Optimal. Leaf size=167 \[ -\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{12 a^4 b^2 f \sqrt{a \sin (e+f x)}}+\frac{1}{12 a^4 b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{1}{30 a^2 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}} \]
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Rubi [A] time = 0.233063, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2597, 2599, 2601, 2641} \[ -\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{12 a^4 b^2 f \sqrt{a \sin (e+f x)}}+\frac{1}{12 a^4 b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}+\frac{1}{30 a^2 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2597
Rule 2599
Rule 2601
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(a \sin (e+f x))^{9/2} (b \tan (e+f x))^{3/2}} \, dx &=-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}}-\frac{\int \frac{\sqrt{b \tan (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx}{10 b^2}\\ &=-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}}+\frac{1}{30 a^2 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}-\frac{\int \frac{\sqrt{b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx}{12 a^2 b^2}\\ &=-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}}+\frac{1}{30 a^2 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}+\frac{1}{12 a^4 b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{a \sin (e+f x)}} \, dx}{24 a^4 b^2}\\ &=-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}}+\frac{1}{30 a^2 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}+\frac{1}{12 a^4 b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\left (\sqrt{\cos (e+f x)} \sqrt{b \tan (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx}{24 a^4 b^2 \sqrt{a \sin (e+f x)}}\\ &=-\frac{1}{5 b f (a \sin (e+f x))^{9/2} \sqrt{b \tan (e+f x)}}+\frac{1}{30 a^2 b f (a \sin (e+f x))^{5/2} \sqrt{b \tan (e+f x)}}+\frac{1}{12 a^4 b f \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}}-\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \tan (e+f x)}}{12 a^4 b^2 f \sqrt{a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.399519, size = 106, normalized size = 0.63 \[ \frac{\sqrt [4]{\cos ^2(e+f x)} \left (-12 \csc ^4(e+f x)+2 \csc ^2(e+f x)+5\right )-5 \sin (e+f x) F\left (\left .\frac{1}{2} \sin ^{-1}(\sin (e+f x))\right |2\right )}{60 a^4 b f \sqrt [4]{\cos ^2(e+f x)} \sqrt{a \sin (e+f x)} \sqrt{b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.218, size = 487, normalized size = 2.9 \begin{align*} \text{result too large to display} \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}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \sin \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )}}{{\left (a^{5} b^{2} \cos \left (f x + e\right )^{4} - 2 \, a^{5} b^{2} \cos \left (f x + e\right )^{2} + a^{5} b^{2}\right )} \sin \left (f x + e\right ) \tan \left (f x + e\right )^{2}}, x\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}{\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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