Optimal. Leaf size=70 \[ -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d f}-\frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3768, 3771, 2639} \[ -\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d f}-\frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2639
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \frac {\csc ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx &=\frac {\int (d \csc (e+f x))^{3/2} \, dx}{d^2}\\ &=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d f}-\int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx\\ &=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d f}-\frac {\int \sqrt {\sin (e+f x)} \, dx}{\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}}\\ &=-\frac {2 \cos (e+f x) \sqrt {d \csc (e+f x)}}{d f}-\frac {2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 52, normalized size = 0.74 \[ \frac {\frac {2 E\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )}{\sqrt {\sin (e+f x)}}-2 \cot (e+f x)}{f \sqrt {d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right )}{d}, x\right ) \]
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 \[ \int \frac {\csc \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 522, normalized size = 7.46 \[ \frac {\left (2 \cos \left (f x +e \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 )}}\, \EllipticE \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 )-\cos \left (f x +e \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 )}}\, \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 )+2 \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 )}}\, \EllipticE \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 )}}\, \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 {2}\right ) \sqrt {2}}{f \sqrt {\frac {d}{\sin \left (f x +e \right )}}\, \sin \left (f x +e \right )} \]
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 \[ \int \frac {\csc \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^2\,\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\csc ^{2}{\left (e + f x \right )}}{\sqrt {d \csc {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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