Optimal. Leaf size=101 \[ \frac{\left (6 a^2+1\right ) x}{4 b^3}+\frac{(a+b x)^3}{12 b^4}-\frac{a (a+b x)^2}{2 b^4}+\frac{(1-a)^4 \log (-a-b x+1)}{8 b^4}-\frac{(a+1)^4 \log (a+b x+1)}{8 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x) \]
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Rubi [A] time = 0.126333, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6112, 5927, 702, 633, 31} \[ \frac{\left (6 a^2+1\right ) x}{4 b^3}+\frac{(a+b x)^3}{12 b^4}-\frac{a (a+b x)^2}{2 b^4}+\frac{(1-a)^4 \log (-a-b x+1)}{8 b^4}-\frac{(a+1)^4 \log (a+b x+1)}{8 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 6112
Rule 5927
Rule 702
Rule 633
Rule 31
Rubi steps
\begin{align*} \int x^3 \coth ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right )^3 \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{4} x^4 \coth ^{-1}(a+b x)-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^4}{1-x^2} \, dx,x,a+b x\right )\\ &=\frac{1}{4} x^4 \coth ^{-1}(a+b x)-\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{1+6 a^2}{b^4}+\frac{4 a x}{b^4}-\frac{x^2}{b^4}+\frac{1+6 a^2+a^4-4 a \left (1+a^2\right ) x}{b^4 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac{\left (1+6 a^2\right ) x}{4 b^3}-\frac{a (a+b x)^2}{2 b^4}+\frac{(a+b x)^3}{12 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \frac{1+6 a^2+a^4-4 a \left (1+a^2\right ) x}{1-x^2} \, dx,x,a+b x\right )}{4 b^4}\\ &=\frac{\left (1+6 a^2\right ) x}{4 b^3}-\frac{a (a+b x)^2}{2 b^4}+\frac{(a+b x)^3}{12 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)-\frac{(1-a)^4 \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,a+b x\right )}{8 b^4}+\frac{(1+a)^4 \operatorname{Subst}\left (\int \frac{1}{-1-x} \, dx,x,a+b x\right )}{8 b^4}\\ &=\frac{\left (1+6 a^2\right ) x}{4 b^3}-\frac{a (a+b x)^2}{2 b^4}+\frac{(a+b x)^3}{12 b^4}+\frac{1}{4} x^4 \coth ^{-1}(a+b x)+\frac{(1-a)^4 \log (1-a-b x)}{8 b^4}-\frac{(1+a)^4 \log (1+a+b x)}{8 b^4}\\ \end{align*}
Mathematica [A] time = 0.038819, size = 81, normalized size = 0.8 \[ \frac{6 \left (3 a^2+1\right ) b x-6 a b^2 x^2+6 b^4 x^4 \coth ^{-1}(a+b x)+3 (a-1)^4 \log (-a-b x+1)-3 (a+1)^4 \log (a+b x+1)+2 b^3 x^3}{24 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 199, normalized size = 2. \begin{align*}{\frac{a}{4\,{b}^{4}}}+{\frac{x}{4\,{b}^{3}}}+{\frac{{x}^{4}{\rm arccoth} \left (bx+a\right )}{4}}-{\frac{\ln \left ( bx+a+1 \right ) }{8\,{b}^{4}}}+{\frac{\ln \left ( bx+a-1 \right ) }{8\,{b}^{4}}}+{\frac{{x}^{3}}{12\,b}}-{\frac{a{x}^{2}}{4\,{b}^{2}}}+{\frac{3\,{a}^{2}x}{4\,{b}^{3}}}+{\frac{13\,{a}^{3}}{12\,{b}^{4}}}+{\frac{\ln \left ( bx+a-1 \right ){a}^{4}}{8\,{b}^{4}}}-{\frac{\ln \left ( bx+a-1 \right ){a}^{3}}{2\,{b}^{4}}}+{\frac{3\,\ln \left ( bx+a-1 \right ){a}^{2}}{4\,{b}^{4}}}-{\frac{\ln \left ( bx+a-1 \right ) a}{2\,{b}^{4}}}-{\frac{\ln \left ( bx+a+1 \right ){a}^{4}}{8\,{b}^{4}}}-{\frac{\ln \left ( bx+a+1 \right ){a}^{3}}{2\,{b}^{4}}}-{\frac{3\,\ln \left ( bx+a+1 \right ){a}^{2}}{4\,{b}^{4}}}-{\frac{\ln \left ( bx+a+1 \right ) a}{2\,{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959818, size = 143, normalized size = 1.42 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arcoth}\left (b x + a\right ) + \frac{1}{24} \, b{\left (\frac{2 \,{\left (b^{2} x^{3} - 3 \, a b x^{2} + 3 \,{\left (3 \, a^{2} + 1\right )} x\right )}}{b^{4}} - \frac{3 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac{3 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55089, size = 279, normalized size = 2.76 \begin{align*} \frac{3 \, b^{4} x^{4} \log \left (\frac{b x + a + 1}{b x + a - 1}\right ) + 2 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 6 \,{\left (3 \, a^{2} + 1\right )} b x - 3 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) + 3 \,{\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{24 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 x^{3} \operatorname{arcoth}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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