Optimal. Leaf size=72 \[ \frac{2 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{3 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.0388401, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {16, 3768, 3771, 2641} \[ \frac{2 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{3 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \csc (e+f x) (d \csc (e+f x))^{3/2} \, dx &=\frac{\int (d \csc (e+f x))^{5/2} \, dx}{d}\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}+\frac{1}{3} d \int \sqrt{d \csc (e+f x)} \, dx\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}+\frac{1}{3} \left (d \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\sin (e+f x)}} \, dx\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}+\frac{2 d \sqrt{d \csc (e+f x)} F\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right ) \sqrt{\sin (e+f x)}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.17595, size = 58, normalized size = 0.81 \[ -\frac{(d \csc (e+f x))^{5/2} \left (\sin (2 (e+f x))+2 \sin ^{\frac{5}{2}}(e+f x) F\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{3 d f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.118, size = 319, normalized size = 4.4 \begin{align*}{\frac{\sqrt{2} \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{3\,f \left ( \sin \left ( fx+e \right ) \right ) ^{4}} \left ( i\cos \left ( fx+e \right ) \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ){\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +i\sqrt{{\frac{-i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}\sqrt{-{\frac{i\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}},{\frac{\sqrt{2}}{2}} \right ) -\sqrt{2}\cos \left ( fx+e \right ) \right ) \left ({\frac{d}{\sin \left ( fx+e \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 \left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}} \csc \left (f x + e\right )\,{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 (\sqrt{d \csc \left (f x + e\right )} d \csc \left (f x + e\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 \left (d \csc{\left (e + f x \right )}\right )^{\frac{3}{2}} \csc{\left (e + f 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 \left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}} \csc \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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