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 = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, 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 12
Rule 203
Rule 206
Rule 298
Rule 329
Rule 2565
Rule 2601
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*}
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Mathematica [A] time = 0.15, 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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fricas [B] time = 1.24, size = 419, normalized size = 3.95 \[ \left [\frac {2 \, \sqrt {-a b} \arctan \left (\frac {2 \, \sqrt {-a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\left (a b \cos \left (f x + e\right ) + a b\right )} \sin \left (f x + e\right )}\right ) - \sqrt {-a b} \log \left (-\frac {a b \cos \left (f x + e\right )^{3} - 5 \, a b \cos \left (f x + e\right )^{2} + 4 \, \sqrt {-a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 5 \, a b \cos \left (f x + e\right ) + a b}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + 3 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, a b f}, -\frac {2 \, \sqrt {a b} \arctan \left (\frac {2 \, \sqrt {a b} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\left (a b \cos \left (f x + e\right ) - a b\right )} \sin \left (f x + e\right )}\right ) - \sqrt {a b} \log \left (\frac {4 \, \sqrt {a b} {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - {\left (a b \cos \left (f x + e\right )^{2} + 6 \, a b \cos \left (f x + e\right ) + a b\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right )}\right )}{4 \, a b f}\right ] \]
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 \[ \int \frac {1}{\sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 177, normalized size = 1.67 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (\arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}}\right )+\ln \left (-\frac {2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2 \cos \left (f x +e \right )-1}{\sin \left (f x +e \right )^{2}}\right )\right )}{2 f \sqrt {a \sin \left (f x +e \right )}\, \sin \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}} \]
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 \[ \int \frac {1}{\sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {a\,\sin \left (e+f\,x\right )}\,\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \sin {\left (e + f x \right )}} \sqrt {b \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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