Optimal. Leaf size=71 \[ -\frac {x \text {PolyLog}\left (2,-e^{a+b x}\right )}{2 b}+\frac {x \text {PolyLog}\left (2,e^{a+b x}\right )}{2 b}+\frac {\text {PolyLog}\left (3,-e^{a+b x}\right )}{2 b^2}-\frac {\text {PolyLog}\left (3,e^{a+b x}\right )}{2 b^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6348, 2611,
2320, 6724} \begin {gather*} \frac {\text {Li}_3\left (-e^{a+b x}\right )}{2 b^2}-\frac {\text {Li}_3\left (e^{a+b x}\right )}{2 b^2}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{2 b}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 6348
Rule 6724
Rubi steps
\begin {align*} \int x \tanh ^{-1}\left (e^{a+b x}\right ) \, dx &=-\left (\frac {1}{2} \int x \log \left (1-e^{a+b x}\right ) \, dx\right )+\frac {1}{2} \int x \log \left (1+e^{a+b x}\right ) \, dx\\ &=-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{2 b}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{2 b}+\frac {\int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{2 b}-\frac {\int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{2 b}\\ &=-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{2 b}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}\\ &=-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{2 b}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{2 b}+\frac {\text {Li}_3\left (-e^{a+b x}\right )}{2 b^2}-\frac {\text {Li}_3\left (e^{a+b x}\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 113, normalized size = 1.59 \begin {gather*} \frac {2 b^2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )+b^2 x^2 \log \left (1-e^{a+b x}\right )-b^2 x^2 \log \left (1+e^{a+b x}\right )-2 b x \text {PolyLog}\left (2,-e^{a+b x}\right )+2 b x \text {PolyLog}\left (2,e^{a+b x}\right )+2 \text {PolyLog}\left (3,-e^{a+b x}\right )-2 \text {PolyLog}\left (3,e^{a+b x}\right )}{4 b^2} \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. \(177\) vs.
\(2(59)=118\).
time = 0.10, size = 178, normalized size = 2.51
method | result | size |
risch | \(\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x a}{2 b}+\frac {x \polylog \left (2, {\mathrm e}^{b x +a}\right )}{2 b}+\frac {a^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {a \dilog \left ({\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {\polylog \left (3, {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {x \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{2 b}+\frac {\dilog \left ({\mathrm e}^{b x +a}+1\right ) a}{2 b^{2}}-\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {\polylog \left (3, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}\) | \(155\) |
default | \(\frac {x^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{2}-\frac {a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )-\frac {\left (b x +a \right )^{2} \ln \left (1-{\mathrm e}^{b x +a}\right )}{2}-\left (b x +a \right ) \polylog \left (2, {\mathrm e}^{b x +a}\right )+\polylog \left (3, {\mathrm e}^{b x +a}\right )+\frac {\left (b x +a \right )^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{2}+\left (b x +a \right ) \polylog \left (2, -{\mathrm e}^{b x +a}\right )-\polylog \left (3, -{\mathrm e}^{b x +a}\right )+a \left (b x +a \right ) \ln \left (1-{\mathrm e}^{b x +a}\right )-a \left (b x +a \right ) \ln \left ({\mathrm e}^{b x +a}+1\right )-\polylog \left (2, -{\mathrm e}^{b x +a}\right ) a +\polylog \left (2, {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}\) | \(178\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 108, normalized size = 1.52 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {artanh}\left (e^{\left (b x + a\right )}\right ) - \frac {1}{4} \, b {\left (\frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} - \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs.
\(2 (57) = 114\).
time = 0.36, size = 199, normalized size = 2.80 \begin {gather*} \frac {b^{2} x^{2} \log \left (-\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1}\right ) - b^{2} x^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, b x {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, b x {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + a^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{4 \, b^{2}} \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 \operatorname {atanh}{\left (e^{a} e^{b 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 = -0.01 \begin {gather*} \int x\,\mathrm {atanh}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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