Optimal. Leaf size=116 \[ \frac {4 i e^2}{9 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{5/2}}+\frac {2 e \sin (c+d x)}{9 a^2 d (e \sec (c+d x))^{3/2}}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \]
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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, 2639} \[ \frac {4 i e^2}{9 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{5/2}}+\frac {2 e \sin (c+d x)}{9 a^2 d (e \sec (c+d x))^{3/2}}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3500
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2} \, dx &=\frac {4 i e^2}{9 d (e \sec (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (5 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{9 a^2}\\ &=\frac {2 e \sin (c+d x)}{9 a^2 d (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{9 d (e \sec (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{3 a^2}\\ &=\frac {2 e \sin (c+d x)}{9 a^2 d (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{9 d (e \sec (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\int \sqrt {\cos (c+d x)} \, dx}{3 a^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e \sin (c+d x)}{9 a^2 d (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{9 d (e \sec (c+d x))^{5/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 1.45, size = 123, normalized size = 1.06 \[ \frac {(\sin (2 (c+d x))+i \cos (2 (c+d x))) \left (2 (7 i \sin (2 (c+d x))+8 \cos (2 (c+d x))+2)-\frac {8 e^{4 i (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}\right )}{18 a^2 d \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-9 i \, e^{\left (7 i \, d x + 7 i \, c\right )} - 15 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 5 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 19 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 36 \, {\left (a^{2} d e e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e e^{\left (5 i \, d x + 5 i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2 i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{3 \, {\left (a^{2} d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a^{2} d e e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d e e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{36 \, {\left (a^{2} d e e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e e^{\left (5 i \, d x + 5 i \, c\right )}\right )}} \]
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 \[ \int \frac {1}{\sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.42, size = 366, normalized size = 3.16 \[ \frac {2 \left (2 i \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-3 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-2 \left (\cos ^{6}\left (d x +c \right )\right )+3 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-3 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+\cos ^{4}\left (d x +c \right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+3 \cos \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \sqrt {\frac {e}{\cos \left (d x +c \right )}}}{9 a^{2} d \sin \left (d x +c \right )^{5} e} \]
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 \[ \int \frac {1}{\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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 \[ - \frac {\int \frac {1}{\sqrt {e \sec {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )} - 2 i \sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )} - \sqrt {e \sec {\left (c + d x \right )}}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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