Integrand size = 12, antiderivative size = 168 \[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=-\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{1+a+b f^{c+d x}}\right )}{d \log (f)}+\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b f^{c+d x}}\right )}{2 d \log (f)}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{2 d \log (f)} \]
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Time = 0.10 (sec) , antiderivative size = 168, 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 = {2320, 6247, 6058, 2449, 2352, 2497} \[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{b f^{c+d x}+a+1}\right )}{2 d \log (f)}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (b f^{c+d x}+a+1\right )}\right )}{2 d \log (f)}-\frac {\log \left (\frac {2}{a+b f^{c+d x}+1}\right ) \coth ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)}+\frac {\log \left (\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right ) \coth ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)} \]
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Rule 2320
Rule 2352
Rule 2449
Rule 2497
Rule 6058
Rule 6247
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\coth ^{-1}(a+b x)}{x} \, dx,x,f^{c+d x}\right )}{d \log (f)} \\ & = \frac {\text {Subst}\left (\int \frac {\coth ^{-1}(x)}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b f^{c+d x}\right )}{b d \log (f)} \\ & = -\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{1+a+b f^{c+d x}}\right )}{d \log (f)}+\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,a+b f^{c+d x}\right )}{d \log (f)}-\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2 \left (-\frac {a}{b}+\frac {x}{b}\right )}{\left (\frac {1}{b}-\frac {a}{b}\right ) (1+x)}\right )}{1-x^2} \, dx,x,a+b f^{c+d x}\right )}{d \log (f)} \\ & = -\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{1+a+b f^{c+d x}}\right )}{d \log (f)}+\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{d \log (f)}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{2 d \log (f)}+\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+a+b f^{c+d x}}\right )}{d \log (f)} \\ & = -\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{1+a+b f^{c+d x}}\right )}{d \log (f)}+\frac {\coth ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b f^{c+d x}}\right )}{2 d \log (f)}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{2 d \log (f)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.64 \[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {d x \log (f) \left (2 \coth ^{-1}\left (a+b f^{c+d x}\right )+\log \left (\frac {-1+a+b f^{c+d x}}{-1+a}\right )-\log \left (\frac {1+a+b f^{c+d x}}{1+a}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{-1+a}\right )-\operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)} \]
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Time = 1.16 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\ln \left (-b \,f^{d x +c}\right ) \operatorname {arccoth}\left (a +b \,f^{d x +c}\right )-\frac {\operatorname {dilog}\left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}-\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}+\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}}{d \ln \left (f \right )}\) | \(160\) |
default | \(\frac {\ln \left (-b \,f^{d x +c}\right ) \operatorname {arccoth}\left (a +b \,f^{d x +c}\right )-\frac {\operatorname {dilog}\left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}-\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {-b \,f^{d x +c}-a -1}{-1-a}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}+\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}}{d \ln \left (f \right )}\) | \(160\) |
risch | \(-\frac {x \ln \left (b \,f^{d x +c}+a -1\right )}{2}+\frac {\operatorname {dilog}\left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right )}{2 \ln \left (f \right ) d}+\frac {\ln \left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right ) x}{2}+\frac {c \ln \left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right )}{2 d}-\frac {c \ln \left (f^{d x} f^{c} b +a -1\right )}{2 d}+\frac {\ln \left (1+a +b \,f^{d x +c}\right ) \ln \left (\frac {f^{d x +c} b}{-1-a}\right )}{2 d \ln \left (f \right )}+\frac {\operatorname {dilog}\left (\frac {f^{d x +c} b}{-1-a}\right )}{2 d \ln \left (f \right )}\) | \(181\) |
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Time = 0.26 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.68 \[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {d x \log \left (f\right ) \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}\right ) + c \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1\right ) \log \left (f\right ) - c \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1\right ) \log \left (f\right ) - {\left (d x + c\right )} \log \left (f\right ) \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1}\right ) + {\left (d x + c\right )} \log \left (f\right ) \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1}\right ) - {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1} + 1\right ) + {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1} + 1\right )}{2 \, d \log \left (f\right )} \]
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\[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int \operatorname {acoth}{\left (a + b f^{c + d x} \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.20 \[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {{\left (d x + c\right )} \operatorname {arcoth}\left (b f^{d x + c} + a\right )}{d} - \frac {{\left (d x + c\right )} b {\left (\frac {\log \left (b f^{d x + c} + a + 1\right )}{b} - \frac {\log \left (b f^{d x + c} + a - 1\right )}{b}\right )} \log \left (f\right ) - b {\left (\frac {\log \left (b f^{d x + c} + a + 1\right ) \log \left (-\frac {b f^{d x + c} + a + 1}{a + 1} + 1\right ) + {\rm Li}_2\left (\frac {b f^{d x + c} + a + 1}{a + 1}\right )}{b} - \frac {\log \left (b f^{d x + c} + a - 1\right ) \log \left (-\frac {b f^{d x + c} + a - 1}{a - 1} + 1\right ) + {\rm Li}_2\left (\frac {b f^{d x + c} + a - 1}{a - 1}\right )}{b}\right )}}{2 \, d \log \left (f\right )} \]
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Exception generated. \[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int \mathrm {acoth}\left (a+b\,f^{c+d\,x}\right ) \,d x \]
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