Integrand size = 12, antiderivative size = 97 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d} \]
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Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6239, 6022, 6132, 6056, 2449, 2352} \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d}-\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{d} \]
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Rule 2352
Rule 2449
Rule 6022
Rule 6056
Rule 6132
Rule 6239
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d} \\ & = \frac {\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac {2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.14 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 (-1+c+d x) \coth ^{-1}(c+d x)^2+2 b \coth ^{-1}(c+d x) \left (a c+a d x-b \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+a \left (a c+a d x-2 b \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )\right )+b^2 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{d} \]
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Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.79
method | result | size |
parts | \(a^{2} x +\frac {b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {2 a b \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\) | \(174\) |
derivativedivides | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+2 a b \left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+a b \ln \left (\left (d x +c \right )^{2}-1\right )}{d}\) | \(175\) |
default | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+2 a b \left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+a b \ln \left (\left (d x +c \right )^{2}-1\right )}{d}\) | \(175\) |
risch | \(a^{2} x -\frac {b^{2} \ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{d}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} c}{4 d}-\ln \left (d x +c -1\right ) a b x +\frac {\ln \left (d x +c -1\right ) a b}{d}+\frac {a b \ln \left (d x +c +1\right )}{d}+\frac {a^{2} c}{d}+\left (-\frac {b^{2} x \ln \left (d x +c -1\right )}{2}+\frac {b \left (2 a d x -b \ln \left (d x +c -1\right ) c +b \ln \left (d x +c -1\right )\right )}{2 d}\right ) \ln \left (d x +c +1\right )+\frac {\left (d x +c +1\right ) b^{2} \ln \left (d x +c +1\right )^{2}}{4 d}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} x}{4}-\frac {\ln \left (d x +c -1\right )^{2} b^{2}}{4 d}-\frac {b^{2} \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{d}-\frac {a^{2}}{d}+\frac {a b \ln \left (d x +c +1\right ) c}{d}-\frac {\ln \left (d x +c -1\right ) a b c}{d}\) | \(260\) |
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2}\, dx \]
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int {\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \]
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