Optimal. Leaf size=47 \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b}-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(a+b x)} \]
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Rubi [A] time = 0.0163912, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2636, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b}-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{\frac{5}{2}}(a+b x)} \, dx &=-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(a+b x)}+\frac{1}{3} \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{3 b}-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.103562, size = 43, normalized size = 0.91 \[ \frac{2 \left (F\left (\left .\frac{1}{4} (2 a+2 b x-\pi )\right |2\right )-\frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(a+b x)}\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 88, normalized size = 1.9 \begin{align*}{\frac{1}{3\,b\cos \left ( bx+a \right ) } \left ( \sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \sin \left ( bx+a \right ) -2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \left ( \sin \left ( bx+a \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{1}{\sin \left (b x + a\right )^{\frac{5}{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{1}{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt{\sin \left (b x + a\right )}}, 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{5}{2}}{\left (a + 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 + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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