Optimal. Leaf size=146 \[ -\frac{2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt{b \tan (e+f x)}}-\frac{16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt{b \tan (e+f x)}}-\frac{64 a^6 \sqrt{a \sin (e+f x)}}{585 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.20672, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2596, 2598, 2589} \[ -\frac{2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt{b \tan (e+f x)}}-\frac{16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt{b \tan (e+f x)}}-\frac{64 a^6 \sqrt{a \sin (e+f x)}}{585 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt{b \tan (e+f x)}} \]
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))^{13/2}}{(b \tan (e+f x))^{3/2}} \, dx &=\frac{2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt{b \tan (e+f x)}}+\frac{a^2 \int (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)} \, dx}{13 b^2}\\ &=-\frac{2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt{b \tan (e+f x)}}+\frac{\left (8 a^4\right ) \int (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)} \, dx}{117 b^2}\\ &=-\frac{16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt{b \tan (e+f x)}}-\frac{2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt{b \tan (e+f x)}}+\frac{\left (32 a^6\right ) \int \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)} \, dx}{585 b^2}\\ &=-\frac{64 a^6 \sqrt{a \sin (e+f x)}}{585 b f \sqrt{b \tan (e+f x)}}-\frac{16 a^4 (a \sin (e+f x))^{5/2}}{585 b f \sqrt{b \tan (e+f x)}}-\frac{2 a^2 (a \sin (e+f x))^{9/2}}{117 b f \sqrt{b \tan (e+f x)}}+\frac{2 (a \sin (e+f x))^{13/2}}{13 b f \sqrt{b \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.423783, size = 67, normalized size = 0.46 \[ \frac{a^6 \cos ^2(e+f x) (340 \cos (2 (e+f x))-45 \cos (4 (e+f x))-551) \sqrt{a \sin (e+f x)}}{2340 b f \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.151, size = 70, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 90\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-260\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+234 \right ) \cos \left ( fx+e \right ) }{585\,f \left ( \sin \left ( fx+e \right ) \right ) ^{5}} \left ( a\sin \left ( fx+e \right ) \right ) ^{{\frac{13}{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{13}{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 = 1.81169, size = 213, normalized size = 1.46 \begin{align*} -\frac{2 \,{\left (45 \, a^{6} \cos \left (f x + e\right )^{7} - 130 \, a^{6} \cos \left (f x + e\right )^{5} + 117 \, a^{6} \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 )}}}{585 \, 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{13}{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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