Optimal. Leaf size=72 \[ \frac {6 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{5 f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 72, 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, 3769, 3771, 2639} \[ \frac {6 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{5 f \sqrt {\sin (e+f x)} \sqrt {d \csc (e+f x)}}-\frac {2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sin ^2(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx &=d^2 \int \frac {1}{(d \csc (e+f x))^{5/2}} \, dx\\ &=-\frac {2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {3}{5} \int \frac {1}{\sqrt {d \csc (e+f x)}} \, dx\\ &=-\frac {2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {3 \int \sqrt {\sin (e+f x)} \, dx}{5 \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}}\\ &=-\frac {2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}+\frac {6 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*}
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Mathematica [A] time = 0.18, size = 57, normalized size = 0.79 \[ \frac {-2 \sin (2 (e+f x))-\frac {12 E\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )}{\sqrt {\sin (e+f x)}}}{10 f \sqrt {d \csc (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {d \csc \left (f x + e\right )}}{d \csc \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\sin \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 547, normalized size = 7.60 \[ -\frac {\left (6 \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 )-3 \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 )+6 \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 )-3 \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 )-\left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}+4 \cos \left (f x +e \right ) \sqrt {2}-3 \sqrt {2}\right ) \sqrt {2}}{5 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 \[ \int \frac {\sin \left (f x + e\right )^{2}}{\sqrt {d \csc \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\sin \left (e+f\,x\right )}^2}{\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\sin ^{2}{\left (e + f x \right )}}{\sqrt {d \csc {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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