Optimal. Leaf size=77 \[ \frac{6 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d^3 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]
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Rubi [A] time = 0.0556424, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3769, 3771, 2639} \[ \frac{6 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d^3 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx &=d^4 \int \frac{1}{(d \csc (e+f x))^{5/2}} \, dx\\ &=-\frac{2 d^3 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac{1}{5} \left (3 d^2\right ) \int \frac{1}{\sqrt{d \csc (e+f x)}} \, dx\\ &=-\frac{2 d^3 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac{\left (3 d^2\right ) \int \sqrt{\sin (e+f x)} \, dx}{5 \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ &=-\frac{2 d^3 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac{6 d^2 E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right )}{5 f \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.188771, size = 62, normalized size = 0.81 \[ -\frac{2 (d \csc (e+f x))^{3/2} \left (\sin ^3(e+f x) \cos (e+f x)+3 \sin ^{\frac{3}{2}}(e+f x) E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{5 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.151, size = 545, 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 \left (d \csc \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )^{4}\,{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 ), 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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