Optimal. Leaf size=141 \[ -\frac {a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \tan ^{-1}\left (\sqrt {\cos (e+f x)}\right )}{b^2 f \sqrt {a \sin (e+f x)}}-\frac {a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \tanh ^{-1}\left (\sqrt {\cos (e+f x)}\right )}{b^2 f \sqrt {a \sin (e+f x)}}+\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}} \]
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Rubi [A] time = 0.14, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2595, 2601, 12, 2565, 329, 212, 206, 203} \[ -\frac {a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \tan ^{-1}\left (\sqrt {\cos (e+f x)}\right )}{b^2 f \sqrt {a \sin (e+f x)}}-\frac {a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)} \tanh ^{-1}\left (\sqrt {\cos (e+f x)}\right )}{b^2 f \sqrt {a \sin (e+f x)}}+\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 206
Rule 212
Rule 329
Rule 2565
Rule 2595
Rule 2601
Rubi steps
\begin {align*} \int \frac {\sqrt {a \sin (e+f x)}}{(b \tan (e+f x))^{3/2}} \, dx &=\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}}+\frac {a^2 \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{3/2}} \, dx}{b^2}\\ &=\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}}+\frac {\left (a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \frac {\csc (e+f x)}{a \sqrt {\cos (e+f x)}} \, dx}{b^2 \sqrt {a \sin (e+f x)}}\\ &=\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}}+\frac {\left (a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \frac {\csc (e+f x)}{\sqrt {\cos (e+f x)}} \, dx}{b^2 \sqrt {a \sin (e+f x)}}\\ &=\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}}-\frac {\left (a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,\cos (e+f x)\right )}{b^2 f \sqrt {a \sin (e+f x)}}\\ &=\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}}-\frac {\left (2 a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {\cos (e+f x)}\right )}{b^2 f \sqrt {a \sin (e+f x)}}\\ &=\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}}-\frac {\left (a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\cos (e+f x)}\right )}{b^2 f \sqrt {a \sin (e+f x)}}-\frac {\left (a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\cos (e+f x)}\right )}{b^2 f \sqrt {a \sin (e+f x)}}\\ &=\frac {2 \sqrt {a \sin (e+f x)}}{b f \sqrt {b \tan (e+f x)}}-\frac {a \tan ^{-1}\left (\sqrt {\cos (e+f x)}\right ) \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}{b^2 f \sqrt {a \sin (e+f x)}}-\frac {a \tanh ^{-1}\left (\sqrt {\cos (e+f x)}\right ) \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}{b^2 f \sqrt {a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 88, normalized size = 0.62 \[ \frac {\sqrt {a \sin (e+f x)} \left (2 \sqrt [4]{\cos ^2(e+f x)}-\tan ^{-1}\left (\sqrt [4]{\cos ^2(e+f x)}\right )-\tanh ^{-1}\left (\sqrt [4]{\cos ^2(e+f x)}\right )\right )}{b f \sqrt [4]{\cos ^2(e+f x)} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.12, size = 529, normalized size = 3.75 \[ \left [\frac {2 \, b \sqrt {-\frac {a}{b}} \arctan \left (\frac {2 \, \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {-\frac {a}{b}} \cos \left (f x + e\right )}{{\left (a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + b \sqrt {-\frac {a}{b}} \log \left (-\frac {a \cos \left (f x + e\right )^{3} + 4 \, \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {-\frac {a}{b}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 5 \, a \cos \left (f x + e\right )^{2} - 5 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + 3 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 8 \, \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 )}{4 \, b^{2} f \sin \left (f x + e\right )}, \frac {2 \, b \sqrt {\frac {a}{b}} \arctan \left (\frac {2 \, \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {a}{b}} \cos \left (f x + e\right )}{{\left (a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + b \sqrt {\frac {a}{b}} \log \left (\frac {4 \, {\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 )}} \sqrt {\frac {a}{b}} - {\left (a \cos \left (f x + e\right )^{2} + 6 \, a \cos \left (f x + e\right ) + a\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 ) \sin \left (f x + e\right ) + 8 \, \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 )}{4 \, b^{2} f \sin \left (f x + e\right )}\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 \[ \int \frac {\sqrt {a \sin \left (f x + e\right )}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 237, normalized size = 1.68 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+\arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}}\right )-\ln \left (-\frac {2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2 \cos \left (f x +e \right )-1}{\sin \left (f x +e \right )^{2}}\right )+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\right ) \sqrt {a \sin \left (f x +e \right )}}{2 f \sin \left (f x +e \right ) \cos \left (f x +e \right ) \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}} \]
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 \[ \int \frac {\sqrt {a \sin \left (f x + e\right )}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {a\,\sin \left (e+f\,x\right )}}{{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {a \sin {\left (e + f x \right )}}}{\left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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