Optimal. Leaf size=133 \[ -\frac{3 b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac{3 b d^3 \text{PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4}+\frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{b (c+d x)^4}{4 d} \]
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Rubi [A] time = 0.259571, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3722, 3716, 2190, 2531, 6609, 2282, 6589} \[ -\frac{3 b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac{3 b d^3 \text{PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4}+\frac{a (c+d x)^4}{4 d}+\frac{b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{b (c+d x)^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 3722
Rule 3716
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^3 (a+b \coth (e+f x)) \, dx &=\int \left (a (c+d x)^3+b (c+d x)^3 \coth (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+b \int (c+d x)^3 \coth (e+f x) \, dx\\ &=\frac{a (c+d x)^4}{4 d}-\frac{b (c+d x)^4}{4 d}-(2 b) \int \frac{e^{2 (e+f x)} (c+d x)^3}{1-e^{2 (e+f x)}} \, dx\\ &=\frac{a (c+d x)^4}{4 d}-\frac{b (c+d x)^4}{4 d}+\frac{b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{(3 b d) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{b (c+d x)^4}{4 d}+\frac{b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac{\left (3 b d^2\right ) \int (c+d x) \text{Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{b (c+d x)^4}{4 d}+\frac{b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac{3 b d^2 (c+d x) \text{Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac{\left (3 b d^3\right ) \int \text{Li}_3\left (e^{2 (e+f x)}\right ) \, dx}{2 f^3}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{b (c+d x)^4}{4 d}+\frac{b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac{3 b d^2 (c+d x) \text{Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac{\left (3 b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4}\\ &=\frac{a (c+d x)^4}{4 d}-\frac{b (c+d x)^4}{4 d}+\frac{b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac{3 b d (c+d x)^2 \text{Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac{3 b d^2 (c+d x) \text{Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac{3 b d^3 \text{Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}\\ \end{align*}
Mathematica [A] time = 0.358417, size = 249, normalized size = 1.87 \[ \frac{1}{4} \left (-\frac{6 b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac{6 b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac{3 b d^3 \text{PolyLog}\left (4,e^{2 (e+f x)}\right )}{f^4}+6 a c^2 d x^2+4 a c^3 x+4 a c d^2 x^3+a d^3 x^4+\frac{12 b c^2 d x \log \left (1-e^{2 (e+f x)}\right )}{f}-6 b c^2 d x^2+\frac{4 b c^3 \log (\tanh (e+f x))}{f}+\frac{4 b c^3 \log (\cosh (e+f x))}{f}+\frac{12 b c d^2 x^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-4 b c d^2 x^3+\frac{4 b d^3 x^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-b d^3 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.153, size = 748, normalized size = 5.6 \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 [B] time = 1.41425, size = 566, normalized size = 4.26 \begin{align*} \frac{1}{4} \, a d^{3} x^{4} + \frac{1}{4} \, b d^{3} x^{4} + a c d^{2} x^{3} + b c d^{2} x^{3} + \frac{3}{2} \, a c^{2} d x^{2} + \frac{3}{2} \, b c^{2} d x^{2} + a c^{3} x + \frac{b c^{3} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac{3 \,{\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} b c^{2} d}{f^{2}} + \frac{3 \,{\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )} b c^{2} d}{f^{2}} + \frac{3 \,{\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x{\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} b c d^{2}}{f^{3}} + \frac{3 \,{\left (f^{2} x^{2} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 2 \, f x{\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (f x + e\right )})\right )} b c d^{2}}{f^{3}} + \frac{{\left (f^{3} x^{3} \log \left (e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2}{\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 6 \, f x{\rm Li}_{3}(-e^{\left (f x + e\right )}) + 6 \,{\rm Li}_{4}(-e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} + \frac{{\left (f^{3} x^{3} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2}{\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 6 \, f x{\rm Li}_{3}(e^{\left (f x + e\right )}) + 6 \,{\rm Li}_{4}(e^{\left (f x + e\right )})\right )} b d^{3}}{f^{4}} - \frac{b d^{3} f^{4} x^{4} + 4 \, b c d^{2} f^{4} x^{3} + 6 \, b c^{2} d f^{4} x^{2}}{2 \, f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.35873, size = 1207, normalized size = 9.08 \begin{align*} \frac{{\left (a - b\right )} d^{3} f^{4} x^{4} + 4 \,{\left (a - b\right )} c d^{2} f^{4} x^{3} + 6 \,{\left (a - b\right )} c^{2} d f^{4} x^{2} + 4 \,{\left (a - b\right )} c^{3} f^{4} x + 24 \, b d^{3}{\rm polylog}\left (4, \cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + 24 \, b d^{3}{\rm polylog}\left (4, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 12 \,{\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )}{\rm Li}_2\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + 12 \,{\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )}{\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + 4 \,{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) - 4 \,{\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1\right ) + 4 \,{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right ) + 1\right ) - 24 \,{\left (b d^{3} f x + b c d^{2} f\right )}{\rm polylog}\left (3, \cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) - 24 \,{\left (b d^{3} f x + b c d^{2} f\right )}{\rm polylog}\left (3, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right )}{4 \, f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \coth{\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \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{\left (d x + c\right )}^{3}{\left (b \coth \left (f x + e\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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