Optimal. Leaf size=38 \[ -\frac {2 x \cos (e+f x)}{f \sqrt {\sin (e+f x)}}+\frac {4 \sqrt {\sin (e+f x)}}{f^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {3396}
\begin {gather*} \frac {4 \sqrt {\sin (e+f x)}}{f^2}-\frac {2 x \cos (e+f x)}{f \sqrt {\sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\sin ^{\frac {3}{2}}(e+f x)}+x \sqrt {\sin (e+f x)}\right ) \, dx &=\int \frac {x}{\sin ^{\frac {3}{2}}(e+f x)} \, dx+\int x \sqrt {\sin (e+f x)} \, dx\\ &=-\frac {2 x \cos (e+f x)}{f \sqrt {\sin (e+f x)}}+\frac {4 \sqrt {\sin (e+f x)}}{f^2}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 33, normalized size = 0.87 \begin {gather*} \frac {-2 f x \cos (e+f x)+4 \sin (e+f x)}{f^2 \sqrt {\sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {x}{\sin \left (f x +e \right )^{\frac {3}{2}}}+x \left (\sqrt {\sin }\left (f x +e \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x \left (\sin ^{2}{\left (e + f x \right )} + 1\right )}{\sin ^{\frac {3}{2}}{\left (e + f x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 36, normalized size = 0.95 \begin {gather*} \frac {4\,{\sin \left (e+f\,x\right )}^2-f\,x\,\sin \left (2\,e+2\,f\,x\right )}{f^2\,{\sin \left (e+f\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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