Optimal. Leaf size=77 \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b c^2 \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0319249, 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, 2639} \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b c^2 \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(c \csc (a+b x))^{5/2}} \, dx &=-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac{3 \int \frac{1}{\sqrt{c \csc (a+b x)}} \, dx}{5 c^2}\\ &=-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac{3 \int \sqrt{\sin (a+b x)} \, dx}{5 c^2 \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ &=-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac{6 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{5 b c^2 \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.143341, size = 60, normalized size = 0.78 \[ \frac{-2 \sin (2 (a+b x))-\frac{12 E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{\sqrt{\sin (a+b x)}}}{10 b c^2 \sqrt{c \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.252, size = 547, normalized size = 7.1 \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{1}{\left (c \csc \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 \csc \left (b x + a\right )}}{c^{3} \csc \left (b x + a\right )^{3}}, 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{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 \csc \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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