Optimal. Leaf size=54 \[ \frac{(a+b x)^2}{6 b}+\frac{\log \left (1-(a+b x)^2\right )}{6 b}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0479171, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6108, 5917, 266, 43} \[ \frac{(a+b x)^2}{6 b}+\frac{\log \left (1-(a+b x)^2\right )}{6 b}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6108
Rule 5917
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac{\operatorname{Subst}\left (\int \frac{x^3}{1-x^2} \, dx,x,a+b x\right )}{3 b}\\ &=\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac{\operatorname{Subst}\left (\int \frac{x}{1-x} \, dx,x,(a+b x)^2\right )}{6 b}\\ &=\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac{\operatorname{Subst}\left (\int \left (-1+\frac{1}{1-x}\right ) \, dx,x,(a+b x)^2\right )}{6 b}\\ &=\frac{(a+b x)^2}{6 b}+\frac{(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}+\frac{\log \left (1-(a+b x)^2\right )}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0345896, size = 42, normalized size = 0.78 \[ \frac{(a+b x)^2+\log \left (1-(a+b x)^2\right )+2 (a+b x)^3 \coth ^{-1}(a+b x)}{6 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.034, size = 95, normalized size = 1.8 \begin{align*}{\frac{{b}^{2}{\rm arccoth} \left (bx+a\right ){x}^{3}}{3}}+b{\rm arccoth} \left (bx+a\right ){x}^{2}a+{\rm arccoth} \left (bx+a\right )x{a}^{2}+{\frac{{\rm arccoth} \left (bx+a\right ){a}^{3}}{3\,b}}+{\frac{b{x}^{2}}{6}}+{\frac{ax}{3}}+{\frac{{a}^{2}}{6\,b}}+{\frac{\ln \left ( bx+a-1 \right ) }{6\,b}}+{\frac{\ln \left ( bx+a+1 \right ) }{6\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.95623, size = 109, normalized size = 2.02 \begin{align*} \frac{1}{6} \, b{\left (\frac{b x^{2} + 2 \, a x}{b} + \frac{{\left (a^{3} + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac{{\left (a^{3} - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} + \frac{1}{3} \,{\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} \operatorname{arcoth}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87158, size = 211, normalized size = 3.91 \begin{align*} \frac{b^{2} x^{2} + 2 \, a b x +{\left (a^{3} + 1\right )} \log \left (b x + a + 1\right ) -{\left (a^{3} - 1\right )} \log \left (b x + a - 1\right ) +{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x\right )} \log \left (\frac{b x + a + 1}{b x + a - 1}\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.13341, size = 97, normalized size = 1.8 \begin{align*} \begin{cases} \frac{a^{3} \operatorname{acoth}{\left (a + b x \right )}}{3 b} + a^{2} x \operatorname{acoth}{\left (a + b x \right )} + a b x^{2} \operatorname{acoth}{\left (a + b x \right )} + \frac{a x}{3} + \frac{b^{2} x^{3} \operatorname{acoth}{\left (a + b x \right )}}{3} + \frac{b x^{2}}{6} + \frac{\log{\left (\frac{a}{b} + x + \frac{1}{b} \right )}}{3 b} - \frac{\operatorname{acoth}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\a^{2} x \operatorname{acoth}{\left (a \right )} & \text{otherwise} \end{cases} \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 (b x + a\right )}^{2} \operatorname{arcoth}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]