Optimal. Leaf size=64 \[ \frac {b \log (x)}{1-a^2}-\frac {b \log (-a-b x+1)}{2 (1-a)}-\frac {b \log (a+b x+1)}{2 (a+1)}-\frac {\coth ^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 64, 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 = {6110, 371, 706, 31, 633} \[ \frac {b \log (x)}{1-a^2}-\frac {b \log (-a-b x+1)}{2 (1-a)}-\frac {b \log (a+b x+1)}{2 (a+1)}-\frac {\coth ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 31
Rule 371
Rule 633
Rule 706
Rule 6110
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx &=-\frac {\coth ^{-1}(a+b x)}{x}+b \int \frac {1}{x \left (1-(a+b x)^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a+b x)}{x}+b \operatorname {Subst}\left (\int \frac {1}{(-a+x) \left (1-x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac {\coth ^{-1}(a+b x)}{x}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-a+x} \, dx,x,a+b x\right )}{1-a^2}+\frac {b \operatorname {Subst}\left (\int \frac {a+x}{1-x^2} \, dx,x,a+b x\right )}{1-a^2}\\ &=-\frac {\coth ^{-1}(a+b x)}{x}+\frac {b \log (x)}{1-a^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,a+b x\right )}{2 (1-a)}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,a+b x\right )}{2 (1+a)}\\ &=-\frac {\coth ^{-1}(a+b x)}{x}+\frac {b \log (x)}{1-a^2}-\frac {b \log (1-a-b x)}{2 (1-a)}-\frac {b \log (1+a+b x)}{2 (1+a)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 55, normalized size = 0.86 \[ \frac {b ((a+1) \log (-a-b x+1)-(a-1) \log (a+b x+1)-2 \log (x))}{2 \left (a^2-1\right )}-\frac {\coth ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 68, normalized size = 1.06 \[ -\frac {{\left (a - 1\right )} b x \log \left (b x + a + 1\right ) - {\left (a + 1\right )} b x \log \left (b x + a - 1\right ) + 2 \, b x \log \relax (x) + {\left (a^{2} - 1\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{2 \, {\left (a^{2} - 1\right )} x} \]
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 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 63, normalized size = 0.98 \[ -\frac {\mathrm {arccoth}\left (b x +a \right )}{x}+\frac {b \ln \left (b x +a -1\right )}{2 a -2}-\frac {b \ln \left (b x +a +1\right )}{2+2 a}-\frac {b \ln \left (b x \right )}{\left (a -1\right ) \left (1+a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 54, normalized size = 0.84 \[ -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a + 1\right )}{a + 1} - \frac {\log \left (b x + a - 1\right )}{a - 1} + \frac {2 \, \log \relax (x)}{a^{2} - 1}\right )} - \frac {\operatorname {arcoth}\left (b x + a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 62, normalized size = 0.97 \[ -\frac {\mathrm {acoth}\left (a+b\,x\right )}{x}-\frac {b\,x\,\ln \relax (x)-\frac {b\,x\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )}{2}+a\,b\,x\,\mathrm {acoth}\left (a+b\,x\right )}{x\,\left (a^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 144, normalized size = 2.25 \[ \begin {cases} \frac {b \operatorname {acoth}{\left (b x - 1 \right )}}{2} - \frac {\operatorname {acoth}{\left (b x - 1 \right )}}{x} - \frac {1}{2 x} & \text {for}\: a = -1 \\- \frac {b \operatorname {acoth}{\left (b x + 1 \right )}}{2} - \frac {\operatorname {acoth}{\left (b x + 1 \right )}}{x} + \frac {1}{2 x} & \text {for}\: a = 1 \\- \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} - \frac {a b x \operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} - \frac {b x \log {\relax (x )}}{a^{2} x - x} + \frac {b x \log {\left (a + b x + 1 \right )}}{a^{2} x - x} - \frac {b x \operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} + \frac {\operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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