Optimal. Leaf size=103 \[ -\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{21 f} \]
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Rubi [A] time = 0.0758318, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3769, 3771, 2641} \[ -\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 d \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{d \csc (e+f x)}}{21 f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx &=d^5 \int \frac{1}{(d \csc (e+f x))^{7/2}} \, dx\\ &=-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}+\frac{1}{7} \left (5 d^3\right ) \int \frac{1}{(d \csc (e+f x))^{3/2}} \, dx\\ &=-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{1}{21} (5 d) \int \sqrt{d \csc (e+f x)} \, dx\\ &=-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{1}{21} \left (5 d \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}\right ) \int \frac{1}{\sqrt{\sin (e+f x)}} \, dx\\ &=-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 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)}}{21 f}\\ \end{align*}
Mathematica [A] time = 0.123485, size = 68, normalized size = 0.66 \[ -\frac{d \sqrt{d \csc (e+f x)} \left (26 \sin (2 (e+f x))-3 \sin (4 (e+f x))+40 \sqrt{\sin (e+f x)} F\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{84 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.113, size = 216, normalized size = 2.1 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{21\,f \left ( -1+\cos \left ( fx+e \right ) \right ) } \left ( 5\,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 ) }}}{\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{{\frac{i\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -i}{\sin \left ( fx+e \right ) }}}-3\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{4}+3\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{3}+8\,\sqrt{2} \left ( \cos \left ( fx+e \right ) \right ) ^{2}-8\,\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}} \sin \left (f x + e\right )^{5}\,{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 ({\left (d \cos \left (f x + e\right )^{4} - 2 \, d \cos \left (f x + e\right )^{2} + d\right )} \sqrt{d \csc \left (f x + e\right )} \csc \left (f x + e\right ) \sin \left (f x + e\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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