Optimal. Leaf size=122 \[ -\frac{d (c+d x)}{2 f^2 (a \coth (e+f x)+a)}-\frac{(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a \coth (e+f x)+a)}+\frac{d^2 x}{4 a f^2} \]
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Rubi [A] time = 0.117448, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3723, 3479, 8} \[ -\frac{d (c+d x)}{2 f^2 (a \coth (e+f x)+a)}-\frac{(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a \coth (e+f x)+a)}+\frac{d^2 x}{4 a f^2} \]
Antiderivative was successfully verified.
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Rule 3723
Rule 3479
Rule 8
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{a+a \coth (e+f x)} \, dx &=\frac{(c+d x)^3}{6 a d}-\frac{(c+d x)^2}{2 f (a+a \coth (e+f x))}+\frac{d \int \frac{c+d x}{a+a \coth (e+f x)} \, dx}{f}\\ &=\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d (c+d x)}{2 f^2 (a+a \coth (e+f x))}-\frac{(c+d x)^2}{2 f (a+a \coth (e+f x))}+\frac{d^2 \int \frac{1}{a+a \coth (e+f x)} \, dx}{2 f^2}\\ &=\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a+a \coth (e+f x))}-\frac{d (c+d x)}{2 f^2 (a+a \coth (e+f x))}-\frac{(c+d x)^2}{2 f (a+a \coth (e+f x))}+\frac{d^2 \int 1 \, dx}{4 a f^2}\\ &=\frac{d^2 x}{4 a f^2}+\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a+a \coth (e+f x))}-\frac{d (c+d x)}{2 f^2 (a+a \coth (e+f x))}-\frac{(c+d x)^2}{2 f (a+a \coth (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.252905, size = 169, normalized size = 1.39 \[ \frac{\text{csch}(e+f x) (\sinh (f x)+\cosh (f x)) \left (\frac{4}{3} f^3 x \left (3 c^2+3 c d x+d^2 x^2\right ) (\sinh (e)+\cosh (e))+(\cosh (e)-\sinh (e)) \cosh (2 f x) \left (2 c^2 f^2+2 c d f (2 f x+1)+d^2 \left (2 f^2 x^2+2 f x+1\right )\right )+(\sinh (e)-\cosh (e)) \sinh (2 f x) \left (2 c^2 f^2+2 c d f (2 f x+1)+d^2 \left (2 f^2 x^2+2 f x+1\right )\right )\right )}{8 a f^3 (\coth (e+f x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 469, normalized size = 3.8 \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 = 1.25439, size = 167, normalized size = 1.37 \begin{align*} \frac{1}{4} \, c^{2}{\left (\frac{2 \,{\left (f x + e\right )}}{a f} + \frac{e^{\left (-2 \, f x - 2 \, e\right )}}{a f}\right )} + \frac{{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} +{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c d e^{\left (-2 \, e\right )}}{4 \, a f^{2}} + \frac{{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 3 \,{\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} d^{2} e^{\left (-2 \, e\right )}}{24 \, a f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08446, size = 419, normalized size = 3.43 \begin{align*} \frac{{\left (4 \, d^{2} f^{3} x^{3} + 6 \, c^{2} f^{2} + 6 \, c d f + 6 \,{\left (2 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + 3 \, d^{2} + 6 \,{\left (2 \, c^{2} f^{3} + 2 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) +{\left (4 \, d^{2} f^{3} x^{3} - 6 \, c^{2} f^{2} - 6 \, c d f + 6 \,{\left (2 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - 3 \, d^{2} + 6 \,{\left (2 \, c^{2} f^{3} - 2 \, c d f^{2} - d^{2} f\right )} x\right )} \sinh \left (f x + e\right )}{24 \,{\left (a f^{3} \cosh \left (f x + e\right ) + a f^{3} \sinh \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.18102, size = 522, normalized size = 4.28 \begin{align*} \begin{cases} \frac{6 c^{2} f^{3} x \tanh{\left (e + f x \right )}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{6 c^{2} f^{3} x}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{6 c^{2} f^{2}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{6 c d f^{3} x^{2} \tanh{\left (e + f x \right )}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{6 c d f^{3} x^{2}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} - \frac{6 c d f^{2} x \tanh{\left (e + f x \right )}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{6 c d f^{2} x}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{6 c d f}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{2 d^{2} f^{3} x^{3} \tanh{\left (e + f x \right )}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{2 d^{2} f^{3} x^{3}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} - \frac{3 d^{2} f^{2} x^{2} \tanh{\left (e + f x \right )}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{3 d^{2} f^{2} x^{2}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} - \frac{3 d^{2} f x \tanh{\left (e + f x \right )}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{3 d^{2} f x}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} + \frac{3 d^{2}}{12 a f^{3} \tanh{\left (e + f x \right )} + 12 a f^{3}} & \text{for}\: f \neq 0 \\\frac{c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}}{a \coth{\left (e \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15187, size = 166, normalized size = 1.36 \begin{align*} \frac{{\left (4 \, d^{2} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c d f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c^{2} f^{3} x e^{\left (2 \, f x + 2 \, e\right )} + 6 \, d^{2} f^{2} x^{2} + 12 \, c d f^{2} x + 6 \, c^{2} f^{2} + 6 \, d^{2} f x + 6 \, c d f + 3 \, d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{24 \, a f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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