Optimal. Leaf size=111 \[ \frac {16 \cos (e+f x)}{27 f^3 \sqrt {\csc (e+f x)}}-\frac {16 \sqrt {\sin (e+f x)} \sqrt {\csc (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{27 f^3}+\frac {8 x}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x^2 \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \]
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Rubi [A] time = 0.21, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4188, 4189, 3769, 3771, 2641} \[ \frac {8 x}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}+\frac {16 \cos (e+f x)}{27 f^3 \sqrt {\csc (e+f x)}}-\frac {16 \sqrt {\sin (e+f x)} \sqrt {\csc (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right )}{27 f^3}-\frac {2 x^2 \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4188
Rule 4189
Rubi steps
\begin {align*} \int \left (\frac {x^2}{\csc ^{\frac {3}{2}}(e+f x)}-\frac {1}{3} x^2 \sqrt {\csc (e+f x)}\right ) \, dx &=-\left (\frac {1}{3} \int x^2 \sqrt {\csc (e+f x)} \, dx\right )+\int \frac {x^2}{\csc ^{\frac {3}{2}}(e+f x)} \, dx\\ &=\frac {8 x}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}-\frac {2 x^2 \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}+\frac {1}{3} \int x^2 \sqrt {\csc (e+f x)} \, dx-\frac {8 \int \frac {1}{\csc ^{\frac {3}{2}}(e+f x)} \, dx}{9 f^2}-\frac {1}{3} \left (\sqrt {\csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {x^2}{\sqrt {\sin (e+f x)}} \, dx\\ &=\frac {8 x}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}+\frac {16 \cos (e+f x)}{27 f^3 \sqrt {\csc (e+f x)}}-\frac {2 x^2 \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}-\frac {8 \int \sqrt {\csc (e+f x)} \, dx}{27 f^2}\\ &=\frac {8 x}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}+\frac {16 \cos (e+f x)}{27 f^3 \sqrt {\csc (e+f x)}}-\frac {2 x^2 \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}-\frac {\left (8 \sqrt {\csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{27 f^2}\\ &=\frac {8 x}{9 f^2 \csc ^{\frac {3}{2}}(e+f x)}+\frac {16 \cos (e+f x)}{27 f^3 \sqrt {\csc (e+f x)}}-\frac {2 x^2 \cos (e+f x)}{3 f \sqrt {\csc (e+f x)}}-\frac {16 \sqrt {\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)}}{27 f^3}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 87, normalized size = 0.78 \[ -\frac {\sqrt {\csc (e+f x)} \left (9 f^2 x^2 \sin (2 (e+f x))-8 \sin (2 (e+f x))+12 f x \cos (2 (e+f x))-16 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )-12 f x\right )}{27 f^3} \]
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 \[ \int -\frac {1}{3} \, x^{2} \sqrt {\csc \left (f x + e\right )} + \frac {x^{2}}{\csc \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\csc \left (f x +e \right )^{\frac {3}{2}}}-\frac {x^{2} \left (\sqrt {\csc }\left (f x +e \right )\right )}{3}\, dx \]
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 \[ \int -\frac {1}{3} \, x^{2} \sqrt {\csc \left (f x + e\right )} + \frac {x^{2}}{\csc \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^{3/2}}-\frac {x^2\,\sqrt {\frac {1}{\sin \left (e+f\,x\right )}}}{3} \,d x \]
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 \[ - \frac {\int \left (- \frac {3 x^{2}}{\csc ^{\frac {3}{2}}{\left (e + f x \right )}}\right )\, dx + \int x^{2} \sqrt {\csc {\left (e + f x \right )}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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