Optimal. Leaf size=269 \[ \frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {x^2 \text {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \text {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac {x \text {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac {x \text {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac {\text {PolyLog}\left (4,\frac {b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac {\text {PolyLog}\left (4,-\frac {b f^{c+d x}}{1+a}\right )}{d^3 \log ^3(f)} \]
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Rubi [A]
time = 1.74, antiderivative size = 269, normalized size of antiderivative = 1.00, 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 = {6349, 2631,
12, 6874, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \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}}{a+1}\right )}{d^3 \log ^3(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}}{a+1}\right )}{d^2 \log ^2(f)}+\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}}{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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 2631
Rule 6349
Rule 6724
Rule 6744
Rule 6874
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 {\text {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 {\text {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*}
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Mathematica [A]
time = 0.05, size = 235, normalized size = 0.87 \begin {gather*} \frac {2 d^3 x^3 \coth ^{-1}\left (a+b f^{c+d x}\right ) \log ^3(f)+d^3 x^3 \log ^3(f) \log \left (1+\frac {b f^{c+d x}}{-1+a}\right )-d^3 x^3 \log ^3(f) \log \left (1+\frac {b f^{c+d x}}{1+a}\right )+3 d^2 x^2 \log ^2(f) \text {PolyLog}\left (2,-\frac {b f^{c+d x}}{-1+a}\right )-3 d^2 x^2 \log ^2(f) \text {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )-6 d x \log (f) \text {PolyLog}\left (3,-\frac {b f^{c+d x}}{-1+a}\right )+6 d x \log (f) \text {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )+6 \text {PolyLog}\left (4,-\frac {b f^{c+d x}}{-1+a}\right )-6 \text {PolyLog}\left (4,-\frac {b f^{c+d x}}{1+a}\right )}{6 d^3 \log ^3(f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs.
\(2(257)=514\).
time = 0.07, size = 666, normalized size = 2.48
method | result | size |
risch | \(-\frac {x^{3} \ln \left (a +b \,f^{d x +c}-1\right )}{6}+\frac {x^{3} \ln \left (1+a +b \,f^{d x +c}\right )}{6}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-a -1}\right ) x^{3}}{6}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-a -1}\right ) x \,c^{2}}{2 d^{2}}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-a -1}\right ) c^{3}}{3 d^{3}}-\frac {\polylog \left (2, \frac {b \,f^{d x} f^{c}}{-a -1}\right ) x^{2}}{2 \ln \left (f \right ) d}+\frac {\polylog \left (2, \frac {b \,f^{d x} f^{c}}{-a -1}\right ) c^{2}}{2 \ln \left (f \right ) d^{3}}+\frac {\polylog \left (3, \frac {b \,f^{d x} f^{c}}{-a -1}\right ) x}{\ln \left (f \right )^{2} d^{2}}-\frac {\polylog \left (4, \frac {b \,f^{d x} f^{c}}{-a -1}\right )}{\ln \left (f \right )^{3} d^{3}}+\frac {c^{3} \ln \left (1+a +b \,f^{d x} f^{c}\right )}{6 d^{3}}-\frac {c^{2} \dilog \left (\frac {1+a +b \,f^{d x} f^{c}}{1+a}\right )}{2 \ln \left (f \right ) d^{3}}-\frac {c^{2} \ln \left (\frac {1+a +b \,f^{d x} f^{c}}{1+a}\right ) x}{2 d^{2}}-\frac {c^{3} \ln \left (\frac {1+a +b \,f^{d x} f^{c}}{1+a}\right )}{2 d^{3}}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) x^{3}}{6}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) x \,c^{2}}{2 d^{2}}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) c^{3}}{3 d^{3}}+\frac {\polylog \left (2, \frac {b \,f^{d x} f^{c}}{1-a}\right ) x^{2}}{2 \ln \left (f \right ) d}-\frac {\polylog \left (2, \frac {b \,f^{d x} f^{c}}{1-a}\right ) c^{2}}{2 \ln \left (f \right ) d^{3}}-\frac {\polylog \left (3, \frac {b \,f^{d x} f^{c}}{1-a}\right ) x}{\ln \left (f \right )^{2} d^{2}}+\frac {\polylog \left (4, \frac {b \,f^{d x} f^{c}}{1-a}\right )}{\ln \left (f \right )^{3} d^{3}}-\frac {c^{3} \ln \left (b \,f^{d x} f^{c}+a -1\right )}{6 d^{3}}+\frac {c^{2} \dilog \left (\frac {b \,f^{d x} f^{c}+a -1}{-1+a}\right )}{2 \ln \left (f \right ) d^{3}}+\frac {c^{2} \ln \left (\frac {b \,f^{d x} f^{c}+a -1}{-1+a}\right ) x}{2 d^{2}}+\frac {c^{3} \ln \left (\frac {b \,f^{d x} f^{c}+a -1}{-1+a}\right )}{2 d^{3}}\) | \(666\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 254, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (b f^{d x + c} + a\right ) - \frac {1}{6} \, b d {\left (\frac {d^{3} x^{3} \log \left (\frac {b f^{d x} f^{c}}{a + 1} + 1\right ) \log \left (f\right )^{3} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b f^{d x} f^{c}}{a + 1}\right ) \log \left (f\right )^{2} - 6 \, d x \log \left (f\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 {d^{3} x^{3} \log \left (\frac {b f^{d x} f^{c}}{a - 1} + 1\right ) \log \left (f\right )^{3} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b f^{d x} f^{c}}{a - 1}\right ) \log \left (f\right )^{2} - 6 \, d x \log \left (f\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 479, normalized size = 1.78 \begin {gather*} \frac {d^{3} x^{3} \log \left (f\right )^{3} \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 ) - 3 \, d^{2} x^{2} {\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} + 3 \, d^{2} x^{2} {\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} + c^{3} \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 )^{3} - c^{3} \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 )^{3} - {\left (d^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} \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^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} \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 ) + 6 \, d x \log \left (f\right ) {\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 ) - 6 \, d x \log \left (f\right ) {\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 ) - 6 \, {\rm polylog}\left (4, -\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 ) + 6 \, {\rm polylog}\left (4, -\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 )}{6 \, d^{3} \log \left (f\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {acoth}{\left (a + b f^{c} f^{d x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\mathrm {acoth}\left (a+b\,f^{c+d\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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