Optimal. Leaf size=411 \[ \frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right )}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right )}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left (-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right )}{2 \sqrt{2} a^{3/2} b^{5/2} f}+\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left (\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right )}{2 \sqrt{2} a^{3/2} b^{5/2} f}-\frac{2}{a b f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.35963, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2581, 2585, 2574, 297, 1162, 617, 204, 1165, 628} \[ \frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right )}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right )}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left (-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right )}{2 \sqrt{2} a^{3/2} b^{5/2} f}+\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left (\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right )}{2 \sqrt{2} a^{3/2} b^{5/2} f}-\frac{2}{a b f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2581
Rule 2585
Rule 2574
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx &=-\frac{2}{a b f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}-\frac{\int \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)} \, dx}{a^2 b^2}\\ &=-\frac{2}{a b f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}-\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}} \, dx}{a^2 b^2}\\ &=-\frac{2}{a b f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}-\frac{\left (2 \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{a^2+b^2 x^4} \, dx,x,\frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}\right )}{a b f}\\ &=-\frac{2}{a b f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}+\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{a-b x^2}{a^2+b^2 x^4} \, dx,x,\frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}\right )}{a b^2 f}-\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{a+b x^2}{a^2+b^2 x^4} \, dx,x,\frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}\right )}{a b^2 f}\\ &=-\frac{2}{a b f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}-\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{b}}+x^2} \, dx,x,\frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}\right )}{2 a b^3 f}-\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{b}}+x^2} \, dx,x,\frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}\right )}{2 a b^3 f}-\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{b}}+2 x}{-\frac{a}{b}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{b}}-x^2} \, dx,x,\frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}\right )}{2 \sqrt{2} a^{3/2} b^{5/2} f}-\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{b}}-2 x}{-\frac{a}{b}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{b}}-x^2} \, dx,x,\frac{\sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}\right )}{2 \sqrt{2} a^{3/2} b^{5/2} f}\\ &=-\frac{\sqrt{b \cos (e+f x)} \log \left (\sqrt{a}-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)\right ) \sqrt{b \sec (e+f x)}}{2 \sqrt{2} a^{3/2} b^{5/2} f}+\frac{\sqrt{b \cos (e+f x)} \log \left (\sqrt{a}+\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)\right ) \sqrt{b \sec (e+f x)}}{2 \sqrt{2} a^{3/2} b^{5/2} f}-\frac{2}{a b f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}-\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right )}{\sqrt{2} a^{3/2} b^{5/2} f}+\frac{\left (\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right )}{\sqrt{2} a^{3/2} b^{5/2} f}\\ &=\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right ) \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right ) \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\sqrt{b \cos (e+f x)} \log \left (\sqrt{a}-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)\right ) \sqrt{b \sec (e+f x)}}{2 \sqrt{2} a^{3/2} b^{5/2} f}+\frac{\sqrt{b \cos (e+f x)} \log \left (\sqrt{a}+\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)\right ) \sqrt{b \sec (e+f x)}}{2 \sqrt{2} a^{3/2} b^{5/2} f}-\frac{2}{a b f \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.20698, size = 66, normalized size = 0.16 \[ -\frac{2 \left (\tan ^2(e+f x) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right )+3\right )}{3 a b f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.111, size = 957, normalized size = 2.3 \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 (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (a \sin \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(-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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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 (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \left (a \sin \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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