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 = 0.0680948, antiderivative size = 96, normalized size of antiderivative = 1., 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 3486
Rule 3769
Rule 3771
Rule 2641
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*}
Mathematica [A] time = 0.394509, 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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Maple [A] time = 0.199, size = 170, normalized size = 1.8 \begin{align*}{\frac{2\,a}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}} \left ( i\cos \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +i\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) -i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{1}{\left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{i \tan{\left (c + d x \right )}}{\left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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