Optimal. Leaf size=292 \[ \frac {d \text {Li}_2\left (-\frac {b (d+c x)}{a c+c-b d}\right )}{2 c^2}-\frac {d \text {Li}_2\left (\frac {b (d+c x)}{-a c+c+b d}\right )}{2 c^2}+\frac {d \log \left (-\frac {-a-b x+1}{a+b x}\right ) \log (c x+d)}{2 c^2}-\frac {d \log (c x+d) \log \left (\frac {c (-a-b x+1)}{-a c+b d+c}\right )}{2 c^2}+\frac {d \log (c x+d) \log \left (\frac {c (a+b x+1)}{a c-b d+c}\right )}{2 c^2}-\frac {d \log \left (\frac {a+b x+1}{a+b x}\right ) \log (c x+d)}{2 c^2}+\frac {(-a-b x+1) \log \left (-\frac {-a-b x+1}{a+b x}\right )}{2 b c}+\frac {\log (a+b x)}{2 b c}+\frac {\log (a+b x+1)}{2 b c}+\frac {(a+b x) \log \left (\frac {a+b x+1}{a+b x}\right )}{2 b c} \]
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Rubi [A] time = 0.50, antiderivative size = 360, normalized size of antiderivative = 1.23, number of steps used = 37, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6116, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 43} \[ \frac {d \text {PolyLog}\left (2,\frac {c (-a-b x+1)}{-a c+b d+c}\right )}{2 c^2}-\frac {d \text {PolyLog}\left (2,\frac {c (a+b x+1)}{a c-b d+c}\right )}{2 c^2}+\frac {d \log (a+b x-1) \log \left (\frac {b (c x+d)}{-a c+b d+c}\right )}{2 c^2}-\frac {d \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \log (c x+d)}{2 c^2}-\frac {d \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \log (c x+d)}{2 c^2}-\frac {d \log (a+b x+1) \log \left (-\frac {b (c x+d)}{a c-b d+c}\right )}{2 c^2}+\frac {(-a-b x+1) \log (a+b x-1)}{2 b c}+\frac {x \left (\log (a+b x-1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right )}{2 c}+\frac {(a+b x+1) \log (a+b x+1)}{2 b c}+\frac {x \left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right )}{2 c} \]
Warning: Unable to verify antiderivative.
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Rule 43
Rule 193
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2513
Rule 6116
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx &=-\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 {\operatorname {Subst}(\int \log (x) \, dx,x,-1+a+b x)}{2 b c}+\frac {\operatorname {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 \operatorname {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 \operatorname {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 \text {Li}_2\left (\frac {c (1-a-b x)}{c-a c+b d}\right )}{2 c^2}-\frac {d \text {Li}_2\left (\frac {c (1+a+b x)}{c+a c-b d}\right )}{2 c^2}\\ \end {align*}
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Mathematica [C] time = 4.49, size = 502, normalized size = 1.72 \[ \frac {b^2 d^2 \sqrt {1-\frac {c^2}{(a c-b d)^2}} \coth ^{-1}(a+b x)^2 e^{\tanh ^{-1}\left (\frac {c}{a c-b d}\right )}-b^2 d^2 \coth ^{-1}(a+b x)^2-a b c d \sqrt {1-\frac {c^2}{(a c-b d)^2}} \coth ^{-1}(a+b x)^2 e^{\tanh ^{-1}\left (\frac {c}{a c-b d}\right )}-2 c^2 \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )+2 a c^2 \coth ^{-1}(a+b x)+2 b c^2 x \coth ^{-1}(a+b x)+b c d \text {Li}_2\left (\exp \left (2 \tanh ^{-1}\left (\frac {c}{a c-b d}\right )-2 \coth ^{-1}(a+b x)\right )\right )-2 b c d \coth ^{-1}(a+b x) \log \left (1-\exp \left (2 \tanh ^{-1}\left (\frac {c}{a c-b d}\right )-2 \coth ^{-1}(a+b x)\right )\right )+2 b c d \tanh ^{-1}\left (\frac {c}{a c-b d}\right ) \log \left (1-\exp \left (2 \tanh ^{-1}\left (\frac {c}{a c-b d}\right )-2 \coth ^{-1}(a+b x)\right )\right )-b c d \text {Li}_2\left (e^{-2 \coth ^{-1}(a+b x)}\right )-i \pi b c d \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right )+a b c d \coth ^{-1}(a+b x)^2+b c d \coth ^{-1}(a+b x)^2-i \pi b c d \coth ^{-1}(a+b x)+2 b c d \coth ^{-1}(a+b x) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )+i \pi b c d \log \left (e^{2 \coth ^{-1}(a+b x)}+1\right )+2 b c d \coth ^{-1}(a+b x) \tanh ^{-1}\left (\frac {c}{a c-b d}\right )-2 b c d \tanh ^{-1}\left (\frac {c}{a c-b d}\right ) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac {c}{a c-b d}\right )\right )\right )}{2 b c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \operatorname {arcoth}\left (b x + a\right )}{c x + d}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{c + \frac {d}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 297, normalized size = 1.02 \[ \frac {\mathrm {arccoth}\left (b x +a \right ) x}{c}+\frac {\mathrm {arccoth}\left (b x +a \right ) a}{b c}-\frac {\mathrm {arccoth}\left (b x +a \right ) d \ln \left (c \left (b x +a \right )-a c +b d \right )}{c^{2}}+\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}+2 \left (c \left (b x +a \right )-a c +b d \right ) a c -2 \left (c \left (b x +a \right )-a c +b d \right ) b d +\left (c \left (b x +a \right )-a c +b d \right )^{2}-c^{2}\right )}{2 b c}-\frac {d \ln \left (c \left (b x +a \right )-a c +b d \right ) \ln \left (\frac {c \left (b x +a \right )-c}{a c -b d -c}\right )}{2 c^{2}}-\frac {d \dilog \left (\frac {c \left (b x +a \right )-c}{a c -b d -c}\right )}{2 c^{2}}+\frac {d \ln \left (c \left (b x +a \right )-a c +b d \right ) \ln \left (\frac {c \left (b x +a \right )+c}{a c -b d +c}\right )}{2 c^{2}}+\frac {d \dilog \left (\frac {c \left (b x +a \right )+c}{a c -b d +c}\right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 192, normalized size = 0.66 \[ \frac {1}{2} \, b {\left (\frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d + c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d + c}\right )\right )} d}{b c^{2}} - \frac {{\left (\log \left (c x + d\right ) \log \left (\frac {b c x + b d}{a c - b d - c} + 1\right ) + {\rm Li}_2\left (-\frac {b c x + b d}{a c - b d - c}\right )\right )} d}{b c^{2}} + \frac {{\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{2} c} - \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{2} c}\right )} + {\left (\frac {x}{c} - \frac {d \log \left (c x + d\right )}{c^{2}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+\frac {d}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \operatorname {acoth}{\left (a + b x \right )}}{c x + d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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