Optimal. Leaf size=116 \[ \frac {4 i e^2}{7 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}+\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 116, 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 = {3500, 3769, 3771, 2641} \[ \frac {4 i e^2}{7 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}+\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3500
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx &=\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \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}\\ &=\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\int \sqrt {e \sec (c+d x)} \, dx}{7 a^2}\\ &=\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 a^2}\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d}+\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 112, normalized size = 0.97 \[ -\frac {\sec ^2(c+d x) \sqrt {e \sec (c+d x)} \left (-\sin (2 (c+d x))+2 i \cos (2 (c+d x))+2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+2 i\right )}{7 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ \frac {{\left (14 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} {\rm integral}\left (-\frac {i \, \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{7 \, a^{2} d}, x\right ) + \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (3 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{14 \, a^{2} d} \]
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 \[ \int \frac {\sqrt {e \sec \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.00, size = 180, normalized size = 1.55 \[ \frac {2 \sqrt {\frac {e}{\cos \left (d x +c \right )}}\, \left (i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right )+2 i \left (\cos ^{4}\left (d x +c \right )\right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+2 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{7 a^{2} d} \]
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 \[ \int \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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 \[ - \frac {\int \frac {\sqrt {e \sec {\left (c + d x \right )}}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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