Optimal. Leaf size=90 \[ \frac {a b^2 \log (x)}{\left (1-a^2\right )^2}-\frac {b}{2 \left (1-a^2\right ) x}-\frac {b^2 \log (-a-b x+1)}{4 (1-a)^2}+\frac {b^2 \log (a+b x+1)}{4 (a+1)^2}-\frac {\coth ^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6110, 371, 710, 801} \[ \frac {a b^2 \log (x)}{\left (1-a^2\right )^2}-\frac {b}{2 \left (1-a^2\right ) x}-\frac {b^2 \log (-a-b x+1)}{4 (1-a)^2}+\frac {b^2 \log (a+b x+1)}{4 (a+1)^2}-\frac {\coth ^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 371
Rule 710
Rule 801
Rule 6110
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx &=-\frac {\coth ^{-1}(a+b x)}{2 x^2}+\frac {1}{2} b \int \frac {1}{x^2 \left (1-(a+b x)^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a+b x)}{2 x^2}+\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {1}{(-a+x)^2 \left (1-x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac {b}{2 \left (1-a^2\right ) x}-\frac {\coth ^{-1}(a+b x)}{2 x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {-a-x}{(-a+x) \left (1-x^2\right )} \, dx,x,a+b x\right )}{2 \left (1-a^2\right )}\\ &=-\frac {b}{2 \left (1-a^2\right ) x}-\frac {\coth ^{-1}(a+b x)}{2 x^2}-\frac {b^2 \operatorname {Subst}\left (\int \left (-\frac {2 a}{\left (-1+a^2\right ) (a-x)}+\frac {-1-a}{2 (-1+a) (-1+x)}+\frac {-1+a}{2 (1+a) (1+x)}\right ) \, dx,x,a+b x\right )}{2 \left (1-a^2\right )}\\ &=-\frac {b}{2 \left (1-a^2\right ) x}-\frac {\coth ^{-1}(a+b x)}{2 x^2}+\frac {a b^2 \log (x)}{\left (1-a^2\right )^2}-\frac {b^2 \log (1-a-b x)}{4 (1-a)^2}+\frac {b^2 \log (1+a+b x)}{4 (1+a)^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 76, normalized size = 0.84 \[ \frac {1}{4} \left (b \left (\frac {4 a b \log (x)}{\left (a^2-1\right )^2}+\frac {2}{\left (a^2-1\right ) x}-\frac {b \log (-a-b x+1)}{(a-1)^2}+\frac {b \log (a+b x+1)}{(a+1)^2}\right )-\frac {2 \coth ^{-1}(a+b x)}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 111, normalized size = 1.23 \[ \frac {{\left (a^{2} - 2 \, a + 1\right )} b^{2} x^{2} \log \left (b x + a + 1\right ) - {\left (a^{2} + 2 \, a + 1\right )} b^{2} x^{2} \log \left (b x + a - 1\right ) + 4 \, a b^{2} x^{2} \log \relax (x) + 2 \, {\left (a^{2} - 1\right )} b x - {\left (a^{4} - 2 \, a^{2} + 1\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{4 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}} \]
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 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 82, normalized size = 0.91 \[ -\frac {\mathrm {arccoth}\left (b x +a \right )}{2 x^{2}}-\frac {b^{2} \ln \left (b x +a -1\right )}{4 \left (a -1\right )^{2}}+\frac {b^{2} \ln \left (b x +a +1\right )}{4 \left (1+a \right )^{2}}+\frac {b}{2 \left (a -1\right ) \left (1+a \right ) x}+\frac {b^{2} a \ln \left (b x \right )}{\left (a -1\right )^{2} \left (1+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 85, normalized size = 0.94 \[ \frac {1}{4} \, {\left (\frac {4 \, a b \log \relax (x)}{a^{4} - 2 \, a^{2} + 1} + \frac {b \log \left (b x + a + 1\right )}{a^{2} + 2 \, a + 1} - \frac {b \log \left (b x + a - 1\right )}{a^{2} - 2 \, a + 1} + \frac {2}{{\left (a^{2} - 1\right )} x}\right )} b - \frac {\operatorname {arcoth}\left (b x + a\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.95, size = 247, normalized size = 2.74 \[ \ln \relax (x)\,\left (\frac {b^2}{4\,{\left (a-1\right )}^2}-\frac {b^2}{4\,{\left (a+1\right )}^2}\right )-\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )\,\left (\frac {b^2}{8\,{\left (a-1\right )}^2}-\frac {b^2}{8\,{\left (a+1\right )}^2}\right )-\frac {\mathrm {acoth}\left (a+b\,x\right )\,\left (\frac {a^2}{2}-\frac {1}{2}\right )-\frac {b\,x}{2}+\frac {b^2\,x^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}+\frac {x^3\,\left (3\,a^2\,b^3+b^3\right )}{2\,{\left (a^2-1\right )}^2}+\frac {a\,b^4\,x^4}{{\left (a^2-1\right )}^2}+a\,b\,x\,\mathrm {acoth}\left (a+b\,x\right )}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4-x^2}-\frac {\mathrm {atan}\left (\frac {2\,x\,b^2+2\,a\,b}{2\,\sqrt {b^2\,\left (a^2-1\right )-a^2\,b^2}}\right )\,\left (a^2\,b^3+b^3\right )}{\sqrt {-b^2}\,\left (2\,a^4-4\,a^2+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.43, size = 410, normalized size = 4.56 \[ \begin {cases} \frac {b^{2} \operatorname {acoth}{\left (b x - 1 \right )}}{8} - \frac {b}{8 x} - \frac {\operatorname {acoth}{\left (b x - 1 \right )}}{2 x^{2}} - \frac {1}{8 x^{2}} & \text {for}\: a = -1 \\\frac {b^{2} \operatorname {acoth}{\left (b x + 1 \right )}}{8} - \frac {b}{8 x} - \frac {\operatorname {acoth}{\left (b x + 1 \right )}}{2 x^{2}} + \frac {1}{8 x^{2}} & \text {for}\: a = 1 \\- \frac {a^{4} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {a^{2} b^{2} x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {a^{2} b x}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {2 a^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {2 a b^{2} x^{2} \log {\relax (x )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} - \frac {2 a b^{2} x^{2} \log {\left (a + b x + 1 \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {2 a b^{2} x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {b^{2} x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} - \frac {b x}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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