Optimal. Leaf size=122 \[ \frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b \left (6 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{6 c^3}+\frac{b d e x \sqrt{\frac{1}{c^2 x^2}+1}}{c}+\frac{b e^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{6 c}-\frac{b d^3 \text{csch}^{-1}(c x)}{3 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25939, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {6290, 1568, 1475, 1807, 844, 215, 266, 63, 208} \[ \frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b \left (6 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{6 c^3}+\frac{b d e x \sqrt{\frac{1}{c^2 x^2}+1}}{c}+\frac{b e^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{6 c}-\frac{b d^3 \text{csch}^{-1}(c x)}{3 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6290
Rule 1568
Rule 1475
Rule 1807
Rule 844
Rule 215
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b \int \frac{(d+e x)^3}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{3 c e}\\ &=\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b \int \frac{\left (e+\frac{d}{x}\right )^3 x}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{3 c e}\\ &=\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{b \operatorname{Subst}\left (\int \frac{(e+d x)^3}{x^3 \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{3 c e}\\ &=\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b \operatorname{Subst}\left (\int \frac{-6 d e^2-e \left (6 d^2-\frac{e^2}{c^2}\right ) x-2 d^3 x^2}{x^2 \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c e}\\ &=\frac{b d e \sqrt{1+\frac{1}{c^2 x^2}} x}{c}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{b \operatorname{Subst}\left (\int \frac{e \left (6 d^2-\frac{e^2}{c^2}\right )+2 d^3 x}{x \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c e}\\ &=\frac{b d e \sqrt{1+\frac{1}{c^2 x^2}} x}{c}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{\left (b d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{3 c e}-\frac{\left (b \left (6 d^2-\frac{e^2}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{6 c}\\ &=\frac{b d e \sqrt{1+\frac{1}{c^2 x^2}} x}{c}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}-\frac{b d^3 \text{csch}^{-1}(c x)}{3 e}+\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{\left (b \left (6 c^2 d^2-e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{12 c^3}\\ &=\frac{b d e \sqrt{1+\frac{1}{c^2 x^2}} x}{c}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}-\frac{b d^3 \text{csch}^{-1}(c x)}{3 e}+\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{\left (b \left (6 c^2 d^2-e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{6 c}\\ &=\frac{b d e \sqrt{1+\frac{1}{c^2 x^2}} x}{c}+\frac{b e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}-\frac{b d^3 \text{csch}^{-1}(c x)}{3 e}+\frac{(d+e x)^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b \left (6 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{6 c^3}\\ \end{align*}
Mathematica [A] time = 0.181265, size = 122, normalized size = 1. \[ \frac{c^2 x \left (2 a c \left (3 d^2+3 d e x+e^2 x^2\right )+b e \sqrt{\frac{1}{c^2 x^2}+1} (6 d+e x)\right )+b \left (6 c^2 d^2-e^2\right ) \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+2 b c^3 x \text{csch}^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )}{6 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.218, size = 204, normalized size = 1.7 \begin{align*}{\frac{1}{c} \left ({\frac{ \left ( cxe+cd \right ) ^{3}a}{3\,{c}^{2}e}}+{\frac{b}{{c}^{2}} \left ({\frac{{e}^{2}{\rm arccsch} \left (cx\right ){c}^{3}{x}^{3}}{3}}+e{\rm arccsch} \left (cx\right ){c}^{3}{x}^{2}d+{\rm arccsch} \left (cx\right ){c}^{3}x{d}^{2}+{\frac{{\rm arccsch} \left (cx\right ){c}^{3}{d}^{3}}{3\,e}}+{\frac{1}{6\,cxe}\sqrt{{c}^{2}{x}^{2}+1} \left ( -2\,{c}^{3}{d}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) +6\,{c}^{2}{d}^{2}e{\it Arcsinh} \left ( cx \right ) +{e}^{3}cx\sqrt{{c}^{2}{x}^{2}+1}+6\,cd{e}^{2}\sqrt{{c}^{2}{x}^{2}+1}-{e}^{3}{\it Arcsinh} \left ( cx \right ) \right ){\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02931, size = 259, normalized size = 2.12 \begin{align*} \frac{1}{3} \, a e^{2} x^{3} + a d e x^{2} +{\left (x^{2} \operatorname{arcsch}\left (c x\right ) + \frac{x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c}\right )} b d e + \frac{1}{12} \,{\left (4 \, x^{3} \operatorname{arcsch}\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 e^{2} + a d^{2} x + \frac{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d^{2}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.17135, size = 709, normalized size = 5.81 \begin{align*} \frac{2 \, a c^{3} e^{2} x^{3} + 6 \, a c^{3} d e x^{2} + 6 \, a c^{3} d^{2} x + 2 \,{\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) -{\left (6 \, b c^{2} d^{2} - b e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 2 \,{\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \,{\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} - 3 \, b c^{3} d e - b c^{3} e^{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^{2} x^{2} + 6 \, b c^{2} d e x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acsch}{\left (c x \right )}\right ) \left (d + e x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]