Optimal. Leaf size=133 \[ -\frac{c+d x}{4 f \left (a^2 \coth (e+f x)+a^2\right )}+\frac{x (c+d x)}{4 a^2}-\frac{3 d}{16 f^2 \left (a^2 \coth (e+f x)+a^2\right )}+\frac{3 d x}{16 a^2 f}-\frac{d x^2}{8 a^2}-\frac{c+d x}{4 f (a \coth (e+f x)+a)^2}-\frac{d}{16 f^2 (a \coth (e+f x)+a)^2} \]
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Rubi [A] time = 0.129456, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3479, 8, 3730} \[ -\frac{c+d x}{4 f \left (a^2 \coth (e+f x)+a^2\right )}+\frac{x (c+d x)}{4 a^2}-\frac{3 d}{16 f^2 \left (a^2 \coth (e+f x)+a^2\right )}+\frac{3 d x}{16 a^2 f}-\frac{d x^2}{8 a^2}-\frac{c+d x}{4 f (a \coth (e+f x)+a)^2}-\frac{d}{16 f^2 (a \coth (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 8
Rule 3730
Rubi steps
\begin{align*} \int \frac{c+d x}{(a+a \coth (e+f x))^2} \, dx &=\frac{x (c+d x)}{4 a^2}-\frac{c+d x}{4 f (a+a \coth (e+f x))^2}-\frac{c+d x}{4 f \left (a^2+a^2 \coth (e+f x)\right )}-d \int \left (\frac{x}{4 a^2}-\frac{1}{4 f (a+a \coth (e+f x))^2}-\frac{1}{4 f \left (a^2+a^2 \coth (e+f x)\right )}\right ) \, dx\\ &=-\frac{d x^2}{8 a^2}+\frac{x (c+d x)}{4 a^2}-\frac{c+d x}{4 f (a+a \coth (e+f x))^2}-\frac{c+d x}{4 f \left (a^2+a^2 \coth (e+f x)\right )}+\frac{d \int \frac{1}{(a+a \coth (e+f x))^2} \, dx}{4 f}+\frac{d \int \frac{1}{a^2+a^2 \coth (e+f x)} \, dx}{4 f}\\ &=-\frac{d x^2}{8 a^2}+\frac{x (c+d x)}{4 a^2}-\frac{d}{16 f^2 (a+a \coth (e+f x))^2}-\frac{c+d x}{4 f (a+a \coth (e+f x))^2}-\frac{d}{8 f^2 \left (a^2+a^2 \coth (e+f x)\right )}-\frac{c+d x}{4 f \left (a^2+a^2 \coth (e+f x)\right )}+\frac{d \int 1 \, dx}{8 a^2 f}+\frac{d \int \frac{1}{a+a \coth (e+f x)} \, dx}{8 a f}\\ &=\frac{d x}{8 a^2 f}-\frac{d x^2}{8 a^2}+\frac{x (c+d x)}{4 a^2}-\frac{d}{16 f^2 (a+a \coth (e+f x))^2}-\frac{c+d x}{4 f (a+a \coth (e+f x))^2}-\frac{3 d}{16 f^2 \left (a^2+a^2 \coth (e+f x)\right )}-\frac{c+d x}{4 f \left (a^2+a^2 \coth (e+f x)\right )}+\frac{d \int 1 \, dx}{16 a^2 f}\\ &=\frac{3 d x}{16 a^2 f}-\frac{d x^2}{8 a^2}+\frac{x (c+d x)}{4 a^2}-\frac{d}{16 f^2 (a+a \coth (e+f x))^2}-\frac{c+d x}{4 f (a+a \coth (e+f x))^2}-\frac{3 d}{16 f^2 \left (a^2+a^2 \coth (e+f x)\right )}-\frac{c+d x}{4 f \left (a^2+a^2 \coth (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.521503, size = 114, normalized size = 0.86 \[ \frac{\text{csch}^2(e+f x) \left (\left (4 c f (4 f x+1)+d \left (8 f^2 x^2+4 f x+1\right )\right ) \sinh (2 (e+f x))+\left (4 c f (4 f x-1)+d \left (8 f^2 x^2-4 f x-1\right )\right ) \cosh (2 (e+f x))+8 (2 c f+2 d f x+d)\right )}{64 a^2 f^2 (\coth (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 74, normalized size = 0.6 \begin{align*}{\frac{d{x}^{2}}{8\,{a}^{2}}}+{\frac{cx}{4\,{a}^{2}}}+{\frac{ \left ( 2\,dfx+2\,cf+d \right ){{\rm e}^{-2\,fx-2\,e}}}{8\,{a}^{2}{f}^{2}}}-{\frac{ \left ( 4\,dfx+4\,cf+d \right ){{\rm e}^{-4\,fx-4\,e}}}{64\,{a}^{2}{f}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50981, size = 144, normalized size = 1.08 \begin{align*} \frac{1}{16} \, c{\left (\frac{4 \,{\left (f x + e\right )}}{a^{2} f} + \frac{4 \, e^{\left (-2 \, f x - 2 \, e\right )} - e^{\left (-4 \, f x - 4 \, e\right )}}{a^{2} f}\right )} + \frac{{\left (8 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 8 \,{\left (2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} -{\left (4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} d e^{\left (-4 \, e\right )}}{64 \, a^{2} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03748, size = 454, normalized size = 3.41 \begin{align*} \frac{16 \, d f x +{\left (8 \, d f^{2} x^{2} - 4 \, c f + 4 \,{\left (4 \, c f^{2} - d f\right )} x - d\right )} \cosh \left (f x + e\right )^{2} + 2 \,{\left (8 \, d f^{2} x^{2} + 4 \, c f + 4 \,{\left (4 \, c f^{2} + d f\right )} x + d\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) +{\left (8 \, d f^{2} x^{2} - 4 \, c f + 4 \,{\left (4 \, c f^{2} - d f\right )} x - d\right )} \sinh \left (f x + e\right )^{2} + 16 \, c f + 8 \, d}{64 \,{\left (a^{2} f^{2} \cosh \left (f x + e\right )^{2} + 2 \, a^{2} f^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a^{2} f^{2} \sinh \left (f x + e\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.84553, size = 700, normalized size = 5.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.145, size = 147, normalized size = 1.11 \begin{align*} \frac{{\left (8 \, d f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} + 16 \, c f^{2} x e^{\left (4 \, f x + 4 \, e\right )} + 16 \, d f x e^{\left (2 \, f x + 2 \, e\right )} - 4 \, d f x + 16 \, c f e^{\left (2 \, f x + 2 \, e\right )} - 4 \, c f + 8 \, d e^{\left (2 \, f x + 2 \, e\right )} - d\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{64 \, a^{2} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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