Optimal. Leaf size=92 \[ \frac {2 \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d (e \cos (c+d x))^{3/2}}+\frac {4 i \cos ^2(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3515, 3500, 3771, 2639} \[ \frac {2 \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d (e \cos (c+d x))^{3/2}}+\frac {4 i \cos ^2(c+d x)}{5 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3500
Rule 3515
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{3/2} (a+i a \tan (c+d x))^2} \, dx &=\frac {\int \frac {(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx}{(e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {4 i \cos ^2(c+d x)}{5 d (e \cos (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {e^2 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a^2 (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}}\\ &=\frac {4 i \cos ^2(c+d x)}{5 d (e \cos (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\cos ^{\frac {3}{2}}(c+d x) \int \sqrt {\cos (c+d x)} \, dx}{5 a^2 (e \cos (c+d x))^{3/2}}\\ &=\frac {2 \cos ^{\frac {3}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d (e \cos (c+d x))^{3/2}}+\frac {4 i \cos ^2(c+d x)}{5 d (e \cos (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.69, size = 244, normalized size = 2.65 \[ \frac {(\sin (c+d x)+i \cos (c+d x)) \left (i \sin (2 (c+d x))+3 \cos (2 (c+d x))-2 \sqrt {\sin (c+d x)-i \cos (c+d x)+1} \sqrt {\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} F\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+2 \sqrt {\sin (c+d x)-i \cos (c+d x)+1} \sqrt {\sin (c+d x)+i \sin (2 (c+d x))-i \cos (c+d x)+\cos (2 (c+d x))} E\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3\right )}{5 a^2 d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ \frac {{\left (5 \, a^{2} d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} {\rm integral}\left (-\frac {2 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, {\left (a^{2} d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e^{2}\right )}}, x\right ) + \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{5 \, a^{2} d e^{2}} \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.72, size = 207, normalized size = 2.25 \[ -\frac {2 \left (16 i \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 e \,a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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