Optimal. Leaf size=65 \[ \frac {(1-a)^2 \log (-a-b x+1)}{4 b^2}-\frac {(a+1)^2 \log (a+b x+1)}{4 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)+\frac {x}{2 b} \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6112, 5927, 702, 633, 31} \[ \frac {(1-a)^2 \log (-a-b x+1)}{4 b^2}-\frac {(a+1)^2 \log (a+b x+1)}{4 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)+\frac {x}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 702
Rule 5927
Rule 6112
Rubi steps
\begin {align*} \int x \coth ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{1-x^2} \, dx,x,a+b x\right )\\ &=\frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{b^2}+\frac {1+a^2-2 a x}{b^2 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac {x}{2 b}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \frac {1+a^2-2 a x}{1-x^2} \, dx,x,a+b x\right )}{2 b^2}\\ &=\frac {x}{2 b}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)-\frac {(1-a)^2 \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,a+b x\right )}{4 b^2}+\frac {(1+a)^2 \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,a+b x\right )}{4 b^2}\\ &=\frac {x}{2 b}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)+\frac {(1-a)^2 \log (1-a-b x)}{4 b^2}-\frac {(1+a)^2 \log (1+a+b x)}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 0.86 \[ \frac {2 b^2 x^2 \coth ^{-1}(a+b x)+(a-1)^2 \log (-a-b x+1)-(a+1)^2 \log (a+b x+1)+2 b x}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 66, normalized size = 1.02 \[ \frac {b^{2} x^{2} \log \left (\frac {b x + a + 1}{b x + a - 1}\right ) + 2 \, b x - {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) + {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{4 \, b^{2}} \]
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 \[ \int x \operatorname {arcoth}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 89, normalized size = 1.37 \[ \frac {x^{2} \mathrm {arccoth}\left (b x +a \right )}{2}-\frac {\mathrm {arccoth}\left (b x +a \right ) a^{2}}{2 b^{2}}+\frac {x}{2 b}+\frac {a}{2 b^{2}}+\frac {\ln \left (b x +a -1\right )}{4 b^{2}}-\frac {\ln \left (b x +a -1\right ) a}{2 b^{2}}-\frac {\ln \left (b x +a +1\right )}{4 b^{2}}-\frac {\ln \left (b x +a +1\right ) a}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 61, normalized size = 0.94 \[ \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (b x + a\right ) + \frac {1}{4} \, b {\left (\frac {2 \, x}{b^{2}} - \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{3}} + \frac {{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.00, size = 62, normalized size = 0.95 \[ \frac {x^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}-\frac {\frac {\mathrm {acoth}\left (a+b\,x\right )}{2}-\frac {b\,x}{2}+\frac {a^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}+\frac {a\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )}{2}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 76, normalized size = 1.17 \[ \begin {cases} - \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 b^{2}} - \frac {a \log {\left (a + b x + 1 \right )}}{b^{2}} + \frac {a \operatorname {acoth}{\left (a + b x \right )}}{b^{2}} + \frac {x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2} + \frac {x}{2 b} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {acoth}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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