Optimal. Leaf size=175 \[ -\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d}+\frac {22 i \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{35 d}+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d} \]
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Rubi [A] time = 0.19, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3498, 3486, 3768, 3771, 2639} \[ -\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d}+\frac {22 i \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{35 d}+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3486
Rule 3498
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx &=\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {1}{7} (11 a) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2 \, dx\\ &=\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac {1}{5} \left (11 a^2\right ) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx\\ &=\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac {1}{5} \left (11 a^3\right ) \int (e \sec (c+d x))^{3/2} \, dx\\ &=\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}-\frac {1}{5} \left (11 a^3 e^2\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx\\ &=\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}-\frac {\left (11 a^3 e^2\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=-\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}\\ \end {align*}
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Mathematica [C] time = 2.73, size = 129, normalized size = 0.74 \[ \frac {a^3 (1+i \tan (c+d x)) (e \sec (c+d x))^{3/2} \left (77 i e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-308 i \cos (2 (c+d x))+17 \tan (c+d x)+77 \sin (3 (c+d x)) \sec (c+d x)-116 i\right )}{210 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} {\left (-462 i \, a^{3} e e^{\left (7 i \, d x + 7 i \, c\right )} - 574 i \, a^{3} e e^{\left (5 i \, d x + 5 i \, c\right )} - 506 i \, a^{3} e e^{\left (3 i \, d x + 3 i \, c\right )} - 154 i \, a^{3} e e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} {\rm integral}\left (\frac {11 i \, \sqrt {2} a^{3} e \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, d}, x\right )}{105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.99, size = 392, normalized size = 2.24 \[ \frac {2 a^{3} \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (231 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-231 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+231 i \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-231 i \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+140 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-231 \left (\cos ^{4}\left (d x +c \right )\right )+294 \left (\cos ^{3}\left (d x +c \right )\right )-15 i \sin \left (d x +c \right )-63 \cos \left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}{105 d \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]
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 \[ - i a^{3} \left (\int i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (- 3 \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan {\left (c + d x \right )}\right )\, dx + \int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (c + d x \right )}\, dx + \int \left (- 3 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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