Integrand size = 10, antiderivative size = 90 \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=-\frac {b}{2 \left (1-a^2\right ) x}-\frac {\coth ^{-1}(a+b x)}{2 x^2}+\frac {a b^2 \log (x)}{\left (1-a^2\right )^2}-\frac {b^2 \log (1-a-b x)}{4 (1-a)^2}+\frac {b^2 \log (1+a+b x)}{4 (1+a)^2} \]
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Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6245, 378, 724, 815} \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=\frac {a b^2 \log (x)}{\left (1-a^2\right )^2}-\frac {b}{2 \left (1-a^2\right ) x}-\frac {b^2 \log (-a-b x+1)}{4 (1-a)^2}+\frac {b^2 \log (a+b x+1)}{4 (a+1)^2}-\frac {\coth ^{-1}(a+b x)}{2 x^2} \]
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Rule 378
Rule 724
Rule 815
Rule 6245
Rubi steps \begin{align*} \text {integral}& = -\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 \text {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 \text {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 \text {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*}
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 \coth ^{-1}(a+b x)}{x^2}+b \left (\frac {2}{\left (-1+a^2\right ) x}+\frac {4 a b \log (x)}{\left (-1+a^2\right )^2}-\frac {b \log (1-a-b x)}{(-1+a)^2}+\frac {b \log (1+a+b x)}{(1+a)^2}\right )\right ) \]
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Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.87
method | result | size |
parts | \(-\frac {\operatorname {arccoth}\left (b x +a \right )}{2 x^{2}}-\frac {b \left (-\frac {1}{\left (-1+a \right ) \left (1+a \right ) x}-\frac {2 a b \ln \left (x \right )}{\left (-1+a \right )^{2} \left (1+a \right )^{2}}+\frac {b \ln \left (b x +a -1\right )}{2 \left (-1+a \right )^{2}}-\frac {b \ln \left (b x +a +1\right )}{2 \left (1+a \right )^{2}}\right )}{2}\) | \(78\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {arccoth}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\ln \left (b x +a +1\right )}{4 \left (1+a \right )^{2}}-\frac {\ln \left (b x +a -1\right )}{4 \left (-1+a \right )^{2}}+\frac {1}{2 \left (-1+a \right ) \left (1+a \right ) b x}+\frac {a \ln \left (-b x \right )}{\left (-1+a \right )^{2} \left (1+a \right )^{2}}\right )\) | \(83\) |
default | \(b^{2} \left (-\frac {\operatorname {arccoth}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\ln \left (b x +a +1\right )}{4 \left (1+a \right )^{2}}-\frac {\ln \left (b x +a -1\right )}{4 \left (-1+a \right )^{2}}+\frac {1}{2 \left (-1+a \right ) \left (1+a \right ) b x}+\frac {a \ln \left (-b x \right )}{\left (-1+a \right )^{2} \left (1+a \right )^{2}}\right )\) | \(83\) |
parallelrisch | \(\frac {x^{2} \operatorname {arccoth}\left (b x +a \right ) a^{2} b^{2}+2 \ln \left (x \right ) a \,b^{2} x^{2}-2 \ln \left (b x +a -1\right ) x^{2} a \,b^{2}-2 x^{2} \operatorname {arccoth}\left (b x +a \right ) a \,b^{2}+\operatorname {arccoth}\left (b x +a \right ) b^{2} x^{2}-2 a \,b^{2} x^{2}-\operatorname {arccoth}\left (b x +a \right ) a^{4}+a^{2} b x +2 \,\operatorname {arccoth}\left (b x +a \right ) a^{2}-b x -\operatorname {arccoth}\left (b x +a \right )}{2 x^{2} \left (a^{2}+2 a +1\right ) \left (a^{2}-2 a +1\right )}\) | \(147\) |
risch | \(-\frac {\ln \left (b x +a +1\right )}{4 x^{2}}-\frac {\ln \left (-b x -a +1\right ) a^{2} b^{2} x^{2}-\ln \left (-b x -a -1\right ) a^{2} b^{2} x^{2}+2 \ln \left (-b x -a +1\right ) a \,b^{2} x^{2}+2 \ln \left (-b x -a -1\right ) a \,b^{2} x^{2}-4 \ln \left (x \right ) a \,b^{2} x^{2}+\ln \left (-b x -a +1\right ) b^{2} x^{2}-\ln \left (-b x -a -1\right ) b^{2} x^{2}-a^{4} \ln \left (b x +a -1\right )-2 a^{2} b x +2 \ln \left (b x +a -1\right ) a^{2}+2 b x -\ln \left (b x +a -1\right )}{4 x^{2} \left (a^{4}-2 a^{2}+1\right )}\) | \(201\) |
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.23 \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=\frac {{\left (a^{2} - 2 \, a + 1\right )} b^{2} x^{2} \log \left (b x + a + 1\right ) - {\left (a^{2} + 2 \, a + 1\right )} b^{2} x^{2} \log \left (b x + a - 1\right ) + 4 \, a b^{2} x^{2} \log \left (x\right ) + 2 \, {\left (a^{2} - 1\right )} b x - {\left (a^{4} - 2 \, a^{2} + 1\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{4 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (73) = 146\).
Time = 0.78 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.56 \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=\begin {cases} \frac {b^{2} \operatorname {acoth}{\left (b x - 1 \right )}}{8} - \frac {b}{8 x} - \frac {\operatorname {acoth}{\left (b x - 1 \right )}}{2 x^{2}} - \frac {1}{8 x^{2}} & \text {for}\: a = -1 \\\frac {b^{2} \operatorname {acoth}{\left (b x + 1 \right )}}{8} - \frac {b}{8 x} - \frac {\operatorname {acoth}{\left (b x + 1 \right )}}{2 x^{2}} + \frac {1}{8 x^{2}} & \text {for}\: a = 1 \\- \frac {a^{4} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {a^{2} b^{2} x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {a^{2} b x}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {2 a^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {2 a b^{2} x^{2} \log {\left (x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} - \frac {2 a b^{2} x^{2} \log {\left (a + b x + 1 \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {2 a b^{2} x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} + \frac {b^{2} x^{2} \operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} - \frac {b x}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{2 a^{4} x^{2} - 4 a^{2} x^{2} + 2 x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=\frac {1}{4} \, {\left (\frac {4 \, a b \log \left (x\right )}{a^{4} - 2 \, a^{2} + 1} + \frac {b \log \left (b x + a + 1\right )}{a^{2} + 2 \, a + 1} - \frac {b \log \left (b x + a - 1\right )}{a^{2} - 2 \, a + 1} + \frac {2}{{\left (a^{2} - 1\right )} x}\right )} b - \frac {\operatorname {arcoth}\left (b x + a\right )}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (76) = 152\).
Time = 0.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 4.00 \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=-\frac {1}{2} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {a b \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{a^{4} - 2 \, a^{2} + 1} - \frac {a b \log \left ({\left | \frac {{\left (b x + a + 1\right )} a}{b x + a - 1} - a - \frac {b x + a + 1}{b x + a - 1} - 1 \right |}\right )}{a^{4} - 2 \, a^{2} + 1} + \frac {{\left (\frac {{\left (b x + a + 1\right )} a b}{b x + a - 1} - a b - \frac {{\left (b x + a + 1\right )} b}{b x + a - 1}\right )} \log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{{\left (a^{2} - 2 \, a + 1\right )} {\left (\frac {{\left (b x + a + 1\right )} a}{b x + a - 1} - a - \frac {b x + a + 1}{b x + a - 1} - 1\right )}^{2}} + \frac {a b + b}{{\left (\frac {{\left (b x + a + 1\right )} a}{b x + a - 1} - a - \frac {b x + a + 1}{b x + a - 1} - 1\right )} {\left (a + 1\right )}^{2} {\left (a - 1\right )}^{2}}\right )} \]
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Time = 5.60 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.74 \[ \int \frac {\coth ^{-1}(a+b x)}{x^3} \, dx=\ln \left (x\right )\,\left (\frac {b^2}{4\,{\left (a-1\right )}^2}-\frac {b^2}{4\,{\left (a+1\right )}^2}\right )-\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )\,\left (\frac {b^2}{8\,{\left (a-1\right )}^2}-\frac {b^2}{8\,{\left (a+1\right )}^2}\right )-\frac {\mathrm {acoth}\left (a+b\,x\right )\,\left (\frac {a^2}{2}-\frac {1}{2}\right )-\frac {b\,x}{2}+\frac {b^2\,x^2\,\mathrm {acoth}\left (a+b\,x\right )}{2}+\frac {x^3\,\left (3\,a^2\,b^3+b^3\right )}{2\,{\left (a^2-1\right )}^2}+\frac {a\,b^4\,x^4}{{\left (a^2-1\right )}^2}+a\,b\,x\,\mathrm {acoth}\left (a+b\,x\right )}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4-x^2}-\frac {\mathrm {atan}\left (\frac {2\,x\,b^2+2\,a\,b}{2\,\sqrt {b^2\,\left (a^2-1\right )-a^2\,b^2}}\right )\,\left (a^2\,b^3+b^3\right )}{\sqrt {-b^2}\,\left (2\,a^4-4\,a^2+2\right )} \]
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