Optimal. Leaf size=39 \[ -\frac {\tanh ^{-1}(a+b x)}{2 b}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}+\frac {x}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6108, 5917, 321, 206} \[ -\frac {\tanh ^{-1}(a+b x)}{2 b}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}+\frac {x}{2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 5917
Rule 6108
Rubi steps
\begin {align*} \int (a+b x) \coth ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}-\frac {\operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,a+b x\right )}{2 b}\\ &=\frac {x}{2}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,a+b x\right )}{2 b}\\ &=\frac {x}{2}+\frac {(a+b x)^2 \coth ^{-1}(a+b x)}{2 b}-\frac {\tanh ^{-1}(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 1.69 \[ \frac {a^2 \log (a+b x+1)-\left (a^2-1\right ) \log (-a-b x+1)-\log (a+b x+1)+2 b x (2 a+b x) \coth ^{-1}(a+b x)+2 b x}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 44, normalized size = 1.13 \[ \frac {2 \, b x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{4 \, b} \]
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 \[ \int {\left (b x + a\right )} \operatorname {arcoth}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 70, normalized size = 1.79 \[ \frac {b \,\mathrm {arccoth}\left (b x +a \right ) x^{2}}{2}+\mathrm {arccoth}\left (b x +a \right ) x a +\frac {\mathrm {arccoth}\left (b x +a \right ) a^{2}}{2 b}+\frac {x}{2}+\frac {a}{2 b}+\frac {\ln \left (b x +a -1\right )}{4 b}-\frac {\ln \left (b x +a +1\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 62, normalized size = 1.59 \[ \frac {1}{4} \, b {\left (\frac {2 \, x}{b} + \frac {{\left (a^{2} - 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac {{\left (a^{2} - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \operatorname {arcoth}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.02, size = 50, normalized size = 1.28 \[ \frac {x}{2}-\frac {\frac {\mathrm {acoth}\left (a+b\,x\right )}{2}-\frac {a^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}}{b}+a\,x\,\mathrm {acoth}\left (a+b\,x\right )+\frac {b\,x^2\,\mathrm {acoth}\left (a+b\,x\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 56, normalized size = 1.44 \[ \begin {cases} \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 b} + a x \operatorname {acoth}{\left (a + b x \right )} + \frac {b x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2} + \frac {x}{2} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\a x \operatorname {acoth}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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