Optimal. Leaf size=77 \[ \frac{2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right ) \sqrt{c \csc (a+b x)}}{3 b c^2}-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}} \]
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Rubi [A] time = 0.032662, 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 = {3769, 3771, 2641} \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \csc (a+b x)}}{3 b c^2}-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(c \csc (a+b x))^{3/2}} \, dx &=-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}}+\frac{\int \sqrt{c \csc (a+b x)} \, dx}{3 c^2}\\ &=-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}}+\frac{\left (\sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{3 c^2}\\ &=-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}}+\frac{2 \sqrt{c \csc (a+b x)} 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}\\ \end{align*}
Mathematica [A] time = 0.0823655, size = 63, normalized size = 0.82 \[ -\frac{\csc ^2(a+b x) \left (2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+\sin (2 (a+b x))\right )}{3 b (c \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.215, size = 189, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2}}{3\,b \left ( -1+\cos \left ( bx+a \right ) \right ) \sin \left ( bx+a \right ) } \left ( i\sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}\sin \left ( bx+a \right ){\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{2}-\sqrt{2}\cos \left ( bx+a \right ) \right ) \left ({\frac{c}{\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}{\left (c \csc \left (b x + a\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c \csc \left (b x + a\right )}}{c^{2} \csc \left (b x + a\right )^{2}}, 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 \csc{\left (a + b x \right )}\right )^{\frac{3}{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 \csc \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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