Optimal. Leaf size=146 \[ \frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}-\frac{b c d^2 x \tan ^{-1}\left (\sqrt{-c^2 x^2-1}\right )}{4 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (2 c^2 d-e\right )}{4 c^3 \sqrt{-c^2 x^2}}-\frac{b e x \left (-c^2 x^2-1\right )^{3/2}}{12 c^3 \sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.105749, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6300, 446, 88, 63, 205} \[ \frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}-\frac{b c d^2 x \tan ^{-1}\left (\sqrt{-c^2 x^2-1}\right )}{4 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (2 c^2 d-e\right )}{4 c^3 \sqrt{-c^2 x^2}}-\frac{b e x \left (-c^2 x^2-1\right )^{3/2}}{12 c^3 \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6300
Rule 446
Rule 88
Rule 63
Rule 205
Rubi steps
\begin{align*} \int x \left (d+e x^2\right ) \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^2}{x \sqrt{-1-c^2 x^2}} \, dx}{4 e \sqrt{-c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^2}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt{-c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}-\frac{(b c x) \operatorname{Subst}\left (\int \left (-\frac{e \left (-2 c^2 d+e\right )}{c^2 \sqrt{-1-c^2 x}}+\frac{d^2}{x \sqrt{-1-c^2 x}}-\frac{e^2 \sqrt{-1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{8 e \sqrt{-c^2 x^2}}\\ &=\frac{b \left (2 c^2 d-e\right ) x \sqrt{-1-c^2 x^2}}{4 c^3 \sqrt{-c^2 x^2}}-\frac{b e x \left (-1-c^2 x^2\right )^{3/2}}{12 c^3 \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}-\frac{\left (b c d^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt{-c^2 x^2}}\\ &=\frac{b \left (2 c^2 d-e\right ) x \sqrt{-1-c^2 x^2}}{4 c^3 \sqrt{-c^2 x^2}}-\frac{b e x \left (-1-c^2 x^2\right )^{3/2}}{12 c^3 \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}+\frac{\left (b d^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{-1-c^2 x^2}\right )}{4 c e \sqrt{-c^2 x^2}}\\ &=\frac{b \left (2 c^2 d-e\right ) x \sqrt{-1-c^2 x^2}}{4 c^3 \sqrt{-c^2 x^2}}-\frac{b e x \left (-1-c^2 x^2\right )^{3/2}}{12 c^3 \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{4 e}-\frac{b c d^2 x \tan ^{-1}\left (\sqrt{-1-c^2 x^2}\right )}{4 e \sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0953076, size = 77, normalized size = 0.53 \[ \frac{x \left (3 a c^3 x \left (2 d+e x^2\right )+b \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 \left (6 d+e x^2\right )-2 e\right )+3 b c^3 x \text{csch}^{-1}(c x) \left (2 d+e x^2\right )\right )}{12 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 115, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{{c}^{2}} \left ({\frac{{c}^{4}{x}^{4}e}{4}}+{\frac{{x}^{2}{c}^{4}d}{2}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsch} \left (cx\right ){c}^{4}{x}^{4}e}{4}}+{\frac{{\rm arccsch} \left (cx\right ){c}^{4}{x}^{2}d}{2}}+{\frac{ \left ({c}^{2}{x}^{2}+1 \right ) \left ({c}^{2}{x}^{2}e+6\,{c}^{2}d-2\,e \right ) }{12\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00803, size = 128, normalized size = 0.88 \begin{align*} \frac{1}{4} \, a e x^{4} + \frac{1}{2} \, a d x^{2} + \frac{1}{2} \,{\left (x^{2} \operatorname{arcsch}\left (c x\right ) + \frac{x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c}\right )} b d + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arcsch}\left (c x\right ) + \frac{c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.7772, size = 266, normalized size = 1.82 \begin{align*} \frac{3 \, a c^{3} e x^{4} + 6 \, a c^{3} d x^{2} + 3 \,{\left (b c^{3} e x^{4} + 2 \, b c^{3} d x^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (b c^{2} e x^{3} + 2 \,{\left (3 \, b c^{2} d - b e\right )} x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{12 \, c^{3}} \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 \left (a + b \operatorname{acsch}{\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{arcsch}\left (c x\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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