Optimal. Leaf size=161 \[ \frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^2 \sqrt{c^2 x^2-1} \left (20 c^2 d+9 e\right )}{120 c^3 \sqrt{c^2 x^2}}+\frac{b x \left (20 c^2 d+9 e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{120 c^4 \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.107796, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {14, 5239, 12, 459, 321, 217, 206} \[ \frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^2 \sqrt{c^2 x^2-1} \left (20 c^2 d+9 e\right )}{120 c^3 \sqrt{c^2 x^2}}+\frac{b x \left (20 c^2 d+9 e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{120 c^4 \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5239
Rule 12
Rule 459
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^2 \left (5 d+3 e x^2\right )}{15 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^2 \left (5 d+3 e x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{15 \sqrt{c^2 x^2}}\\ &=\frac{b e x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+\frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (b c \left (-20 d-\frac{9 e}{c^2}\right ) x\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{60 \sqrt{c^2 x^2}}\\ &=\frac{b \left (20 c^2 d+9 e\right ) x^2 \sqrt{-1+c^2 x^2}}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+\frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (b \left (-20 d-\frac{9 e}{c^2}\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{120 c \sqrt{c^2 x^2}}\\ &=\frac{b \left (20 c^2 d+9 e\right ) x^2 \sqrt{-1+c^2 x^2}}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+\frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac{\left (b \left (-20 d-\frac{9 e}{c^2}\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{120 c \sqrt{c^2 x^2}}\\ &=\frac{b \left (20 c^2 d+9 e\right ) x^2 \sqrt{-1+c^2 x^2}}{120 c^3 \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{-1+c^2 x^2}}{20 c \sqrt{c^2 x^2}}+\frac{1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac{b \left (20 c^2 d+9 e\right ) x \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{120 c^4 \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.166245, size = 121, normalized size = 0.75 \[ \frac{c^2 x^2 \left (8 a c^3 x \left (5 d+3 e x^2\right )+b \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 \left (20 d+6 e x^2\right )+9 e\right )\right )+b \left (20 c^2 d+9 e\right ) \log \left (x \left (\sqrt{1-\frac{1}{c^2 x^2}}+1\right )\right )+8 b c^5 x^3 \csc ^{-1}(c x) \left (5 d+3 e x^2\right )}{120 c^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.184, size = 282, normalized size = 1.8 \begin{align*}{\frac{ae{x}^{5}}{5}}+{\frac{ad{x}^{3}}{3}}+{\frac{b{\rm arccsc} \left (cx\right )e{x}^{5}}{5}}+{\frac{b{\rm arccsc} \left (cx\right )d{x}^{3}}{3}}+{\frac{b{x}^{4}e}{20\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{be{x}^{2}}{40\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{bd{x}^{2}}{6\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{bd}{6\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{bd}{6\,{c}^{4}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{3\,be}{40\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{3\,be}{40\,{c}^{6}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986601, size = 313, normalized size = 1.94 \begin{align*} \frac{1}{5} \, a e x^{5} + \frac{1}{3} \, a d x^{3} + \frac{1}{12} \,{\left (4 \, x^{3} \operatorname{arccsc}\left (c x\right ) + \frac{\frac{2 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac{\log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d + \frac{1}{80} \,{\left (16 \, x^{5} \operatorname{arccsc}\left (c x\right ) - \frac{\frac{2 \,{\left (3 \,{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.56358, size = 398, normalized size = 2.47 \begin{align*} \frac{24 \, a c^{5} e x^{5} + 40 \, a c^{5} d x^{3} + 8 \,{\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3} - 5 \, b c^{5} d - 3 \, b c^{5} e\right )} \operatorname{arccsc}\left (c x\right ) - 16 \,{\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (20 \, b c^{2} d + 9 \, b e\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (6 \, b c^{3} e x^{3} +{\left (20 \, b c^{3} d + 9 \, b c e\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{120 \, c^{5}} \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*} \int x^{2} \left (a + b \operatorname{acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \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{\left (e x^{2} + d\right )}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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