Optimal. Leaf size=168 \[ \frac{3 x^2 \text{PolyLog}\left (3,-(1-d) e^{2 a+2 b x}\right )}{8 b^2}-\frac{3 x \text{PolyLog}\left (4,-(1-d) e^{2 a+2 b x}\right )}{8 b^3}+\frac{3 \text{PolyLog}\left (5,-(1-d) e^{2 a+2 b x}\right )}{16 b^4}-\frac{x^3 \text{PolyLog}\left (2,-(1-d) e^{2 a+2 b x}\right )}{4 b}-\frac{1}{8} x^4 \log \left ((1-d) e^{2 a+2 b x}+1\right )+\frac{1}{4} x^4 \tanh ^{-1}(d (-\tanh (a+b x))-d+1)+\frac{b x^5}{20} \]
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Rubi [A] time = 0.296893, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6239, 2184, 2190, 2531, 6609, 2282, 6589} \[ \frac{3 x^2 \text{PolyLog}\left (3,-(1-d) e^{2 a+2 b x}\right )}{8 b^2}-\frac{3 x \text{PolyLog}\left (4,-(1-d) e^{2 a+2 b x}\right )}{8 b^3}+\frac{3 \text{PolyLog}\left (5,-(1-d) e^{2 a+2 b x}\right )}{16 b^4}-\frac{x^3 \text{PolyLog}\left (2,-(1-d) e^{2 a+2 b x}\right )}{4 b}-\frac{1}{8} x^4 \log \left ((1-d) e^{2 a+2 b x}+1\right )+\frac{1}{4} x^4 \tanh ^{-1}(d (-\tanh (a+b x))-d+1)+\frac{b x^5}{20} \]
Antiderivative was successfully verified.
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Rule 6239
Rule 2184
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \tanh ^{-1}(1-d-d \tanh (a+b x)) \, dx &=\frac{1}{4} x^4 \tanh ^{-1}(1-d-d \tanh (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 \tanh ^{-1}(1-d-d \tanh (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 \tanh ^{-1}(1-d-d \tanh (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 \tanh ^{-1}(1-d-d \tanh (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 \tanh ^{-1}(1-d-d \tanh (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 \tanh ^{-1}(1-d-d \tanh (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 \tanh ^{-1}(1-d-d \tanh (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 \tanh ^{-1}(1-d-d \tanh (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*}
Mathematica [A] time = 4.37523, size = 144, normalized size = 0.86 \[ \frac{1}{16} \left (\frac{6 x^2 \text{PolyLog}\left (3,\frac{e^{-2 (a+b x)}}{d-1}\right )}{b^2}+\frac{6 x \text{PolyLog}\left (4,\frac{e^{-2 (a+b x)}}{d-1}\right )}{b^3}+\frac{3 \text{PolyLog}\left (5,\frac{e^{-2 (a+b x)}}{d-1}\right )}{b^4}+\frac{4 x^3 \text{PolyLog}\left (2,\frac{e^{-2 (a+b x)}}{d-1}\right )}{b}-2 x^4 \log \left (1-\frac{e^{-2 (a+b x)}}{d-1}\right )+4 x^4 \tanh ^{-1}(d (-\tanh (a+b x))-d+1)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 15.613, size = 1773, normalized size = 10.6 \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 = 3.27785, size = 197, normalized size = 1.17 \begin{align*} -\frac{1}{4} \, x^{4} \operatorname{artanh}\left (d \tanh \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.00922, size = 1281, normalized size = 7.62 \begin{align*} \frac{2 \, b^{5} x^{5} - 5 \, b^{4} x^{4} \log \left (-\frac{d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}{{\left (d - 2\right )} \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}} \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^{3} \operatorname{artanh}\left (d \tanh \left (b x + a\right ) + d - 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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