Optimal. Leaf size=109 \[ -\frac{2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt{b \tan (e+f x)}}-\frac{8 a^4 \sqrt{a \sin (e+f x)}}{45 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.150827, antiderivative size = 109, 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 = {2596, 2598, 2589} \[ -\frac{2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt{b \tan (e+f x)}}-\frac{8 a^4 \sqrt{a \sin (e+f x)}}{45 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2596
Rule 2598
Rule 2589
Rubi steps
\begin{align*} \int \frac{(a \sin (e+f x))^{9/2}}{(b \tan (e+f x))^{3/2}} \, dx &=\frac{2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt{b \tan (e+f x)}}+\frac{a^2 \int (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)} \, dx}{9 b^2}\\ &=-\frac{2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt{b \tan (e+f x)}}+\frac{\left (4 a^4\right ) \int \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} \, dx}{45 b^2}\\ &=-\frac{8 a^4 \sqrt{a \sin (e+f x)}}{45 b f \sqrt{b \tan (e+f x)}}-\frac{2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt{b \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.212795, size = 57, normalized size = 0.52 \[ \frac{a^4 \cos ^2(e+f x) (5 \cos (2 (e+f x))-13) \sqrt{a \sin (e+f x)}}{45 b f \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 60, normalized size = 0.6 \begin{align*}{\frac{ \left ( 10\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-18 \right ) \cos \left ( fx+e \right ) }{45\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{9}{2}}} \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{\left (a \sin \left (f x + e\right )\right )^{\frac{9}{2}}}{\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 [A] time = 1.70836, size = 173, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (5 \, a^{4} \cos \left (f x + e\right )^{5} - 9 \, a^{4} \cos \left (f x + e\right )^{3}\right )} \sqrt{a \sin \left (f x + e\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{45 \, b^{2} f \sin \left (f x + e\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{\left (a \sin \left (f x + e\right )\right )^{\frac{9}{2}}}{\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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