Optimal. Leaf size=111 \[ \frac{8 x}{9 f^2 \csc ^{\frac{3}{2}}(e+f x)}+\frac{16 \cos (e+f x)}{27 f^3 \sqrt{\csc (e+f x)}}-\frac{16 \sqrt{\sin (e+f x)} \sqrt{\csc (e+f x)} F\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{27 f^3}-\frac{2 x^2 \cos (e+f x)}{3 f \sqrt{\csc (e+f x)}} \]
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Rubi [A] time = 0.209395, antiderivative size = 111, normalized size of antiderivative = 1., 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 4188
Rule 4189
Rule 3769
Rule 3771
Rule 2641
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*}
Mathematica [A] time = 0.556044, 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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Maple [F] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( \csc \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{{x}^{2}}{3}\sqrt{\csc \left ( fx+e \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int - \frac{3 x^{2}}{\csc ^{\frac{3}{2}}{\left (e + f x \right )}}\, dx + \int x^{2} \sqrt{\csc{\left (e + f x \right )}}\, dx}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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