Optimal. Leaf size=96 \[ \frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3486, 3769, 3771, 2641} \[ \frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3486
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{3/2}} \, dx &=-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}+a \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx\\ &=-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}+\frac {a \int \sqrt {e \sec (c+d x)} \, dx}{3 e^2}\\ &=-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}+\frac {\left (a \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 e^2}\\ &=-\frac {2 i a}{3 d (e \sec (c+d x))^{3/2}}+\frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 d e^2}+\frac {2 a \sin (c+d x)}{3 d e \sqrt {e \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 62, normalized size = 0.65 \[ \frac {2 a \left (\sin (c+d x)-i \cos (c+d x)+\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)}}\right )}{3 d e \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ \frac {3 \, d e^{2} {\rm integral}\left (-\frac {i \, \sqrt {2} a \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{3 \, d e^{2}}, x\right ) + \sqrt {2} {\left (-i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{3 \, d e^{2}} \]
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 \[ \int \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.78, size = 170, normalized size = 1.77 \[ \frac {2 a \left (i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-i \left (\cos ^{2}\left (d x +c \right )\right )+\cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{2} \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ i a \left (\int \left (- \frac {i}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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