Optimal. Leaf size=168 \[ -\frac {\tanh ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2}{1+a+b f^{c+d x}}\right )}{d \log (f)}+\frac {\tanh ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac {\text {PolyLog}\left (2,1-\frac {2}{1+a+b f^{c+d x}}\right )}{2 d \log (f)}-\frac {\text {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (1+a+b f^{c+d x}\right )}\right )}{2 d \log (f)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 6246,
6057, 2449, 2352, 2497} \begin {gather*} \frac {\text {Li}_2\left (1-\frac {2}{b f^{c+d x}+a+1}\right )}{2 d \log (f)}-\frac {\text {Li}_2\left (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 ) \tanh ^{-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 ) \tanh ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2320
Rule 2352
Rule 2449
Rule 2497
Rule 6057
Rule 6246
Rubi steps
\begin {align*} \int \tanh ^{-1}\left (a+b f^{c+d x}\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {\tanh ^{-1}(a+b x)}{x} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=\frac {\text {Subst}\left (\int \frac {\tanh ^{-1}(x)}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b f^{c+d x}\right )}{b d \log (f)}\\ &=-\frac {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {\tanh ^{-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 {Li}_2\left (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 {\tanh ^{-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 {\tanh ^{-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 {Li}_2\left (1-\frac {2}{1+a+b f^{c+d x}}\right )}{2 d \log (f)}-\frac {\text {Li}_2\left (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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.50, size = 108, normalized size = 0.64 \begin {gather*} \frac {d x \log (f) \left (2 \tanh ^{-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 )+\text {PolyLog}\left (2,-\frac {b f^{c+d x}}{-1+a}\right )-\text {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.31, size = 160, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\ln \left (-b \,f^{d x +c}\right ) \arctanh \left (a +b \,f^{d x +c}\right )-\frac {\dilog \left (\frac {-b \,f^{d x +c}-a -1}{-a -1}\right )}{2}-\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {-b \,f^{d x +c}-a -1}{-a -1}\right )}{2}+\frac {\dilog \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}+\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}}{d \ln \left (f \right )}\) | \(160\) |
default | \(\frac {\ln \left (-b \,f^{d x +c}\right ) \arctanh \left (a +b \,f^{d x +c}\right )-\frac {\dilog \left (\frac {-b \,f^{d x +c}-a -1}{-a -1}\right )}{2}-\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {-b \,f^{d x +c}-a -1}{-a -1}\right )}{2}+\frac {\dilog \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}+\frac {\ln \left (-b \,f^{d x +c}\right ) \ln \left (\frac {1-a -b \,f^{d x +c}}{1-a}\right )}{2}}{d \ln \left (f \right )}\) | \(160\) |
risch | \(\frac {x \ln \left (1+a +b \,f^{d x +c}\right )}{2}-\frac {\dilog \left (\frac {1+a +b \,f^{d x} f^{c}}{1+a}\right )}{2 \ln \left (f \right ) d}-\frac {\ln \left (\frac {1+a +b \,f^{d x} f^{c}}{1+a}\right ) x}{2}-\frac {\ln \left (\frac {1+a +b \,f^{d x} f^{c}}{1+a}\right ) c}{2 d}+\frac {c \ln \left (1+a +b \,f^{d x} f^{c}\right )}{2 d}-\frac {\ln \left (1-a -b \,f^{d x +c}\right ) \ln \left (-\frac {b \,f^{d x +c}}{-1+a}\right )}{2 d \ln \left (f \right )}-\frac {\dilog \left (-\frac {b \,f^{d x +c}}{-1+a}\right )}{2 d \ln \left (f \right )}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 202, normalized size = 1.20 \begin {gather*} \frac {{\left (d x + c\right )} \operatorname {artanh}\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 284, normalized size = 1.69 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {atanh}{\left (a + b f^{c + d x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {atanh}\left (a+b\,f^{c+d\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________