Optimal. Leaf size=133 \[ \frac {35 i a^4 \sec (c+d x)}{8 d}+\frac {35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {35 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{24 d}+\frac {7 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.10, antiderivative size = 133, 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 = {3498, 3486, 3770} \[ \frac {35 i a^4 \sec (c+d x)}{8 d}+\frac {35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {7 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac {35 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{24 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3498
Rule 3770
Rubi steps
\begin {align*} \int \sec (c+d x) (a+i a \tan (c+d x))^4 \, dx &=\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {1}{4} (7 a) \int \sec (c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {7 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac {1}{12} \left (35 a^2\right ) \int \sec (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {7 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac {35 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{24 d}+\frac {1}{8} \left (35 a^3\right ) \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx\\ &=\frac {35 i a^4 \sec (c+d x)}{8 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {7 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac {35 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{24 d}+\frac {1}{8} \left (35 a^4\right ) \int \sec (c+d x) \, dx\\ &=\frac {35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {35 i a^4 \sec (c+d x)}{8 d}+\frac {i a \sec (c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac {7 i \sec (c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac {35 i \sec (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{24 d}\\ \end {align*}
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Mathematica [A] time = 1.48, size = 237, normalized size = 1.78 \[ -\frac {a^4 \sec ^4(c+d x) \left (3 \left (42 \sin (c+d x)+58 \sin (3 (c+d x))-128 i \cos (3 (c+d x))+35 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+140 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-35 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-896 i \cos (c+d x)\right )}{192 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 256, normalized size = 1.92 \[ \frac {558 i \, a^{4} e^{\left (7 i \, d x + 7 i \, c\right )} + 1022 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} + 770 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, a^{4} e^{\left (i \, d x + i \, c\right )} + 105 \, {\left (a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, {\left (a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{24 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.62, size = 173, normalized size = 1.30 \[ \frac {105 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 105 \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - \frac {2 \, {\left (81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 480 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 544 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 81 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 160 i \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 231, normalized size = 1.74 \[ \frac {a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {27 a^{4} \sin \left (d x +c \right )}{8 d}+\frac {35 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}-\frac {4 i a^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}+\frac {4 i a^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )}+\frac {4 i a^{4} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3 d}+\frac {8 i a^{4} \cos \left (d x +c \right )}{3 d}-\frac {3 a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {4 i a^{4}}{d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 180, normalized size = 1.35 \[ \frac {3 \, a^{4} {\left (\frac {2 \, {\left (5 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 72 \, a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + \frac {192 i \, a^{4}}{\cos \left (d x + c\right )} + \frac {64 i \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{4}}{\cos \left (d x + c\right )^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.77, size = 198, normalized size = 1.49 \[ \frac {35\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {27\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,8{}\mathrm {i}-\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,40{}\mathrm {i}-\frac {35\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,136{}\mathrm {i}}{3}+\frac {27\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {a^4\,40{}\mathrm {i}}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{4} \left (\int \left (- 6 \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\right )\, dx + \int \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 i \tan {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \left (- 4 i \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\right )\, dx + \int \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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