Optimal. Leaf size=269 \[ -\frac{x \text{PolyLog}\left (3,\frac{b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac{x \text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a+1}\right )}{d^2 \log ^2(f)}+\frac{\text{PolyLog}\left (4,\frac{b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac{\text{PolyLog}\left (4,-\frac{b f^{c+d x}}{a+1}\right )}{d^3 \log ^3(f)}+\frac{x^2 \text{PolyLog}\left (2,\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x^2 \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a+1}\right )}{2 d \log (f)}+\frac{1}{6} x^3 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{6} x^3 \log \left (\frac{b f^{c+d x}}{a+1}+1\right )-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (\frac{1}{a+b f^{c+d x}}+1\right ) \]
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Rubi [A] time = 2.58565, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {6214, 2551, 12, 6742, 2190, 2531, 6609, 2282, 6589} \[ -\frac{x \text{PolyLog}\left (3,\frac{b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac{x \text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a+1}\right )}{d^2 \log ^2(f)}+\frac{\text{PolyLog}\left (4,\frac{b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac{\text{PolyLog}\left (4,-\frac{b f^{c+d x}}{a+1}\right )}{d^3 \log ^3(f)}+\frac{x^2 \text{PolyLog}\left (2,\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x^2 \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a+1}\right )}{2 d \log (f)}+\frac{1}{6} x^3 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{6} x^3 \log \left (\frac{b f^{c+d x}}{a+1}+1\right )-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (\frac{1}{a+b f^{c+d x}}+1\right ) \]
Antiderivative was successfully verified.
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Rule 6214
Rule 2551
Rule 12
Rule 6742
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx &=-\left (\frac{1}{2} \int x^2 \log \left (1-\frac{1}{a+b f^{c+d x}}\right ) \, dx\right )+\frac{1}{2} \int x^2 \log \left (1+\frac{1}{a+b f^{c+d x}}\right ) \, dx\\ &=-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} \int \frac{b d f^{c+d x} x^3 \log (f)}{\left (-1+a+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx-\frac{1}{6} \int \frac{b d f^{c+d x} x^3 \log (f)}{\left (-a-b f^{c+d x}\right ) \left (1+a+b f^{c+d x}\right )} \, dx\\ &=-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} (b d \log (f)) \int \frac{f^{c+d x} x^3}{\left (-1+a+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx-\frac{1}{6} (b d \log (f)) \int \frac{f^{c+d x} x^3}{\left (-a-b f^{c+d x}\right ) \left (1+a+b f^{c+d x}\right )} \, dx\\ &=-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} (b d \log (f)) \int \left (\frac{f^{c+d x} x^3}{-a-b f^{c+d x}}+\frac{f^{c+d x} x^3}{-1+a+b f^{c+d x}}\right ) \, dx-\frac{1}{6} (b d \log (f)) \int \left (\frac{f^{c+d x} x^3}{-a-b f^{c+d x}}+\frac{f^{c+d x} x^3}{1+a+b f^{c+d x}}\right ) \, dx\\ &=-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} (b d \log (f)) \int \frac{f^{c+d x} x^3}{-1+a+b f^{c+d x}} \, dx-\frac{1}{6} (b d \log (f)) \int \frac{f^{c+d x} x^3}{1+a+b f^{c+d x}} \, dx\\ &=\frac{1}{6} x^3 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{6} x^3 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )-\frac{1}{2} \int x^2 \log \left (1+\frac{b f^{c+d x}}{-1+a}\right ) \, dx+\frac{1}{2} \int x^2 \log \left (1+\frac{b f^{c+d x}}{1+a}\right ) \, dx\\ &=\frac{1}{6} x^3 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{6} x^3 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{x^2 \text{Li}_2\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x^2 \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac{\int x \text{Li}_2\left (-\frac{b f^{c+d x}}{-1+a}\right ) \, dx}{d \log (f)}+\frac{\int x \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right ) \, dx}{d \log (f)}\\ &=\frac{1}{6} x^3 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{6} x^3 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{x^2 \text{Li}_2\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x^2 \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac{x \text{Li}_3\left (\frac{b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac{x \text{Li}_3\left (-\frac{b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac{\int \text{Li}_3\left (-\frac{b f^{c+d x}}{-1+a}\right ) \, dx}{d^2 \log ^2(f)}-\frac{\int \text{Li}_3\left (-\frac{b f^{c+d x}}{1+a}\right ) \, dx}{d^2 \log ^2(f)}\\ &=\frac{1}{6} x^3 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{6} x^3 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{x^2 \text{Li}_2\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x^2 \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac{x \text{Li}_3\left (\frac{b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac{x \text{Li}_3\left (-\frac{b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{-1+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{1+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}\\ &=\frac{1}{6} x^3 \log \left (1-\frac{b f^{c+d x}}{1-a}\right )-\frac{1}{6} x^3 \log \left (1+\frac{b f^{c+d x}}{1+a}\right )-\frac{1}{6} x^3 \log \left (1-\frac{1}{a+b f^{c+d x}}\right )+\frac{1}{6} x^3 \log \left (1+\frac{1}{a+b f^{c+d x}}\right )+\frac{x^2 \text{Li}_2\left (\frac{b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac{x^2 \text{Li}_2\left (-\frac{b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac{x \text{Li}_3\left (\frac{b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac{x \text{Li}_3\left (-\frac{b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac{\text{Li}_4\left (\frac{b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac{\text{Li}_4\left (-\frac{b f^{c+d x}}{1+a}\right )}{d^3 \log ^3(f)}\\ \end{align*}
Mathematica [A] time = 0.0697542, size = 235, normalized size = 0.87 \[ \frac{3 d^2 x^2 \log ^2(f) \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a-1}\right )-3 d^2 x^2 \log ^2(f) \text{PolyLog}\left (2,-\frac{b f^{c+d x}}{a+1}\right )+6 \text{PolyLog}\left (4,-\frac{b f^{c+d x}}{a-1}\right )-6 \text{PolyLog}\left (4,-\frac{b f^{c+d x}}{a+1}\right )-6 d x \log (f) \text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a-1}\right )+6 d x \log (f) \text{PolyLog}\left (3,-\frac{b f^{c+d x}}{a+1}\right )+d^3 x^3 \log ^3(f) \log \left (\frac{b f^{c+d x}}{a-1}+1\right )-d^3 x^3 \log ^3(f) \log \left (\frac{b f^{c+d x}}{a+1}+1\right )+2 d^3 x^3 \log ^3(f) \coth ^{-1}\left (a+b f^{c+d x}\right )}{6 d^3 \log ^3(f)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.175, size = 666, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27328, size = 338, normalized size = 1.26 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcoth}\left (b f^{d x + c} + a\right ) - \frac{1}{6} \, b d{\left (\frac{\log \left (\frac{b f^{d x} f^{c}}{a + 1} + 1\right ) \log \left (f^{d x}\right )^{3} + 3 \,{\rm Li}_2\left (-\frac{b f^{d x} f^{c}}{a + 1}\right ) \log \left (f^{d x}\right )^{2} - 6 \, \log \left (f^{d x}\right ){\rm Li}_{3}(-\frac{b f^{d x} f^{c}}{a + 1}) + 6 \,{\rm Li}_{4}(-\frac{b f^{d x} f^{c}}{a + 1})}{b d^{4} \log \left (f\right )^{4}} - \frac{\log \left (\frac{b f^{d x} f^{c}}{a - 1} + 1\right ) \log \left (f^{d x}\right )^{3} + 3 \,{\rm Li}_2\left (-\frac{b f^{d x} f^{c}}{a - 1}\right ) \log \left (f^{d x}\right )^{2} - 6 \, \log \left (f^{d x}\right ){\rm Li}_{3}(-\frac{b f^{d x} f^{c}}{a - 1}) + 6 \,{\rm Li}_{4}(-\frac{b f^{d x} f^{c}}{a - 1})}{b d^{4} \log \left (f\right )^{4}}\right )} \log \left (f\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.91391, size = 1453, normalized size = 5.4 \begin{align*} \text{result too large to display} \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^{2} \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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