Optimal. Leaf size=77 \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b c^2 \sqrt{c \sin (a+b x)}}-\frac{2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0325813, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2642, 2641} \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b c^2 \sqrt{c \sin (a+b x)}}-\frac{2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(c \sin (a+b x))^{5/2}} \, dx &=-\frac{2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}+\frac{\int \frac{1}{\sqrt{c \sin (a+b x)}} \, dx}{3 c^2}\\ &=-\frac{2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}+\frac{\sqrt{\sin (a+b x)} \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{3 c^2 \sqrt{c \sin (a+b x)}}\\ &=-\frac{2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}+\frac{2 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{3 b c^2 \sqrt{c \sin (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0752905, size = 55, normalized size = 0.71 \[ -\frac{2 \left (\cos (a+b x)+\sin ^{\frac{3}{2}}(a+b x) F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{3 b c (c \sin (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 105, normalized size = 1.4 \begin{align*} -{\frac{1}{3\,{c}^{2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}\cos \left ( bx+a \right ) b} \left ( \sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2} \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}{\it EllipticF} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}+2\,\sin \left ( bx+a \right ) \right ){\frac{1}{\sqrt{c\sin \left ( bx+a \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}{\left (c \sin \left (b x + a\right )\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{\sqrt{c \sin \left (b x + a\right )}}{{\left (c^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \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}{\left (c \sin{\left (a + b x \right )}\right )^{\frac{5}{2}}}\, 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}{\left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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