Optimal. Leaf size=216 \[ -\frac{\text{PolyLog}\left (3,\frac{b f^{c+d x}}{1-a}\right )}{2 d^2 \log ^2(f)}+\frac{\text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a+1}\right )}{2 d^2 \log ^2(f)}+\frac{x \text{PolyLog}\left (2,\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a+1}\right )}{2 d \log (f)}+\frac{1}{4} x^2 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{4} x^2 \log \left (\frac{b f^{c+d x}}{a+1}+1\right )-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (\frac{1}{a+b f^{c+d x}}+1\right ) \]
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Rubi [A] time = 2.71421, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6214, 2551, 12, 6742, 2190, 2531, 2282, 6589} \[ -\frac{\text{PolyLog}\left (3,\frac{b f^{c+d x}}{1-a}\right )}{2 d^2 \log ^2(f)}+\frac{\text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a+1}\right )}{2 d^2 \log ^2(f)}+\frac{x \text{PolyLog}\left (2,\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a+1}\right )}{2 d \log (f)}+\frac{1}{4} x^2 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{4} x^2 \log \left (\frac{b f^{c+d x}}{a+1}+1\right )-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (\frac{1}{a+b f^{c+d x}}+1\right ) \]
Antiderivative was successfully verified.
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Rule 6214
Rule 2551
Rule 12
Rule 6742
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx &=-\left (\frac{1}{2} \int x \log \left (1-\frac{1}{a+b f^{c+d x}}\right ) \, dx\right )+\frac{1}{2} \int x \log \left (1+\frac{1}{a+b f^{c+d x}}\right ) \, dx\\ &=-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} \int \frac{b d f^{c+d x} x^2 \log (f)}{\left (-1+a+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx-\frac{1}{4} \int \frac{b d f^{c+d x} x^2 \log (f)}{\left (-a-b f^{c+d x}\right ) \left (1+a+b f^{c+d x}\right )} \, dx\\ &=-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} (b d \log (f)) \int \frac{f^{c+d x} x^2}{\left (-1+a+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx-\frac{1}{4} (b d \log (f)) \int \frac{f^{c+d x} x^2}{\left (-a-b f^{c+d x}\right ) \left (1+a+b f^{c+d x}\right )} \, dx\\ &=-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} (b d \log (f)) \int \left (\frac{f^{c+d x} x^2}{-a-b f^{c+d x}}+\frac{f^{c+d x} x^2}{-1+a+b f^{c+d x}}\right ) \, dx-\frac{1}{4} (b d \log (f)) \int \left (\frac{f^{c+d x} x^2}{-a-b f^{c+d x}}+\frac{f^{c+d x} x^2}{1+a+b f^{c+d x}}\right ) \, dx\\ &=-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} (b d \log (f)) \int \frac{f^{c+d x} x^2}{-1+a+b f^{c+d x}} \, dx-\frac{1}{4} (b d \log (f)) \int \frac{f^{c+d x} x^2}{1+a+b f^{c+d x}} \, dx\\ &=\frac{1}{4} x^2 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{4} x^2 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )-\frac{1}{2} \int x \log \left (1+\frac{b f^{c+d x}}{-1+a}\right ) \, dx+\frac{1}{2} \int x \log \left (1+\frac{b f^{c+d x}}{1+a}\right ) \, dx\\ &=\frac{1}{4} x^2 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{4} x^2 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{x \text{Li}_2\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac{\int \text{Li}_2\left (-\frac{b f^{c+d x}}{-1+a}\right ) \, dx}{2 d \log (f)}+\frac{\int \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right ) \, dx}{2 d \log (f)}\\ &=\frac{1}{4} x^2 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{4} x^2 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{x \text{Li}_2\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-1+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{2 d^2 \log ^2(f)}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{1+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{2 d^2 \log ^2(f)}\\ &=\frac{1}{4} x^2 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{4} x^2 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{4} x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{4} x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{x \text{Li}_2\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac{\text{Li}_3\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d^2 \log ^2(f)}+\frac{\text{Li}_3\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d^2 \log ^2(f)}\\ \end{align*}
Mathematica [A] time = 0.104145, size = 177, normalized size = 0.82 \[ \frac{-2 \text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a-1}\right )+2 \text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a+1}\right )+2 d x \log (f) \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a-1}\right )-2 d x \log (f) \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a+1}\right )+d^2 x^2 \log ^2(f) \log \left (\frac{b f^{c+d x}}{a-1}+1\right )-d^2 x^2 \log ^2(f) \log \left (\frac{b f^{c+d x}}{a+1}+1\right )+2 d^2 x^2 \log ^2(f) \coth ^{-1}\left (a+b f^{c+d x}\right )}{4 d^2 \log ^2(f)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.174, size = 590, normalized size = 2.7 \begin{align*} -{\frac{{x}^{2}\ln \left ( b{f}^{dx+c}+a-1 \right ) }{4}}+{\frac{{x}^{2}\ln \left ( 1+a+b{f}^{dx+c} \right ) }{4}}-{\frac{{x}^{2}}{4}\ln \left ( 1-{\frac{b{f}^{dx}{f}^{c}}{-1-a}} \right ) }-{\frac{cx}{2\,d}\ln \left ( 1-{\frac{b{f}^{dx}{f}^{c}}{-1-a}} \right ) }-{\frac{{c}^{2}}{4\,{d}^{2}}\ln \left ( 1-{\frac{b{f}^{dx}{f}^{c}}{-1-a}} \right ) }-{\frac{x}{2\,d\ln \left ( f \right ) }{\it polylog} \left ( 2,{\frac{b{f}^{dx}{f}^{c}}{-1-a}} \right ) }-{\frac{c}{2\,\ln \left ( f \right ){d}^{2}}{\it polylog} \left ( 2,{\frac{b{f}^{dx}{f}^{c}}{-1-a}} \right ) }+{\frac{1}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}{\it polylog} \left ( 3,{\frac{b{f}^{dx}{f}^{c}}{-1-a}} \right ) }-{\frac{{c}^{2}\ln \left ( 1+a+b{f}^{dx}{f}^{c} \right ) }{4\,{d}^{2}}}+{\frac{c}{2\,\ln \left ( f \right ){d}^{2}}{\it dilog} \left ({\frac{1+a+b{f}^{dx}{f}^{c}}{1+a}} \right ) }+{\frac{cx}{2\,d}\ln \left ({\frac{1+a+b{f}^{dx}{f}^{c}}{1+a}} \right ) }+{\frac{{c}^{2}}{2\,{d}^{2}}\ln \left ({\frac{1+a+b{f}^{dx}{f}^{c}}{1+a}} \right ) }+{\frac{{x}^{2}}{4}\ln \left ( 1-{\frac{b{f}^{dx}{f}^{c}}{1-a}} \right ) }+{\frac{cx}{2\,d}\ln \left ( 1-{\frac{b{f}^{dx}{f}^{c}}{1-a}} \right ) }+{\frac{{c}^{2}}{4\,{d}^{2}}\ln \left ( 1-{\frac{b{f}^{dx}{f}^{c}}{1-a}} \right ) }+{\frac{x}{2\,d\ln \left ( f \right ) }{\it polylog} \left ( 2,{\frac{b{f}^{dx}{f}^{c}}{1-a}} \right ) }+{\frac{c}{2\,\ln \left ( f \right ){d}^{2}}{\it polylog} \left ( 2,{\frac{b{f}^{dx}{f}^{c}}{1-a}} \right ) }-{\frac{1}{2\, \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}{\it polylog} \left ( 3,{\frac{b{f}^{dx}{f}^{c}}{1-a}} \right ) }+{\frac{{c}^{2}\ln \left ( b{f}^{dx}{f}^{c}+a-1 \right ) }{4\,{d}^{2}}}-{\frac{c}{2\,\ln \left ( f \right ){d}^{2}}{\it dilog} \left ({\frac{b{f}^{dx}{f}^{c}+a-1}{a-1}} \right ) }-{\frac{cx}{2\,d}\ln \left ({\frac{b{f}^{dx}{f}^{c}+a-1}{a-1}} \right ) }-{\frac{{c}^{2}}{2\,{d}^{2}}\ln \left ({\frac{b{f}^{dx}{f}^{c}+a-1}{a-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28441, size = 262, normalized size = 1.21 \begin{align*} -\frac{1}{4} \, b d{\left (\frac{\log \left (\frac{b f^{d x} f^{c}}{a + 1} + 1\right ) \log \left (f^{d x}\right )^{2} + 2 \,{\rm Li}_2\left (-\frac{b f^{d x} f^{c}}{a + 1}\right ) \log \left (f^{d x}\right ) - 2 \,{\rm Li}_{3}(-\frac{b f^{d x} f^{c}}{a + 1})}{b d^{3} \log \left (f\right )^{3}} - \frac{\log \left (\frac{b f^{d x} f^{c}}{a - 1} + 1\right ) \log \left (f^{d x}\right )^{2} + 2 \,{\rm Li}_2\left (-\frac{b f^{d x} f^{c}}{a - 1}\right ) \log \left (f^{d x}\right ) - 2 \,{\rm Li}_{3}(-\frac{b f^{d x} f^{c}}{a - 1})}{b d^{3} \log \left (f\right )^{3}}\right )} \log \left (f\right ) + \frac{1}{2} \, x^{2} \operatorname{arcoth}\left (b f^{d x + c} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.6202, size = 1193, normalized size = 5.52 \begin{align*} \frac{d^{2} x^{2} \log \left (f\right )^{2} \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^{2} \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 )^{2} + c^{2} \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 )^{2} - 2 \, d x{\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 ) \log \left (f\right ) + 2 \, d x{\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 ) \log \left (f\right ) -{\left (d^{2} x^{2} - c^{2}\right )} \log \left (f\right )^{2} \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^{2} x^{2} - c^{2}\right )} \log \left (f\right )^{2} \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 ) + 2 \,{\rm polylog}\left (3, -\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}\right ) - 2 \,{\rm polylog}\left (3, -\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}\right )}{4 \, d^{2} \log \left (f\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arcoth}\left (b f^{d x + c} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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