Optimal. Leaf size=152 \[ \frac {3 \text {Li}_5\left ((d+1) e^{2 a+2 b x}\right )}{16 b^4}-\frac {3 x \text {Li}_4\left ((d+1) e^{2 a+2 b x}\right )}{8 b^3}+\frac {3 x^2 \text {Li}_3\left ((d+1) e^{2 a+2 b x}\right )}{8 b^2}-\frac {x^3 \text {Li}_2\left ((d+1) e^{2 a+2 b x}\right )}{4 b}-\frac {1}{8} x^4 \log \left (1-(d+1) e^{2 a+2 b x}\right )+\frac {1}{4} x^4 \coth ^{-1}(d \coth (a+b x)+d+1)+\frac {b x^5}{20} \]
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Rubi [A] time = 0.31, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6242, 2184, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 x^2 \text {PolyLog}\left (3,(d+1) e^{2 a+2 b x}\right )}{8 b^2}-\frac {3 x \text {PolyLog}\left (4,(d+1) e^{2 a+2 b x}\right )}{8 b^3}+\frac {3 \text {PolyLog}\left (5,(d+1) e^{2 a+2 b x}\right )}{16 b^4}-\frac {x^3 \text {PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )}{4 b}-\frac {1}{8} x^4 \log \left (1-(d+1) e^{2 a+2 b x}\right )+\frac {1}{4} x^4 \coth ^{-1}(d \coth (a+b x)+d+1)+\frac {b x^5}{20} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2282
Rule 2531
Rule 6242
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx &=\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))+\frac {1}{4} b \int \frac {x^4}{1+(-1-d) e^{2 a+2 b x}} \, dx\\ &=\frac {b x^5}{20}+\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))+\frac {1}{4} (b (1+d)) \int \frac {e^{2 a+2 b x} x^4}{1+(-1-d) e^{2 a+2 b x}} \, dx\\ &=\frac {b x^5}{20}+\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))-\frac {1}{8} x^4 \log \left (1-(1+d) e^{2 a+2 b x}\right )+\frac {1}{2} \int x^3 \log \left (1+(-1-d) e^{2 a+2 b x}\right ) \, dx\\ &=\frac {b x^5}{20}+\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))-\frac {1}{8} x^4 \log \left (1-(1+d) e^{2 a+2 b x}\right )-\frac {x^3 \text {Li}_2\left ((1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {3 \int x^2 \text {Li}_2\left (-(-1-d) e^{2 a+2 b x}\right ) \, dx}{4 b}\\ &=\frac {b x^5}{20}+\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))-\frac {1}{8} x^4 \log \left (1-(1+d) e^{2 a+2 b x}\right )-\frac {x^3 \text {Li}_2\left ((1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {3 x^2 \text {Li}_3\left ((1+d) e^{2 a+2 b x}\right )}{8 b^2}-\frac {3 \int x \text {Li}_3\left ((1+d) e^{2 a+2 b x}\right ) \, dx}{4 b^2}\\ &=\frac {b x^5}{20}+\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))-\frac {1}{8} x^4 \log \left (1-(1+d) e^{2 a+2 b x}\right )-\frac {x^3 \text {Li}_2\left ((1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {3 x^2 \text {Li}_3\left ((1+d) e^{2 a+2 b x}\right )}{8 b^2}-\frac {3 x \text {Li}_4\left ((1+d) e^{2 a+2 b x}\right )}{8 b^3}+\frac {3 \int \text {Li}_4\left ((1+d) e^{2 a+2 b x}\right ) \, dx}{8 b^3}\\ &=\frac {b x^5}{20}+\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))-\frac {1}{8} x^4 \log \left (1-(1+d) e^{2 a+2 b x}\right )-\frac {x^3 \text {Li}_2\left ((1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {3 x^2 \text {Li}_3\left ((1+d) e^{2 a+2 b x}\right )}{8 b^2}-\frac {3 x \text {Li}_4\left ((1+d) e^{2 a+2 b x}\right )}{8 b^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_4((1+d) x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{16 b^4}\\ &=\frac {b x^5}{20}+\frac {1}{4} x^4 \coth ^{-1}(1+d+d \coth (a+b x))-\frac {1}{8} x^4 \log \left (1-(1+d) e^{2 a+2 b x}\right )-\frac {x^3 \text {Li}_2\left ((1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {3 x^2 \text {Li}_3\left ((1+d) e^{2 a+2 b x}\right )}{8 b^2}-\frac {3 x \text {Li}_4\left ((1+d) e^{2 a+2 b x}\right )}{8 b^3}+\frac {3 \text {Li}_5\left ((1+d) e^{2 a+2 b x}\right )}{16 b^4}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 141, normalized size = 0.93 \[ \frac {1}{16} \left (\frac {3 \text {Li}_5\left (\frac {e^{-2 (a+b x)}}{d+1}\right )}{b^4}+\frac {6 x \text {Li}_4\left (\frac {e^{-2 (a+b x)}}{d+1}\right )}{b^3}+\frac {6 x^2 \text {Li}_3\left (\frac {e^{-2 (a+b x)}}{d+1}\right )}{b^2}+\frac {4 x^3 \text {Li}_2\left (\frac {e^{-2 (a+b x)}}{d+1}\right )}{b}-2 x^4 \log \left (1-\frac {e^{-2 (a+b x)}}{d+1}\right )+4 x^4 \coth ^{-1}(d \coth (a+b x)+d+1)\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.53, size = 423, normalized size = 2.78 \[ \frac {2 \, b^{5} x^{5} + 5 \, b^{4} x^{4} \log \left (\frac {d \cosh \left (b x + a\right ) + {\left (d + 2\right )} \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) - 20 \, b^{3} x^{3} {\rm Li}_2\left (\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 20 \, b^{3} x^{3} {\rm Li}_2\left (-\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 5 \, a^{4} \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) + 2 \, \sqrt {d + 1}\right ) - 5 \, a^{4} \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) - 2 \, \sqrt {d + 1}\right ) + 60 \, b^{2} x^{2} {\rm polylog}\left (3, \sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 60 \, b^{2} x^{2} {\rm polylog}\left (3, -\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 120 \, b x {\rm polylog}\left (4, \sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 120 \, b x {\rm polylog}\left (4, -\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 5 \, {\left (b^{4} x^{4} - a^{4}\right )} \log \left (\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - 5 \, {\left (b^{4} x^{4} - a^{4}\right )} \log \left (-\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + 120 \, {\rm polylog}\left (5, \sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 120 \, {\rm polylog}\left (5, -\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{40 \, b^{4}} \]
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 \[ \int x^{3} \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + d + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.86, size = 1698, normalized size = 11.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 146, normalized size = 0.96 \[ \frac {1}{4} \, x^{4} \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + d + 1\right ) + \frac {1}{40} \, {\left (\frac {2 \, x^{5}}{d} - \frac {5 \, {\left (2 \, b^{4} x^{4} \log \left (-{\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 4 \, b^{3} x^{3} {\rm Li}_2\left ({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b^{2} x^{2} {\rm Li}_{3}({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}) + 6 \, b x {\rm Li}_{4}({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}) - 3 \, {\rm Li}_{5}({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )})\right )}}{b^{5} d}\right )} b d \]
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 \[ \int x^3\,\mathrm {acoth}\left (d+d\,\mathrm {coth}\left (a+b\,x\right )+1\right ) \,d x \]
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 \[ \int x^{3} \operatorname {acoth}{\left (d \coth {\left (a + b x \right )} + d + 1 \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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