Optimal. Leaf size=86 \[ -\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{a \sin (e+f x)}}-\frac{1}{b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.104479, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2597, 2601, 2641} \[ -\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{a \sin (e+f x)}}-\frac{1}{b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2597
Rule 2601
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sin (e+f x)} (b \tan (e+f x))^{3/2}} \, dx &=-\frac{1}{b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{a \sin (e+f x)}} \, dx}{2 b^2}\\ &=-\frac{1}{b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\left (\sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx}{2 b^2 \sqrt{a \sin (e+f x)}}\\ &=-\frac{1}{b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{b^2 f \sqrt{a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.184657, size = 79, normalized size = 0.92 \[ \frac{\sin (e+f x) \left (-F\left (\left .\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right |2\right )\right )-\sqrt [4]{\cos ^2(e+f x)}}{b f \sqrt [4]{\cos ^2(e+f x)} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.16, size = 185, normalized size = 2.2 \begin{align*} -{\frac{\sin \left ( fx+e \right ) }{f \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) \right ){\frac{1}{\sqrt{a\sin \left ( fx+e \right ) }}} \left ({\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \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{1}{\sqrt{a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\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*}{\rm integral}\left (\frac{\sqrt{a \sin \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )}}{a b^{2} \sin \left (f x + e\right ) \tan \left (f x + e\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\sqrt{a \sin \left (f x + e\right )} \left (b \tan \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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