Optimal. Leaf size=35 \[ \frac {\log \left (1-(a+b x)^2\right )}{2 b}+\frac {(a+b x) \coth ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6104, 5911, 260} \[ \frac {\log \left (1-(a+b x)^2\right )}{2 b}+\frac {(a+b x) \coth ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5911
Rule 6104
Rubi steps
\begin {align*} \int \coth ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b}-\frac {\operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b}+\frac {\log \left (1-(a+b x)^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.23 \[ \frac {(a+1) \log (a+b x+1)-(a-1) \log (-a-b x+1)}{2 b}+x \coth ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 48, normalized size = 1.37 \[ \frac {b x \log \left (\frac {b x + a + 1}{b x + a - 1}\right ) + {\left (a + 1\right )} \log \left (b x + a + 1\right ) - {\left (a - 1\right )} \log \left (b x + a - 1\right )}{2 \, b} \]
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 \[ \int \operatorname {arcoth}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 36, normalized size = 1.03 \[ x \,\mathrm {arccoth}\left (b x +a \right )+\frac {\mathrm {arccoth}\left (b x +a \right ) a}{b}+\frac {\ln \left (\left (b x +a \right )^{2}-1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 31, normalized size = 0.89 \[ \frac {2 \, {\left (b x + a\right )} \operatorname {arcoth}\left (b x + a\right ) + \log \left (-{\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 42, normalized size = 1.20 \[ \frac {\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )}{2}+a\,\mathrm {acoth}\left (a+b\,x\right )}{b}+x\,\mathrm {acoth}\left (a+b\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 41, normalized size = 1.17 \[ \begin {cases} \frac {a \operatorname {acoth}{\left (a + b x \right )}}{b} + x \operatorname {acoth}{\left (a + b x \right )} + \frac {\log {\left (a + b x + 1 \right )}}{b} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acoth}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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