Optimal. Leaf size=106 \[ \frac{\sqrt{a \sin (e+f x)} \tan ^{-1}\left (\sqrt{\cos (e+f x)}\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\sqrt{a \sin (e+f x)} \tanh ^{-1}\left (\sqrt{\cos (e+f x)}\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.0829008, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2601, 12, 2565, 329, 298, 203, 206} \[ \frac{\sqrt{a \sin (e+f x)} \tan ^{-1}\left (\sqrt{\cos (e+f x)}\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\sqrt{a \sin (e+f x)} \tanh ^{-1}\left (\sqrt{\cos (e+f x)}\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2601
Rule 12
Rule 2565
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}} \, dx &=\frac{\sqrt{a \sin (e+f x)} \int \frac{\sqrt{\cos (e+f x)} \csc (e+f x)}{a} \, dx}{\sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=\frac{\sqrt{a \sin (e+f x)} \int \sqrt{\cos (e+f x)} \csc (e+f x) \, dx}{a \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=-\frac{\sqrt{a \sin (e+f x)} \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-x^2} \, dx,x,\cos (e+f x)\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=-\frac{\left (2 \sqrt{a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,\sqrt{\cos (e+f x)}\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=-\frac{\sqrt{a \sin (e+f x)} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\cos (e+f x)}\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{\sqrt{a \sin (e+f x)} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\cos (e+f x)}\right )}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ &=\frac{\tan ^{-1}\left (\sqrt{\cos (e+f x)}\right ) \sqrt{a \sin (e+f x)}}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\tanh ^{-1}\left (\sqrt{\cos (e+f x)}\right ) \sqrt{a \sin (e+f x)}}{a f \sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.144943, size = 80, normalized size = 0.75 \[ \frac{\sin (2 (e+f x)) \left (\tan ^{-1}\left (\sqrt [4]{\cos ^2(e+f x)}\right )-\tanh ^{-1}\left (\sqrt [4]{\cos ^2(e+f x)}\right )\right )}{2 f \cos ^2(e+f x)^{3/4} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 177, normalized size = 1.7 \begin{align*} -{\frac{\cos \left ( fx+e \right ) -1}{2\,f\sin \left ( fx+e \right ) } \left ( \ln \left ( -{\frac{1}{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \left ( 2\,\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}} \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\,\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}}+2\,\cos \left ( fx+e \right ) -1 \right ) } \right ) +\arctan \left ({\frac{1}{2}{\frac{1}{\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}}}}} \right ) \right ){\frac{1}{\sqrt{a\sin \left ( fx+e \right ) }}}{\frac{1}{\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{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}{\sqrt{a \sin \left (f x + e\right )} \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 [B] time = 3.72479, size = 1072, normalized size = 10.11 \begin{align*} \left [\frac{2 \, \sqrt{-a b} \arctan \left (\frac{2 \, \sqrt{-a b} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\left (a b \cos \left (f x + e\right ) + a b\right )} \sin \left (f x + e\right )}\right ) - \sqrt{-a b} \log \left (-\frac{a b \cos \left (f x + e\right )^{3} - 5 \, a b \cos \left (f x + e\right )^{2} + 4 \, \sqrt{-a b} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 5 \, a b \cos \left (f x + e\right ) + a b}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + 3 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, a b f}, -\frac{2 \, \sqrt{a b} \arctan \left (\frac{2 \, \sqrt{a b} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\left (a b \cos \left (f x + e\right ) - a b\right )} \sin \left (f x + e\right )}\right ) - \sqrt{a b} \log \left (\frac{4 \, \sqrt{a b}{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} -{\left (a b \cos \left (f x + e\right )^{2} + 6 \, a b \cos \left (f x + e\right ) + a b\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right )}\right )}{4 \, a b f}\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 )} \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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