Optimal. Leaf size=37 \[ \frac{2 E\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{b}-\frac{2 \cos (b x)}{b \sqrt{\sin (b x)}} \]
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Rubi [A] time = 0.0131666, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2639} \[ \frac{2 E\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{b}-\frac{2 \cos (b x)}{b \sqrt{\sin (b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{\frac{3}{2}}(b x)} \, dx &=-\frac{2 \cos (b x)}{b \sqrt{\sin (b x)}}-\int \sqrt{\sin (b x)} \, dx\\ &=\frac{2 E\left (\left .\frac{\pi }{4}-\frac{b x}{2}\right |2\right )}{b}-\frac{2 \cos (b x)}{b \sqrt{\sin (b x)}}\\ \end{align*}
Mathematica [A] time = 0.0533583, size = 32, normalized size = 0.86 \[ \frac{2 \left (E\left (\left .\frac{1}{4} (\pi -2 b x)\right |2\right )-\frac{\cos (b x)}{\sqrt{\sin (b x)}}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 110, normalized size = 3. \begin{align*}{\frac{1}{b\cos \left ( bx \right ) } \left ( 2\,\sqrt{\sin \left ( bx \right ) +1}\sqrt{-2\,\sin \left ( bx \right ) +2}\sqrt{-\sin \left ( bx \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( bx \right ) +1},1/2\,\sqrt{2} \right ) -\sqrt{\sin \left ( bx \right ) +1}\sqrt{-2\,\sin \left ( bx \right ) +2}\sqrt{-\sin \left ( bx \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx \right ) +1},{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \cos \left ( bx \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{\sin \left ( bx \right ) }}}} \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}{\sin \left (b x\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{\sin \left (b x\right )}}{\cos \left (b x\right )^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin ^{\frac{3}{2}}{\left (b x \right )}}\, dx \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}{\sin \left (b x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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