Optimal. Leaf size=120 \[ \frac {2 i}{7 d \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt {e \cos (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {2 i}{7 d (a+i a \tan (c+d x))^2 \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.16, antiderivative size = 126, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3515, 3500, 3769, 3771, 2641} \[ \frac {4 i \cos ^2(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt {e \cos (c+d x)}}+\frac {2 \sin (c+d x) \cos (c+d x)}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3500
Rule 3515
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cos (c+d x)} (a+i a \tan (c+d x))^2} \, dx &=\frac {\int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx}{\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {4 i \cos ^2(c+d x)}{7 d \sqrt {e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (3 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a^2 \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {2 \cos (c+d x) \sin (c+d x)}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {4 i \cos ^2(c+d x)}{7 d \sqrt {e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\int \sqrt {e \sec (c+d x)} \, dx}{7 a^2 \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {2 \cos (c+d x) \sin (c+d x)}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {4 i \cos ^2(c+d x)}{7 d \sqrt {e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 a^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {2 \cos (c+d x) \sin (c+d x)}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {4 i \cos ^2(c+d x)}{7 d \sqrt {e \cos (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 158, normalized size = 1.32 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )-i \cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sqrt {\cos (c+d x)} \left (4 i \sin ^3\left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {3}{2} (c+d x)\right )\right )+2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (\sin \left (\frac {3}{2} (c+d x)\right )-i \cos \left (\frac {3}{2} (c+d x)\right )\right )\right )}{7 a^2 d \cos ^{\frac {3}{2}}(c+d x) (\tan (c+d x)-i)^2 \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ \frac {{\left (7 \, a^{2} d e e^{\left (3 i \, d x + 3 i \, c\right )} {\rm integral}\left (-\frac {2 i \, \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{7 \, {\left (a^{2} d e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e\right )}}, x\right ) + \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{7 \, a^{2} d e} \]
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 \[ \int \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 6.02, size = 240, normalized size = 2.00 \[ \frac {\frac {64 i \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {64 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {128 i \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {96 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {96 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {32 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{7}+\frac {12 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{7}+\frac {4 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{7}}{a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \frac {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{\sqrt {e \cos {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )} - 2 i \sqrt {e \cos {\left (c + d x \right )}} \tan {\left (c + d x \right )} - \sqrt {e \cos {\left (c + d x \right )}}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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