Integrand size = 16, antiderivative size = 292 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {(1-a-b x) \log \left (-\frac {1-a-b x}{a+b x}\right )}{2 b c}+\frac {\log (a+b x)}{2 b c}+\frac {\log (1+a+b x)}{2 b c}+\frac {(a+b x) \log \left (\frac {1+a+b x}{a+b x}\right )}{2 b c}-\frac {d \log \left (\frac {c (1-a-b x)}{c-a c+b d}\right ) \log (d+c x)}{2 c^2}+\frac {d \log \left (-\frac {1-a-b x}{a+b x}\right ) \log (d+c x)}{2 c^2}+\frac {d \log \left (\frac {c (1+a+b x)}{c+a c-b d}\right ) \log (d+c x)}{2 c^2}-\frac {d \log \left (\frac {1+a+b x}{a+b x}\right ) \log (d+c x)}{2 c^2}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b (d+c x)}{c+a c-b d}\right )}{2 c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {b (d+c x)}{c-a c+b d}\right )}{2 c^2} \]
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Time = 0.40 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.23, number of steps used = 37, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6251, 2593, 2456, 2436, 2332, 2441, 2440, 2438, 199, 45} \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {d \operatorname {PolyLog}\left (2,\frac {c (-a-b x+1)}{-a c+c+b d}\right )}{2 c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {c (a+b x+1)}{a c+c-b d}\right )}{2 c^2}+\frac {d \log (a+b x-1) \log \left (\frac {b (c x+d)}{-a c+b d+c}\right )}{2 c^2}-\frac {d \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \log (c x+d)}{2 c^2}-\frac {d \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \log (c x+d)}{2 c^2}-\frac {d \log (a+b x+1) \log \left (-\frac {b (c x+d)}{a c-b d+c}\right )}{2 c^2}+\frac {(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac {x \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}+\frac {x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right )}{2 c} \]
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Rule 45
Rule 199
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 2593
Rule 6251
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {\log \left (\frac {-1+a+b x}{a+b x}\right )}{c+\frac {d}{x}} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {1+a+b x}{a+b x}\right )}{c+\frac {d}{x}} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\log (-1+a+b x)}{c+\frac {d}{x}} \, dx\right )+\frac {1}{2} \int \frac {\log (1+a+b x)}{c+\frac {d}{x}} \, dx-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac {1}{c+\frac {d}{x}} \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \frac {1}{c+\frac {d}{x}} \, dx \\ & = -\left (\frac {1}{2} \int \left (\frac {\log (-1+a+b x)}{c}-\frac {d \log (-1+a+b x)}{c (d+c x)}\right ) \, dx\right )+\frac {1}{2} \int \left (\frac {\log (1+a+b x)}{c}-\frac {d \log (1+a+b x)}{c (d+c x)}\right ) \, dx-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \frac {x}{d+c x} \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \frac {x}{d+c x} \, dx \\ & = -\frac {\int \log (-1+a+b x) \, dx}{2 c}+\frac {\int \log (1+a+b x) \, dx}{2 c}+\frac {d \int \frac {\log (-1+a+b x)}{d+c x} \, dx}{2 c}-\frac {d \int \frac {\log (1+a+b x)}{d+c x} \, dx}{2 c}-\frac {1}{2} \left (-\log (-1+a+b x)+\log \left (\frac {-1+a+b x}{a+b x}\right )+\log (a+b x)\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right ) \, dx+\frac {1}{2} \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right ) \, dx \\ & = \frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {d \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right ) \log (d+c x)}{2 c^2}-\frac {d \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}-\frac {d \log (1+a+b x) \log \left (-\frac {b (d+c x)}{c+a c-b d}\right )}{2 c^2}+\frac {d \log (-1+a+b x) \log \left (\frac {b (d+c x)}{c-a c+b d}\right )}{2 c^2}-\frac {\text {Subst}(\int \log (x) \, dx,x,-1+a+b x)}{2 b c}+\frac {\text {Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}-\frac {(b d) \int \frac {\log \left (\frac {b (d+c x)}{-((-1+a) c)+b d}\right )}{-1+a+b x} \, dx}{2 c^2}+\frac {(b d) \int \frac {\log \left (\frac {b (d+c x)}{-((1+a) c)+b d}\right )}{1+a+b x} \, dx}{2 c^2} \\ & = \frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {d \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right ) \log (d+c x)}{2 c^2}-\frac {d \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}-\frac {d \log (1+a+b x) \log \left (-\frac {b (d+c x)}{c+a c-b d}\right )}{2 c^2}+\frac {d \log (-1+a+b x) \log \left (\frac {b (d+c x)}{c-a c+b d}\right )}{2 c^2}-\frac {d \text {Subst}\left (\int \frac {\log \left (1+\frac {c x}{-((-1+a) c)+b d}\right )}{x} \, dx,x,-1+a+b x\right )}{2 c^2}+\frac {d \text {Subst}\left (\int \frac {\log \left (1+\frac {c x}{-((1+a) c)+b d}\right )}{x} \, dx,x,1+a+b x\right )}{2 c^2} \\ & = \frac {(1-a-b x) \log (-1+a+b x)}{2 b c}+\frac {x \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {(1+a+b x) \log (1+a+b x)}{2 b c}+\frac {x \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right )}{2 c}-\frac {d \left (\log (-1+a+b x)-\log \left (-\frac {1-a-b x}{a+b x}\right )-\log (a+b x)\right ) \log (d+c x)}{2 c^2}-\frac {d \left (\log (a+b x)-\log (1+a+b x)+\log \left (\frac {1+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}-\frac {d \log (1+a+b x) \log \left (-\frac {b (d+c x)}{c+a c-b d}\right )}{2 c^2}+\frac {d \log (-1+a+b x) \log \left (\frac {b (d+c x)}{c-a c+b d}\right )}{2 c^2}+\frac {d \operatorname {PolyLog}\left (2,\frac {c (1-a-b x)}{c-a c+b d}\right )}{2 c^2}-\frac {d \operatorname {PolyLog}\left (2,\frac {c (1+a+b x)}{c+a c-b d}\right )}{2 c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.76 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.74 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {2 a c^2 \coth ^{-1}(a+b x)-i b c d \pi \coth ^{-1}(a+b x)+2 b c^2 x \coth ^{-1}(a+b x)+b c d \coth ^{-1}(a+b x)^2+a b c d \coth ^{-1}(a+b x)^2-b^2 d^2 \coth ^{-1}(a+b x)^2-a b c d \sqrt {1-\frac {c^2}{(a c-b d)^2}} e^{\text {arctanh}\left (\frac {c}{a c-b d}\right )} \coth ^{-1}(a+b x)^2+b^2 d^2 \sqrt {1-\frac {c^2}{(a c-b d)^2}} e^{\text {arctanh}\left (\frac {c}{a c-b d}\right )} \coth ^{-1}(a+b x)^2+2 b c d \coth ^{-1}(a+b x) \text {arctanh}\left (\frac {c}{a c-b d}\right )+2 b c d \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )+i b c d \pi \log \left (1+e^{2 \coth ^{-1}(a+b x)}\right )-2 b c d \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {c}{a c-b d}\right )}\right )+2 b c d \text {arctanh}\left (\frac {c}{a c-b d}\right ) \log \left (1-e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {c}{a c-b d}\right )}\right )-2 c^2 \log \left (\frac {1}{a+b x}\right )-2 c^2 \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )-i b c d \pi \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )-2 b c d \text {arctanh}\left (\frac {c}{a c-b d}\right ) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\text {arctanh}\left (\frac {c}{a c-b d}\right )\right )\right )-b c d \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )+b c d \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)+2 \text {arctanh}\left (\frac {c}{a c-b d}\right )}\right )}{2 b c^3} \]
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Time = 1.16 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.89
method | result | size |
parts | \(\frac {\operatorname {arccoth}\left (b x +a \right ) x}{c}-\frac {\operatorname {arccoth}\left (b x +a \right ) d \ln \left (c x +d \right )}{c^{2}}+\frac {b \left (-\frac {\left (-1-a \right ) \ln \left (a c -d b +b \left (c x +d \right )+c \right )}{2 b^{2}}-\frac {\left (-1+a \right ) \ln \left (a c -d b +b \left (c x +d \right )-c \right )}{2 b^{2}}-d \left (-\frac {\frac {\operatorname {dilog}\left (\frac {a c -d b +b \left (c x +d \right )+c}{a c -d b +c}\right )}{b}+\frac {\ln \left (c x +d \right ) \ln \left (\frac {a c -d b +b \left (c x +d \right )+c}{a c -d b +c}\right )}{b}}{2 c}+\frac {\frac {\operatorname {dilog}\left (\frac {a c -d b +b \left (c x +d \right )-c}{a c -d b -c}\right )}{b}+\frac {\ln \left (c x +d \right ) \ln \left (\frac {a c -d b +b \left (c x +d \right )-c}{a c -d b -c}\right )}{b}}{2 c}\right )\right )}{c}\) | \(259\) |
risch | \(\frac {\ln \left (b x +a +1\right ) x}{2 c}+\frac {\ln \left (b x +a +1\right ) a}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c}-\frac {1}{b c}-\frac {d \operatorname {dilog}\left (\frac {\left (b x +a +1\right ) c -a c +d b -c}{-a c +d b -c}\right )}{2 c^{2}}-\frac {d \ln \left (b x +a +1\right ) \ln \left (\frac {\left (b x +a +1\right ) c -a c +d b -c}{-a c +d b -c}\right )}{2 c^{2}}-\frac {\ln \left (b x +a -1\right ) x}{2 c}-\frac {\ln \left (b x +a -1\right ) a}{2 b c}+\frac {\ln \left (b x +a -1\right )}{2 b c}+\frac {d \operatorname {dilog}\left (\frac {\left (b x +a -1\right ) c -a c +d b +c}{-a c +d b +c}\right )}{2 c^{2}}+\frac {d \ln \left (b x +a -1\right ) \ln \left (\frac {\left (b x +a -1\right ) c -a c +d b +c}{-a c +d b +c}\right )}{2 c^{2}}\) | \(264\) |
derivativedivides | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arccoth}\left (b x +a \right ) d b \ln \left (a c -d b -c \left (b x +a \right )\right )}{c^{2}}-\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -d b -c \left (b x +a \right )\right )+2 b d \left (a c -d b -c \left (b x +a \right )\right )-c^{2}+\left (a c -d b -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )-c}{-a c +d b -c}\right )+\ln \left (a c -d b -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )-c}{-a c +d b -c}\right )}{2 c}-\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )+c}{-a c +d b +c}\right )+\ln \left (a c -d b -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )+c}{-a c +d b +c}\right )}{2 c}\right )}{c}}{b}\) | \(297\) |
default | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )}{c}-\frac {\operatorname {arccoth}\left (b x +a \right ) d b \ln \left (a c -d b -c \left (b x +a \right )\right )}{c^{2}}-\frac {-\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}-2 a c \left (a c -d b -c \left (b x +a \right )\right )+2 b d \left (a c -d b -c \left (b x +a \right )\right )-c^{2}+\left (a c -d b -c \left (b x +a \right )\right )^{2}\right )}{2}-b d \left (\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )-c}{-a c +d b -c}\right )+\ln \left (a c -d b -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )-c}{-a c +d b -c}\right )}{2 c}-\frac {\operatorname {dilog}\left (\frac {-c \left (b x +a \right )+c}{-a c +d b +c}\right )+\ln \left (a c -d b -c \left (b x +a \right )\right ) \ln \left (\frac {-c \left (b x +a \right )+c}{-a c +d b +c}\right )}{2 c}\right )}{c}}{b}\) | \(297\) |
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {x \operatorname {acoth}{\left (a + b x \right )}}{c x + d}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.66 \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\frac {1}{2} \, b {\left (\frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d + c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d + c}\right )\right )} d}{b c^{2}} - \frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d - c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d - c}\right )\right )} d}{b c^{2}} + \frac {{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2} c} - \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2} c}\right )} + {\left (\frac {x}{c} - \frac {d \log \left (c x + d\right )}{c^{2}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
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\[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{x}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+\frac {d}{x}} \,d x \]
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