Optimal. Leaf size=46 \[ \frac {2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{d^2 f} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {16, 3771, 2641} \[ \frac {2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{d^2 f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \frac {\csc ^2(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx &=\frac {\int \sqrt {d \csc (e+f x)} \, dx}{d^2}\\ &=\frac {\left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{d^2}\\ &=\frac {2 \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)}}{d^2 f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.98 \[ -\frac {2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right ) \sqrt {d \csc (e+f x)}}{d^2 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \csc \left (f x + e\right )}}{d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{2}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 165, normalized size = 3.59 \[ -\frac {i \sqrt {2}\, \left (\cos \left (f x +e \right )+1\right )^{2} \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right )}{f \left (\frac {d}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{2}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^2\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (d \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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