Optimal. Leaf size=100 \[ -\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}-\frac{6 \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{6 d E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
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Rubi [A] time = 0.0584073, antiderivative size = 100, 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, 3768, 3771, 2639} \[ -\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}-\frac{6 \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{6 d E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \csc ^3(e+f x) \sqrt{d \csc (e+f x)} \, dx &=\frac{\int (d \csc (e+f x))^{7/2} \, dx}{d^3}\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}+\frac{3 \int (d \csc (e+f x))^{3/2} \, dx}{5 d}\\ &=-\frac{6 \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}-\frac{1}{5} (3 d) \int \frac{1}{\sqrt{d \csc (e+f x)}} \, dx\\ &=-\frac{6 \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}-\frac{(3 d) \int \sqrt{\sin (e+f x)} \, dx}{5 \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ &=-\frac{6 \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}-\frac{6 d 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 = 0.139827, size = 68, normalized size = 0.68 \[ -\frac{2 \sqrt{d \csc (e+f x)} \left (3 \cos (e+f x)+\cot (e+f x) \csc (e+f x)-3 \sqrt{\sin (e+f x)} E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{5 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.158, size = 1054, normalized size = 10.5 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \csc \left (f x + e\right )} \csc \left (f x + e\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 \sqrt{d \csc{\left (e + f x \right )}} \csc ^{3}{\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 \sqrt{d \csc \left (f x + e\right )} \csc \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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