Optimal. Leaf size=78 \[ \frac {(a+b x)^2}{6 b^3}+\frac {(1-a)^3 \log (-a-b x+1)}{6 b^3}+\frac {(a+1)^3 \log (a+b x+1)}{6 b^3}-\frac {a x}{b^2}+\frac {1}{3} x^3 \coth ^{-1}(a+b x) \]
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Rubi [A] time = 0.10, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6112, 5927, 702, 633, 31} \[ \frac {(a+b x)^2}{6 b^3}-\frac {a x}{b^2}+\frac {(1-a)^3 \log (-a-b x+1)}{6 b^3}+\frac {(a+1)^3 \log (a+b x+1)}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 702
Rule 5927
Rule 6112
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \coth ^{-1}(a+b x)-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3}{1-x^2} \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \coth ^{-1}(a+b x)-\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {3 a}{b^3}-\frac {x}{b^3}-\frac {a \left (3+a^2\right )-\left (1+3 a^2\right ) x}{b^3 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)+\frac {\operatorname {Subst}\left (\int \frac {a \left (3+a^2\right )-\left (1+3 a^2\right ) x}{1-x^2} \, dx,x,a+b x\right )}{3 b^3}\\ &=-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)-\frac {(1-a)^3 \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,a+b x\right )}{6 b^3}-\frac {(1+a)^3 \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,a+b x\right )}{6 b^3}\\ &=-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)+\frac {(1-a)^3 \log (1-a-b x)}{6 b^3}+\frac {(1+a)^3 \log (1+a+b x)}{6 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 92, normalized size = 1.18 \[ \frac {\left (-a^3+3 a^2-3 a+1\right ) \log (-a-b x+1)}{6 b^3}+\frac {\left (a^3+3 a^2+3 a+1\right ) \log (a+b x+1)}{6 b^3}-\frac {2 a x}{3 b^2}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)+\frac {x^2}{6 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 84, normalized size = 1.08 \[ \frac {b^{3} x^{3} \log \left (\frac {b x + a + 1}{b x + a - 1}\right ) + b^{2} x^{2} - 4 \, a b x + {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right ) - {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arcoth}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 146, normalized size = 1.87 \[ \frac {x^{3} \mathrm {arccoth}\left (b x +a \right )}{3}+\frac {x^{2}}{6 b}-\frac {2 a x}{3 b^{2}}-\frac {5 a^{2}}{6 b^{3}}-\frac {\ln \left (b x +a -1\right ) a^{3}}{6 b^{3}}+\frac {\ln \left (b x +a -1\right ) a^{2}}{2 b^{3}}-\frac {\ln \left (b x +a -1\right ) a}{2 b^{3}}+\frac {\ln \left (b x +a -1\right )}{6 b^{3}}+\frac {\ln \left (b x +a +1\right ) a^{3}}{6 b^{3}}+\frac {\ln \left (b x +a +1\right ) a^{2}}{2 b^{3}}+\frac {\ln \left (b x +a +1\right ) a}{2 b^{3}}+\frac {\ln \left (b x +a +1\right )}{6 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 79, normalized size = 1.01 \[ \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (b x + a\right ) + \frac {1}{6} \, b {\left (\frac {b x^{2} - 4 \, a x}{b^{3}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} - \frac {{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 98, normalized size = 1.26 \[ \frac {x^3\,\ln \left (\frac {1}{a+b\,x}+1\right )}{6}-\frac {x^3\,\ln \left (1-\frac {1}{a+b\,x}\right )}{6}+\frac {x^2}{6\,b}-\frac {\ln \left (a+b\,x-1\right )\,\left (a^3-3\,a^2+3\,a-1\right )}{6\,b^3}+\frac {\ln \left (a+b\,x+1\right )\,\left (a^3+3\,a^2+3\,a+1\right )}{6\,b^3}-\frac {2\,a\,x}{3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.27, size = 117, normalized size = 1.50 \[ \begin {cases} \frac {a^{3} \operatorname {acoth}{\left (a + b x \right )}}{3 b^{3}} + \frac {a^{2} \log {\left (a + b x + 1 \right )}}{b^{3}} - \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{b^{3}} - \frac {2 a x}{3 b^{2}} + \frac {a \operatorname {acoth}{\left (a + b x \right )}}{b^{3}} + \frac {x^{3} \operatorname {acoth}{\left (a + b x \right )}}{3} + \frac {x^{2}}{6 b} + \frac {\log {\left (a + b x + 1 \right )}}{3 b^{3}} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{3 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {acoth}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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