Optimal. Leaf size=128 \[ \frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-\frac {1}{6} x^3 \log \left (1+(1+d) e^{2 a+2 b x}\right )-\frac {x^2 \text {PolyLog}\left (2,-\left ((1+d) e^{2 a+2 b x}\right )\right )}{4 b}+\frac {x \text {PolyLog}\left (3,-\left ((1+d) e^{2 a+2 b x}\right )\right )}{4 b^2}-\frac {\text {PolyLog}\left (4,-\left ((1+d) e^{2 a+2 b x}\right )\right )}{8 b^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6375, 2215,
2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {\text {Li}_4\left (-\left ((d+1) e^{2 a+2 b x}\right )\right )}{8 b^3}+\frac {x \text {Li}_3\left (-\left ((d+1) e^{2 a+2 b x}\right )\right )}{4 b^2}-\frac {x^2 \text {Li}_2\left (-\left ((d+1) e^{2 a+2 b x}\right )\right )}{4 b}-\frac {1}{6} x^3 \log \left ((d+1) e^{2 a+2 b x}+1\right )+\frac {1}{3} x^3 \coth ^{-1}(d \tanh (a+b x)+d+1)+\frac {b x^4}{12} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2320
Rule 2611
Rule 6375
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}(1+d+d \tanh (a+b x)) \, dx &=\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))+\frac {1}{3} b \int \frac {x^3}{1+(1+d) e^{2 a+2 b x}} \, dx\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-\frac {1}{3} (b (1+d)) \int \frac {e^{2 a+2 b x} x^3}{1+(1+d) e^{2 a+2 b x}} \, dx\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-\frac {1}{6} x^3 \log \left (1+(1+d) e^{2 a+2 b x}\right )+\frac {1}{2} \int x^2 \log \left (1+(1+d) e^{2 a+2 b x}\right ) \, dx\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-\frac {1}{6} x^3 \log \left (1+(1+d) e^{2 a+2 b x}\right )-\frac {x^2 \text {Li}_2\left (-(1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {\int x \text {Li}_2\left (-(1+d) e^{2 a+2 b x}\right ) \, dx}{2 b}\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-\frac {1}{6} x^3 \log \left (1+(1+d) e^{2 a+2 b x}\right )-\frac {x^2 \text {Li}_2\left (-(1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {x \text {Li}_3\left (-(1+d) e^{2 a+2 b x}\right )}{4 b^2}-\frac {\int \text {Li}_3\left ((-1-d) e^{2 a+2 b x}\right ) \, dx}{4 b^2}\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-\frac {1}{6} x^3 \log \left (1+(1+d) e^{2 a+2 b x}\right )-\frac {x^2 \text {Li}_2\left (-(1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {x \text {Li}_3\left (-(1+d) e^{2 a+2 b x}\right )}{4 b^2}-\frac {\text {Subst}\left (\int \frac {\text {Li}_3((-1-d) x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^3}\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-\frac {1}{6} x^3 \log \left (1+(1+d) e^{2 a+2 b x}\right )-\frac {x^2 \text {Li}_2\left (-(1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {x \text {Li}_3\left (-(1+d) e^{2 a+2 b x}\right )}{4 b^2}-\frac {\text {Li}_4\left (-(1+d) e^{2 a+2 b x}\right )}{8 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 118, normalized size = 0.92 \begin {gather*} \frac {1}{24} \left (8 x^3 \coth ^{-1}(1+d+d \tanh (a+b x))-4 x^3 \log \left (1+\frac {e^{-2 (a+b x)}}{1+d}\right )+\frac {6 x^2 \text {PolyLog}\left (2,-\frac {e^{-2 (a+b x)}}{1+d}\right )}{b}+\frac {6 x \text {PolyLog}\left (3,-\frac {e^{-2 (a+b x)}}{1+d}\right )}{b^2}+\frac {3 \text {PolyLog}\left (4,-\frac {e^{-2 (a+b x)}}{1+d}\right )}{b^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.52, size = 1667, normalized size = 13.02
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1667\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.69, size = 125, normalized size = 0.98 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (d \tanh \left (b x + a\right ) + d + 1\right ) + \frac {1}{36} \, {\left (\frac {3 \, x^{4}}{d} - \frac {2 \, {\left (4 \, b^{3} x^{3} \log \left ({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-{\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-{\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}) + 3 \, {\rm Li}_{4}(-{\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )})\right )}}{b^{4} d}\right )} b d \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. 381 vs.
\(2 (111) = 222\).
time = 0.36, size = 381, normalized size = 2.98 \begin {gather*} \frac {b^{4} x^{4} + 2 \, b^{3} x^{3} \log \left (\frac {{\left (d + 2\right )} \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) - 6 \, b^{2} x^{2} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 6 \, b^{2} x^{2} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 \, a^{3} \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) + \sqrt {-4 \, d - 4}\right ) + 2 \, a^{3} \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) - \sqrt {-4 \, d - 4}\right ) + 12 \, b x {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 12 \, b x {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - 12 \, {\rm polylog}\left (4, \frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 12 \, {\rm polylog}\left (4, -\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{12 \, b^{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 (d \tanh {\left (a + b x \right )} + d + 1 \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^2\,\mathrm {acoth}\left (d+d\,\mathrm {tanh}\left (a+b\,x\right )+1\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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