Optimal. Leaf size=363 \[ -\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}+\frac{\sqrt{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}+1\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \log \left (\sqrt{b} \cot (e+f x)-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{\sqrt{b} \log \left (\sqrt{b} \cot (e+f x)+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.267997, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2583, 2585, 2575, 297, 1162, 617, 204, 1165, 628} \[ -\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}+\frac{\sqrt{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}+1\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \log \left (\sqrt{b} \cot (e+f x)-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{\sqrt{b} \log \left (\sqrt{b} \cot (e+f x)+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2583
Rule 2585
Rule 2575
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sin ^{\frac{3}{2}}(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx &=-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}+\frac{1}{4} \int \frac{1}{\sqrt{b \sec (e+f x)} \sqrt{\sin (e+f x)}} \, dx\\ &=-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}+\frac{\int \frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}} \, dx}{4 \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{b^2+x^4} \, dx,x,\frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{2 f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}+\frac{b \operatorname{Subst}\left (\int \frac{b-x^2}{b^2+x^4} \, dx,x,\frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{4 f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \operatorname{Subst}\left (\int \frac{b+x^2}{b^2+x^4} \, dx,x,\frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{4 f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}+2 x}{-b-\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}-2 x}{-b+\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{b-\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{b+\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\frac{\sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{\sqrt{b} \log \left (\sqrt{b}+\sqrt{b} \cot (e+f x)-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{\sqrt{b} \log \left (\sqrt{b}+\sqrt{b} \cot (e+f x)+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=\frac{\sqrt{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right )}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \log \left (\sqrt{b}+\sqrt{b} \cot (e+f x)-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{\sqrt{b} \log \left (\sqrt{b}+\sqrt{b} \cot (e+f x)+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}\right )}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.54861, size = 218, normalized size = 0.6 \[ -\frac{\sqrt{\sin (e+f x)} \sqrt{b \sec (e+f x)} \left (2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{\tan ^2(e+f x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt [4]{\tan ^2(e+f x)}+1\right )+4 \sqrt [4]{\tan ^2(e+f x)}+\sqrt{2} \log \left (\sqrt{\tan ^2(e+f x)}-\sqrt{2} \sqrt [4]{\tan ^2(e+f x)}+1\right )-\sqrt{2} \log \left (\sqrt{\tan ^2(e+f x)}+\sqrt{2} \sqrt [4]{\tan ^2(e+f x)}+1\right )+4 \cos (2 (e+f x)) \sqrt [4]{\tan ^2(e+f x)}\right )}{16 b f \sqrt [4]{\tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.129, size = 648, normalized size = 1.8 \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{\sin \left (f x + e\right )^{\frac{3}{2}}}{\sqrt{b \sec \left (f x + e\right )}}\,{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{\sin \left (f x + e\right )^{\frac{3}{2}}}{\sqrt{b \sec \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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