Optimal. Leaf size=62 \[ -\frac {16 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f^3}+\frac {8 x \sqrt {\sin (e+f x)}}{f^2}-\frac {2 x^2 \cos (e+f x)}{f \sqrt {\sin (e+f x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {3316, 2639} \[ \frac {8 x \sqrt {\sin (e+f x)}}{f^2}-\frac {16 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{f^3}-\frac {2 x^2 \cos (e+f x)}{f \sqrt {\sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3316
Rubi steps
\begin {align*} \int \left (\frac {x^2}{\sin ^{\frac {3}{2}}(e+f x)}+x^2 \sqrt {\sin (e+f x)}\right ) \, dx &=\int \frac {x^2}{\sin ^{\frac {3}{2}}(e+f x)} \, dx+\int x^2 \sqrt {\sin (e+f x)} \, dx\\ &=-\frac {2 x^2 \cos (e+f x)}{f \sqrt {\sin (e+f x)}}+\frac {8 x \sqrt {\sin (e+f x)}}{f^2}-\frac {8 \int \sqrt {\sin (e+f x)} \, dx}{f^2}\\ &=-\frac {16 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f^3}-\frac {2 x^2 \cos (e+f x)}{f \sqrt {\sin (e+f x)}}+\frac {8 x \sqrt {\sin (e+f x)}}{f^2}\\ \end {align*}
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Mathematica [C] time = 4.69, size = 185, normalized size = 2.98 \[ -\frac {\sec (e) \left (\left (f^2 x^2-8\right ) \cos (2 e+f x)-8 f x \cos (e) \sin (e+f x)+\left (f^2 x^2+8\right ) \cos (f x)\right )}{f^3 \sqrt {\sin (e+f x)}}+\frac {8 \sec (e) e^{-i f x} \sqrt {2-2 e^{2 i (e+f x)}} \left (3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};e^{2 i (e+f x)}\right )+e^{2 i f x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 i (e+f x)}\right )\right )}{3 f^3 \sqrt {-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int x^{2} \sqrt {\sin \left (f x + e\right )} + \frac {x^{2}}{\sin \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sin \left (f x +e \right )^{\frac {3}{2}}}+x^{2} \left (\sqrt {\sin }\left (f x +e \right )\right )\, dx \]
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 \[ \int x^{2} \sqrt {\sin \left (f x + e\right )} + \frac {x^{2}}{\sin \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int x^2\,\sqrt {\sin \left (e+f\,x\right )}+\frac {x^2}{{\sin \left (e+f\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \left (\sin ^{2}{\left (e + f x \right )} + 1\right )}{\sin ^{\frac {3}{2}}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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