Optimal. Leaf size=87 \[ -\frac{b \sqrt{a \sin (e+f x)}}{a^2 f (b \tan (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{a^2 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.105374, antiderivative size = 87, 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 = {2599, 2601, 2639} \[ -\frac{b \sqrt{a \sin (e+f x)}}{a^2 f (b \tan (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{a^2 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2599
Rule 2601
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(a \sin (e+f x))^{3/2} \sqrt{b \tan (e+f x)}} \, dx &=-\frac{b \sqrt{a \sin (e+f x)}}{a^2 f (b \tan (e+f x))^{3/2}}-\frac{\int \frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \tan (e+f x)}} \, dx}{2 a^2}\\ &=-\frac{b \sqrt{a \sin (e+f x)}}{a^2 f (b \tan (e+f x))^{3/2}}-\frac{\sqrt{a \sin (e+f x)} \int \sqrt{\cos (e+f x)} \, dx}{2 a^2 \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=-\frac{b \sqrt{a \sin (e+f x)}}{a^2 f (b \tan (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{a \sin (e+f x)}}{a^2 f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.337826, size = 89, normalized size = 1.02 \[ -\frac{b \sqrt{a \sin (e+f x)} \left (\sin ^2(e+f x) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\sin ^2(e+f x)\right )+2 \cos ^2(e+f x)^{3/4}\right )}{2 a^2 f \cos ^2(e+f x)^{3/4} (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.171, size = 315, normalized size = 3.6 \begin{align*} -{\frac{\sin \left ( fx+e \right ) }{f\cos \left ( fx+e \right ) } \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\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( fx+e \right ) -1 \right ) }{\sin \left ( fx+e \right ) }},i \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 ) -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 EllipticE} \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 ) \left ( a\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \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}{\left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{b \tan \left (f x + e\right )}}\,{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 )}}{{\left (a^{2} b \cos \left (f x + e\right )^{2} - a^{2} b\right )} \tan \left (f x + e\right )}, 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}{\left (a \sin \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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