (a+i a \tan (c+d x))^2}{d}+\frac {2 i a^2 (a+i a \tan (c+d x))^3}{3 d}+\frac {i a (a+i a \tan (c+d x))^4}{4 d}+\left (16 i a^5\right ) \int \tan (c+d x) \, dx\\ &=16 a^5 x-\frac {16 i a^5 \log (\cos (c+d x))}{d}-\frac {8 a^5 \tan (c+d x)}{d}+\frac {2 i a^3 (a+i a \tan (c+d x))^2}{d}+\frac {2 i a^2 (a+i a \tan (c+d x))^3}{3 d}+\frac {i a (a+i a \tan (c+d x))^4}{4 d}\\ \end {align*}
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Mathematica [A] time = 2.87, size = 228, normalized size = 1.95 \[ \frac {a^5 \sec (c) \sec ^4(c+d x) \left (-70 \sin (c+2 d x)+30 \sin (3 c+2 d x)-25 \sin (3 c+4 d x)+48 d x \cos (3 c+2 d x)-18 i \cos (3 c+2 d x)+12 d x \cos (3 c+4 d x)+12 d x \cos (5 c+4 d x)-24 i \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )+6 \cos (c+2 d x) \left (-4 i \log \left (\cos ^2(c+d x)\right )+8 d x-3 i\right )+\cos (c) \left (-36 i \log \left (\cos ^2(c+d x)\right )+72 d x-33 i\right )-6 i \cos (3 c+4 d x) \log \left (\cos ^2(c+d x)\right )-6 i \cos (5 c+4 d x) \log \left (\cos ^2(c+d x)\right )+75 \sin (c)\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 176, normalized size = 1.50 \[ \frac {-192 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 432 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 352 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - 100 i \, a^{5} + {\left (-48 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 192 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 288 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 192 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - 48 i \, a^{5}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.77, size = 222, normalized size = 1.90 \[ \frac {-48 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 192 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 288 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 192 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 192 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 432 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 352 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - 48 i \, a^{5} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 100 i \, a^{5}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 101, normalized size = 0.86 \[ -\frac {15 a^{5} \tan \left (d x +c \right )}{d}+\frac {i a^{5} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {5 a^{5} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {11 i a^{5} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {8 i a^{5} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {16 a^{5} \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 165, normalized size = 1.41 \[ a^{5} x + \frac {5 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{5}}{3 \, d} + \frac {10 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{5}}{d} + \frac {i \, a^{5} {\left (\frac {4 \, \sin \left (d x + c\right )^{2} - 3}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{4 \, d} + \frac {5 i \, a^{5} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} + \frac {5 i \, a^{5} \log \left (\sec \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.27, size = 73, normalized size = 0.62 \[ \frac {a^5\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,16{}\mathrm {i}-15\,a^5\,\mathrm {tan}\left (c+d\,x\right )-\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\,11{}\mathrm {i}}{2}+\frac {5\,a^5\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^4\,1{}\mathrm {i}}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 182, normalized size = 1.56 \[ - \frac {16 i a^{5} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 192 a^{5} e^{6 i c} e^{6 i d x} - 432 a^{5} e^{4 i c} e^{4 i d x} - 352 a^{5} e^{2 i c} e^{2 i d x} - 100 a^{5}}{- 3 i d e^{8 i c} e^{8 i d x} - 12 i d e^{6 i c} e^{6 i d x} - 18 i d e^{4 i c} e^{4 i d x} - 12 i d e^{2 i c} e^{2 i d x} - 3 i d} \]
Verification of antiderivative is not currently implemented for this CAS.
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