Optimal. Leaf size=168 \[ \frac {\text {Li}_2\left (1-\frac {2}{b f^{c+d x}+a+1}\right )}{2 d \log (f)}-\frac {\text {Li}_2\left (1-\frac {2 b f^{c+d x}}{(1-a) \left (b f^{c+d x}+a+1\right )}\right )}{2 d \log (f)}-\frac {\log \left (\frac {2}{a+b f^{c+d x}+1}\right ) \tanh ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)}+\frac {\log \left (\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right ) \tanh ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)} \]
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Rubi [A] time = 0.13, 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 = {2282, 6111, 5920, 2402, 2315, 2447} \[ \frac {\text {PolyLog}\left (2,1-\frac {2}{a+b f^{c+d x}+1}\right )}{2 d \log (f)}-\frac {\text {PolyLog}\left (2,1-\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right )}{2 d \log (f)}-\frac {\log \left (\frac {2}{a+b f^{c+d x}+1}\right ) \tanh ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)}+\frac {\log \left (\frac {2 b f^{c+d x}}{(1-a) \left (a+b f^{c+d x}+1\right )}\right ) \tanh ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2315
Rule 2402
Rule 2447
Rule 5920
Rule 6111
Rubi steps
\begin {align*} \int \tanh ^{-1}\left (a+b f^{c+d x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\tanh ^{-1}(a+b x)}{x} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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*}
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Mathematica [A] time = 0.08, size = 108, normalized size = 0.64 \[ \frac {\text {Li}_2\left (-\frac {b f^{c+d x}}{a-1}\right )-\text {Li}_2\left (-\frac {b f^{c+d x}}{a+1}\right )+d x \log (f) \left (\log \left (\frac {a+b f^{c+d x}-1}{a-1}\right )-\log \left (\frac {a+b f^{c+d x}+1}{a+1}\right )+2 \tanh ^{-1}\left (a+b f^{c+d x}\right )\right )}{2 d \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 284, normalized size = 1.69 \[ \frac {d x \log \relax (f) \log \left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a + 1}{b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a - 1}\right ) + c \log \left (b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a + 1\right ) \log \relax (f) - c \log \left (b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a - 1\right ) \log \relax (f) - {\left (d x + c\right )} \log \relax (f) \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a + 1}{a + 1}\right ) + {\left (d x + c\right )} \log \relax (f) \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a - 1}{a - 1}\right ) - {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a + 1}{a + 1} + 1\right ) + {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \relax (f)\right ) + b \sinh \left ({\left (d x + c\right )} \log \relax (f)\right ) + a - 1}{a - 1} + 1\right )}{2 \, d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 164, normalized size = 0.98 \[ \frac {\ln \left (b \,f^{d x +c}\right ) \arctanh \left (a +b \,f^{d x +c}\right )}{d \ln \relax (f )}-\frac {\dilog \left (\frac {1+a +b \,f^{d x +c}}{1+a}\right )}{2 d \ln \relax (f )}-\frac {\ln \left (b \,f^{d x +c}\right ) \ln \left (\frac {1+a +b \,f^{d x +c}}{1+a}\right )}{2 d \ln \relax (f )}+\frac {\dilog \left (\frac {b \,f^{d x +c}+a -1}{a -1}\right )}{2 d \ln \relax (f )}+\frac {\ln \left (b \,f^{d x +c}\right ) \ln \left (\frac {b \,f^{d x +c}+a -1}{a -1}\right )}{2 d \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 202, normalized size = 1.20 \[ \frac {{\left (d x + c\right )} \operatorname {artanh}\left (b f^{d x + c} + a\right )}{d} - \frac {{\left (d x + c\right )} b {\left (\frac {\log \left (b f^{d x + c} + a + 1\right )}{b} - \frac {\log \left (b f^{d x + c} + a - 1\right )}{b}\right )} \log \relax (f) - b {\left (\frac {\log \left (b f^{d x + c} + a + 1\right ) \log \left (-\frac {b f^{d x + c} + a + 1}{a + 1} + 1\right ) + {\rm Li}_2\left (\frac {b f^{d x + c} + a + 1}{a + 1}\right )}{b} - \frac {\log \left (b f^{d x + c} + a - 1\right ) \log \left (-\frac {b f^{d x + c} + a - 1}{a - 1} + 1\right ) + {\rm Li}_2\left (\frac {b f^{d x + c} + a - 1}{a - 1}\right )}{b}\right )}}{2 \, d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {atanh}\left (a+b\,f^{c+d\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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