Optimal. Leaf size=156 \[ \frac{b d (c+d x) \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f^2 \left (a^2-b^2\right )}+\frac{b d^2 \text{PolyLog}\left (3,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f^3 \left (a^2-b^2\right )}-\frac{b (c+d x)^2 \log \left (1-\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f \left (a^2-b^2\right )}+\frac{(c+d x)^3}{3 d (a+b)} \]
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Rubi [A] time = 0.284352, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3731, 2190, 2531, 2282, 6589} \[ \frac{b d (c+d x) \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f^2 \left (a^2-b^2\right )}+\frac{b d^2 \text{PolyLog}\left (3,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f^3 \left (a^2-b^2\right )}-\frac{b (c+d x)^2 \log \left (1-\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f \left (a^2-b^2\right )}+\frac{(c+d x)^3}{3 d (a+b)} \]
Antiderivative was successfully verified.
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Rule 3731
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{a+b \coth (e+f x)} \, dx &=\frac{(c+d x)^3}{3 (a+b) d}-(2 b) \int \frac{e^{-2 (e+f x)} (c+d x)^2}{(a+b)^2+\left (-a^2+b^2\right ) e^{-2 (e+f x)}} \, dx\\ &=\frac{(c+d x)^3}{3 (a+b) d}-\frac{b (c+d x)^2 \log \left (1-\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac{(2 b d) \int (c+d x) \log \left (1+\frac{\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{\left (a^2-b^2\right ) f}\\ &=\frac{(c+d x)^3}{3 (a+b) d}-\frac{b (c+d x)^2 \log \left (1-\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac{b d (c+d x) \text{Li}_2\left (\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f^2}-\frac{\left (b d^2\right ) \int \text{Li}_2\left (-\frac{\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{\left (a^2-b^2\right ) f^2}\\ &=\frac{(c+d x)^3}{3 (a+b) d}-\frac{b (c+d x)^2 \log \left (1-\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac{b d (c+d x) \text{Li}_2\left (\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f^2}+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{(a-b) x}{a+b}\right )}{x} \, dx,x,e^{-2 (e+f x)}\right )}{2 \left (a^2-b^2\right ) f^3}\\ &=\frac{(c+d x)^3}{3 (a+b) d}-\frac{b (c+d x)^2 \log \left (1-\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac{b d (c+d x) \text{Li}_2\left (\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f^2}+\frac{b d^2 \text{Li}_3\left (\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^3}\\ \end{align*}
Mathematica [A] time = 3.59568, size = 191, normalized size = 1.22 \[ \frac{b \left (\frac{3 d \left (2 f (c+d x) \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )+d \text{PolyLog}\left (3,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )}{f^3 (a-b)}-\frac{6 (c+d x)^2 \log \left (\frac{(b-a) e^{-2 (e+f x)}}{a+b}+1\right )}{f (a-b)}+\frac{4 (c+d x)^3}{d \left (a \left (e^{2 e}-1\right )+b \left (e^{2 e}+1\right )\right )}\right )}{6 (a+b)}+\frac{x \sinh (e) \left (3 c^2+3 c d x+d^2 x^2\right )}{3 (a \sinh (e)+b \cosh (e))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.147, size = 722, normalized size = 4.6 \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 = 1.63411, size = 448, normalized size = 2.87 \begin{align*} -\frac{{\left (2 \, f x \log \left (-\frac{{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) +{\rm Li}_2\left (\frac{{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right )\right )} b c d}{a^{2} f^{2} - b^{2} f^{2}} - \frac{{\left (2 \, f^{2} x^{2} \log \left (-\frac{{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) + 2 \, f x{\rm Li}_2\left (\frac{{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right ) -{\rm Li}_{3}(\frac{{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b})\right )} b d^{2}}{2 \,{\left (a^{2} f^{3} - b^{2} f^{3}\right )}} - c^{2}{\left (\frac{b \log \left (-{\left (a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + a + b\right )}{{\left (a^{2} - b^{2}\right )} f} - \frac{f x + e}{{\left (a + b\right )} f}\right )} + \frac{2 \,{\left (b d^{2} f^{3} x^{3} + 3 \, b c d f^{3} x^{2}\right )}}{3 \,{\left (a^{2} f^{3} - b^{2} f^{3}\right )}} + \frac{d^{2} x^{3} + 3 \, c d x^{2}}{3 \,{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.3046, size = 1258, normalized size = 8.06 \begin{align*} \frac{{\left (a + b\right )} d^{2} f^{3} x^{3} + 3 \,{\left (a + b\right )} c d f^{3} x^{2} + 3 \,{\left (a + b\right )} c^{2} f^{3} x + 6 \, b d^{2}{\rm polylog}\left (3, \sqrt{\frac{a + b}{a - b}}{\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) + 6 \, b d^{2}{\rm polylog}\left (3, -\sqrt{\frac{a + b}{a - b}}{\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) - 6 \,{\left (b d^{2} f x + b c d f\right )}{\rm Li}_2\left (\sqrt{\frac{a + b}{a - b}}{\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) - 6 \,{\left (b d^{2} f x + b c d f\right )}{\rm Li}_2\left (-\sqrt{\frac{a + b}{a - b}}{\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) - 3 \,{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (2 \,{\left (a + b\right )} \cosh \left (f x + e\right ) + 2 \,{\left (a + b\right )} \sinh \left (f x + e\right ) + 2 \,{\left (a - b\right )} \sqrt{\frac{a + b}{a - b}}\right ) - 3 \,{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (2 \,{\left (a + b\right )} \cosh \left (f x + e\right ) + 2 \,{\left (a + b\right )} \sinh \left (f x + e\right ) - 2 \,{\left (a - b\right )} \sqrt{\frac{a + b}{a - b}}\right ) - 3 \,{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (\sqrt{\frac{a + b}{a - b}}{\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 1\right ) - 3 \,{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (-\sqrt{\frac{a + b}{a - b}}{\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 1\right )}{3 \,{\left (a^{2} - b^{2}\right )} f^{3}} \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 \frac{\left (c + d x\right )^{2}}{a + b \coth{\left (e + f x \right )}}\, 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 \frac{{\left (d x + c\right )}^{2}}{b \coth \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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