Optimal. Leaf size=183 \[ -\frac {c+d x}{8 f \left (a^3 \coth (e+f x)+a^3\right )}+\frac {x (c+d x)}{8 a^3}-\frac {11 d}{96 f^2 \left (a^3 \coth (e+f x)+a^3\right )}+\frac {11 d x}{96 a^3 f}-\frac {d x^2}{16 a^3}-\frac {c+d x}{8 a f (a \coth (e+f x)+a)^2}-\frac {c+d x}{6 f (a \coth (e+f x)+a)^3}-\frac {5 d}{96 a f^2 (a \coth (e+f x)+a)^2}-\frac {d}{36 f^2 (a \coth (e+f x)+a)^3} \]
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Rubi [A] time = 0.21, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3479, 8, 3730} \[ -\frac {c+d x}{8 f \left (a^3 \coth (e+f x)+a^3\right )}+\frac {x (c+d x)}{8 a^3}-\frac {11 d}{96 f^2 \left (a^3 \coth (e+f x)+a^3\right )}+\frac {11 d x}{96 a^3 f}-\frac {d x^2}{16 a^3}-\frac {c+d x}{8 a f (a \coth (e+f x)+a)^2}-\frac {c+d x}{6 f (a \coth (e+f x)+a)^3}-\frac {5 d}{96 a f^2 (a \coth (e+f x)+a)^2}-\frac {d}{36 f^2 (a \coth (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3479
Rule 3730
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx &=\frac {x (c+d x)}{8 a^3}-\frac {c+d x}{6 f (a+a \coth (e+f x))^3}-\frac {c+d x}{8 a f (a+a \coth (e+f x))^2}-\frac {c+d x}{8 f \left (a^3+a^3 \coth (e+f x)\right )}-d \int \left (\frac {x}{8 a^3}-\frac {1}{6 f (a+a \coth (e+f x))^3}-\frac {1}{8 a f (a+a \coth (e+f x))^2}-\frac {1}{8 f \left (a^3+a^3 \coth (e+f x)\right )}\right ) \, dx\\ &=-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}-\frac {c+d x}{6 f (a+a \coth (e+f x))^3}-\frac {c+d x}{8 a f (a+a \coth (e+f x))^2}-\frac {c+d x}{8 f \left (a^3+a^3 \coth (e+f x)\right )}+\frac {d \int \frac {1}{a^3+a^3 \coth (e+f x)} \, dx}{8 f}+\frac {d \int \frac {1}{(a+a \coth (e+f x))^3} \, dx}{6 f}+\frac {d \int \frac {1}{(a+a \coth (e+f x))^2} \, dx}{8 a f}\\ &=-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}-\frac {d}{36 f^2 (a+a \coth (e+f x))^3}-\frac {c+d x}{6 f (a+a \coth (e+f x))^3}-\frac {d}{32 a f^2 (a+a \coth (e+f x))^2}-\frac {c+d x}{8 a f (a+a \coth (e+f x))^2}-\frac {d}{16 f^2 \left (a^3+a^3 \coth (e+f x)\right )}-\frac {c+d x}{8 f \left (a^3+a^3 \coth (e+f x)\right )}+\frac {d \int 1 \, dx}{16 a^3 f}+\frac {d \int \frac {1}{a+a \coth (e+f x)} \, dx}{16 a^2 f}+\frac {d \int \frac {1}{(a+a \coth (e+f x))^2} \, dx}{12 a f}\\ &=\frac {d x}{16 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}-\frac {d}{36 f^2 (a+a \coth (e+f x))^3}-\frac {c+d x}{6 f (a+a \coth (e+f x))^3}-\frac {5 d}{96 a f^2 (a+a \coth (e+f x))^2}-\frac {c+d x}{8 a f (a+a \coth (e+f x))^2}-\frac {3 d}{32 f^2 \left (a^3+a^3 \coth (e+f x)\right )}-\frac {c+d x}{8 f \left (a^3+a^3 \coth (e+f x)\right )}+\frac {d \int 1 \, dx}{32 a^3 f}+\frac {d \int \frac {1}{a+a \coth (e+f x)} \, dx}{24 a^2 f}\\ &=\frac {3 d x}{32 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}-\frac {d}{36 f^2 (a+a \coth (e+f x))^3}-\frac {c+d x}{6 f (a+a \coth (e+f x))^3}-\frac {5 d}{96 a f^2 (a+a \coth (e+f x))^2}-\frac {c+d x}{8 a f (a+a \coth (e+f x))^2}-\frac {11 d}{96 f^2 \left (a^3+a^3 \coth (e+f x)\right )}-\frac {c+d x}{8 f \left (a^3+a^3 \coth (e+f x)\right )}+\frac {d \int 1 \, dx}{48 a^3 f}\\ &=\frac {11 d x}{96 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}-\frac {d}{36 f^2 (a+a \coth (e+f x))^3}-\frac {c+d x}{6 f (a+a \coth (e+f x))^3}-\frac {5 d}{96 a f^2 (a+a \coth (e+f x))^2}-\frac {c+d x}{8 a f (a+a \coth (e+f x))^2}-\frac {11 d}{96 f^2 \left (a^3+a^3 \coth (e+f x)\right )}-\frac {c+d x}{8 f \left (a^3+a^3 \coth (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 185, normalized size = 1.01 \[ \frac {\text {csch}^3(e+f x) \left (4 \left (6 c f (6 f x+1)+d \left (18 f^2 x^2+6 f x+1\right )\right ) \cosh (3 (e+f x))+27 (4 c f+d (4 f x+3)) \cosh (e+f x)+144 c f^2 x \sinh (3 (e+f x))+324 c f \sinh (e+f x)-24 c f \sinh (3 (e+f x))+72 d f^2 x^2 \sinh (3 (e+f x))+324 d f x \sinh (e+f x)-24 d f x \sinh (3 (e+f x))+135 d \sinh (e+f x)-4 d \sinh (3 (e+f x))\right )}{1152 a^3 f^2 (\coth (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 286, normalized size = 1.56 \[ \frac {4 \, {\left (18 \, d f^{2} x^{2} + 6 \, c f + 6 \, {\left (6 \, c f^{2} + d f\right )} x + d\right )} \cosh \left (f x + e\right )^{3} + 12 \, {\left (18 \, d f^{2} x^{2} + 6 \, c f + 6 \, {\left (6 \, c f^{2} + d f\right )} x + d\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + 4 \, {\left (18 \, d f^{2} x^{2} - 6 \, c f + 6 \, {\left (6 \, c f^{2} - d f\right )} x - d\right )} \sinh \left (f x + e\right )^{3} + 27 \, {\left (4 \, d f x + 4 \, c f + 3 \, d\right )} \cosh \left (f x + e\right ) + 3 \, {\left (108 \, d f x + 4 \, {\left (18 \, d f^{2} x^{2} - 6 \, c f + 6 \, {\left (6 \, c f^{2} - d f\right )} x - d\right )} \cosh \left (f x + e\right )^{2} + 108 \, c f + 45 \, d\right )} \sinh \left (f x + e\right )}{1152 \, {\left (a^{3} f^{2} \cosh \left (f x + e\right )^{3} + 3 \, a^{3} f^{2} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, a^{3} f^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + a^{3} f^{2} \sinh \left (f x + e\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 151, normalized size = 0.83 \[ \frac {{\left (72 \, d f^{2} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 144 \, c f^{2} x e^{\left (6 \, f x + 6 \, e\right )} + 216 \, d f x e^{\left (4 \, f x + 4 \, e\right )} - 108 \, d f x e^{\left (2 \, f x + 2 \, e\right )} + 24 \, d f x + 216 \, c f e^{\left (4 \, f x + 4 \, e\right )} - 108 \, c f e^{\left (2 \, f x + 2 \, e\right )} + 24 \, c f + 108 \, d e^{\left (4 \, f x + 4 \, e\right )} - 27 \, d e^{\left (2 \, f x + 2 \, e\right )} + 4 \, d\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{1152 \, a^{3} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.63, size = 651, normalized size = 3.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 138, normalized size = 0.75 \[ \frac {1}{96} \, c {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} + \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} - 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} + 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d e^{\left (-6 \, e\right )}}{1152 \, a^{3} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 127, normalized size = 0.69 \[ {\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {d+6\,c\,f}{288\,a^3\,f^2}+\frac {d\,x}{48\,a^3\,f}\right )+{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {3\,d+6\,c\,f}{32\,a^3\,f^2}+\frac {3\,d\,x}{16\,a^3\,f}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {3\,d+12\,c\,f}{128\,a^3\,f^2}+\frac {3\,d\,x}{32\,a^3\,f}\right )+\frac {d\,x^2}{16\,a^3}+\frac {c\,x}{8\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.40, size = 1287, normalized size = 7.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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