Optimal. Leaf size=120 \[ \frac {2 i \sqrt {e \cos (c+d x)}}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{3 a^2 d \sqrt {\cos (c+d x)}}+\frac {2 i \sqrt {e \cos (c+d x)}}{9 d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.17, 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, 2639} \[ \frac {4 i \cos ^2(c+d x) \sqrt {e \cos (c+d x)}}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {2 \sin (c+d x) \cos (c+d x) \sqrt {e \cos (c+d x)}}{9 a^2 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{3 a^2 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3500
Rule 3515
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sqrt {e \cos (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx &=\left (\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2} \, dx\\ &=\frac {4 i \cos ^2(c+d x) \sqrt {e \cos (c+d x)}}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 e^2 \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{9 a^2}\\ &=\frac {2 \cos (c+d x) \sqrt {e \cos (c+d x)} \sin (c+d x)}{9 a^2 d}+\frac {4 i \cos ^2(c+d x) \sqrt {e \cos (c+d x)}}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{3 a^2}\\ &=\frac {2 \cos (c+d x) \sqrt {e \cos (c+d x)} \sin (c+d x)}{9 a^2 d}+\frac {4 i \cos ^2(c+d x) \sqrt {e \cos (c+d x)}}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{3 a^2 \sqrt {\cos (c+d x)}}\\ &=\frac {2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d \sqrt {\cos (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {e \cos (c+d x)} \sin (c+d x)}{9 a^2 d}+\frac {4 i \cos ^2(c+d x) \sqrt {e \cos (c+d x)}}{9 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 1.72, size = 420, normalized size = 3.50 \[ \frac {(\cos (d x)+i \sin (d x))^2 \sqrt {e \cos (c+d x)} \left (\frac {2 \sqrt {2} \csc (c) e^{-i d x} (\cos (2 c)+i \sin (2 c)) \left (e^{2 i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+3 e^{2 i (c+d x)}-3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} F\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3 \sqrt {1-i e^{i (c+d x)}} \sqrt {e^{i (c+d x)} \left (e^{i (c+d x)}-i\right )} E\left (\left .\sin ^{-1}\left (\sqrt {\sin (c+d x)-i \cos (c+d x)}\right )\right |-1\right )+3\right )}{9 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}-\frac {1}{9} \csc (c) \sqrt {\cos (c+d x)} (\cos (2 d x)-i \sin (2 d x)) (-4 i (-2 \sin (c+2 d x)-\sin (3 c+2 d x)+\sin (c))+7 \cos (c+2 d x)+5 \cos (3 c+2 d x))\right )}{2 d \cos ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ \frac {\sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (-9 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 15 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 18 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (-2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 4 i \, e^{\left (i \, d x + i \, c\right )} - 2 i\right )} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{3 \, {\left (a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )} + 2 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, a^{2} d e^{\left (i \, d x + i \, c\right )} + a^{2} d\right )}}, x\right )}{18 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
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 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.95, size = 277, normalized size = 2.31 \[ -\frac {2 e \left (64 i \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-64 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-160 i \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 i \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-104 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 i \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+20 i \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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 \[ - \frac {\int \frac {\sqrt {e \cos {\left (c + d x \right )}}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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