\(\int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx\) [29]

Optimal result

Integrand size = 18, antiderivative size = 183 \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx=\frac {11 d x}{96 a^3 f}-\frac {d x^2}{16 a^3}+\frac {x (c+d x)}{8 a^3}-\frac {d}{36 f^2 (a+a \coth (e+f x))^3}-\frac {c+d x}{6 f (a+a \coth (e+f x))^3}-\frac {5 d}{96 a f^2 (a+a \coth (e+f x))^2}-\frac {c+d x}{8 a f (a+a \coth (e+f x))^2}-\frac {11 d}{96 f^2 \left (a^3+a^3 \coth (e+f x)\right )}-\frac {c+d x}{8 f \left (a^3+a^3 \coth (e+f x)\right )} \]

[Out]

11/96*d*x/a^3/f-1/16*d*x^2/a^3+1/8*x*(d*x+c)/a^3-1/36*d/f^2/(a+a*coth(f*x+e))^3+1/6*(-d*x-c)/f/(a+a*coth(f*x+e
))^3-5/96*d/a/f^2/(a+a*coth(f*x+e))^2+1/8*(-d*x-c)/a/f/(a+a*coth(f*x+e))^2-11/96*d/f^2/(a^3+a^3*coth(f*x+e))+1
/8*(-d*x-c)/f/(a^3+a^3*coth(f*x+e))

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3560, 8, 3811} \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx=-\frac {c+d x}{8 f \left (a^3 \coth (e+f x)+a^3\right )}+\frac {x (c+d x)}{8 a^3}-\frac {11 d}{96 f^2 \left (a^3 \coth (e+f x)+a^3\right )}+\frac {11 d x}{96 a^3 f}-\frac {d x^2}{16 a^3}-\frac {c+d x}{8 a f (a \coth (e+f x)+a)^2}-\frac {c+d x}{6 f (a \coth (e+f x)+a)^3}-\frac {5 d}{96 a f^2 (a \coth (e+f x)+a)^2}-\frac {d}{36 f^2 (a \coth (e+f x)+a)^3} \]

[In]

Int[(c + d*x)/(a + a*Coth[e + f*x])^3,x]

[Out]

(11*d*x)/(96*a^3*f) - (d*x^2)/(16*a^3) + (x*(c + d*x))/(8*a^3) - d/(36*f^2*(a + a*Coth[e + f*x])^3) - (c + d*x
)/(6*f*(a + a*Coth[e + f*x])^3) - (5*d)/(96*a*f^2*(a + a*Coth[e + f*x])^2) - (c + d*x)/(8*a*f*(a + a*Coth[e +
f*x])^2) - (11*d)/(96*f^2*(a^3 + a^3*Coth[e + f*x])) - (c + d*x)/(8*f*(a^3 + a^3*Coth[e + f*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3560

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 3811

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{u = IntHide[(a
+ b*Tan[e + f*x])^n, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[Dist[(c + d*x)^(m - 1), u, x], x], x]] /; Fr
eeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && ILtQ[n, -1] && GtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx=\frac {\text {csch}^3(e+f x) \left (27 (4 c f+d (3+4 f x)) \cosh (e+f x)+4 \left (6 c f (1+6 f x)+d \left (1+6 f x+18 f^2 x^2\right )\right ) \cosh (3 (e+f x))+135 d \sinh (e+f x)+324 c f \sinh (e+f x)+324 d f x \sinh (e+f x)-4 d \sinh (3 (e+f x))-24 c f \sinh (3 (e+f x))-24 d f x \sinh (3 (e+f x))+144 c f^2 x \sinh (3 (e+f x))+72 d f^2 x^2 \sinh (3 (e+f x))\right )}{1152 a^3 f^2 (1+\coth (e+f x))^3} \]

[In]

Integrate[(c + d*x)/(a + a*Coth[e + f*x])^3,x]

[Out]

(Csch[e + f*x]^3*(27*(4*c*f + d*(3 + 4*f*x))*Cosh[e + f*x] + 4*(6*c*f*(1 + 6*f*x) + d*(1 + 6*f*x + 18*f^2*x^2)
)*Cosh[3*(e + f*x)] + 135*d*Sinh[e + f*x] + 324*c*f*Sinh[e + f*x] + 324*d*f*x*Sinh[e + f*x] - 4*d*Sinh[3*(e +
f*x)] - 24*c*f*Sinh[3*(e + f*x)] - 24*d*f*x*Sinh[3*(e + f*x)] + 144*c*f^2*x*Sinh[3*(e + f*x)] + 72*d*f^2*x^2*S
inh[3*(e + f*x)]))/(1152*a^3*f^2*(1 + Coth[e + f*x])^3)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {d \,x^{2}}{16 a^{3}}+\frac {c x}{8 a^{3}}+\frac {3 \left (2 d x f +2 c f +d \right ) {\mathrm e}^{-2 f x -2 e}}{32 a^{3} f^{2}}-\frac {3 \left (4 d x f +4 c f +d \right ) {\mathrm e}^{-4 f x -4 e}}{128 a^{3} f^{2}}+\frac {\left (6 d x f +6 c f +d \right ) {\mathrm e}^{-6 f x -6 e}}{288 a^{3} f^{2}}\)	\(102\)
parallelrisch	\(\frac {36 \left (\left (\frac {d x}{2}+c \right ) f -\frac {29 d}{12}\right ) x f \tanh \left (f x +e \right )^{3}+\left (\left (54 d \,x^{2}+108 c x \right ) f^{2}+\left (-9 d x +252 c \right ) f +87 d \right ) \tanh \left (f x +e \right )^{2}+\left (\left (54 d \,x^{2}+108 c x \right ) f^{2}+\left (63 d x +324 c \right ) f +135 d \right ) \tanh \left (f x +e \right )+\left (18 d \,x^{2}+36 c x \right ) f^{2}+\left (33 d x +120 c \right ) f +56 d}{288 f^{2} a^{3} \left (1+\tanh \left (f x +e \right )\right )^{3}}\)	\(146\)

[In]

int((d*x+c)/(a+a*coth(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/16*d*x^2/a^3+1/8/a^3*c*x+3/32*(2*d*f*x+2*c*f+d)/a^3/f^2*exp(-2*f*x-2*e)-3/128*(4*d*f*x+4*c*f+d)/a^3/f^2*exp(
-4*f*x-4*e)+1/288*(6*d*f*x+6*c*f+d)/a^3/f^2*exp(-6*f*x-6*e)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.56 \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx=\frac {4 \, {\left (18 \, d f^{2} x^{2} + 6 \, c f + 6 \, {\left (6 \, c f^{2} + d f\right )} x + d\right )} \cosh \left (f x + e\right )^{3} + 12 \, {\left (18 \, d f^{2} x^{2} + 6 \, c f + 6 \, {\left (6 \, c f^{2} + d f\right )} x + d\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + 4 \, {\left (18 \, d f^{2} x^{2} - 6 \, c f + 6 \, {\left (6 \, c f^{2} - d f\right )} x - d\right )} \sinh \left (f x + e\right )^{3} + 27 \, {\left (4 \, d f x + 4 \, c f + 3 \, d\right )} \cosh \left (f x + e\right ) + 3 \, {\left (108 \, d f x + 4 \, {\left (18 \, d f^{2} x^{2} - 6 \, c f + 6 \, {\left (6 \, c f^{2} - d f\right )} x - d\right )} \cosh \left (f x + e\right )^{2} + 108 \, c f + 45 \, d\right )} \sinh \left (f x + e\right )}{1152 \, {\left (a^{3} f^{2} \cosh \left (f x + e\right )^{3} + 3 \, a^{3} f^{2} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, a^{3} f^{2} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + a^{3} f^{2} \sinh \left (f x + e\right )^{3}\right )}} \]

[In]

integrate((d*x+c)/(a+a*coth(f*x+e))^3,x, algorithm="fricas")

[Out]

1/1152*(4*(18*d*f^2*x^2 + 6*c*f + 6*(6*c*f^2 + d*f)*x + d)*cosh(f*x + e)^3 + 12*(18*d*f^2*x^2 + 6*c*f + 6*(6*c
*f^2 + d*f)*x + d)*cosh(f*x + e)*sinh(f*x + e)^2 + 4*(18*d*f^2*x^2 - 6*c*f + 6*(6*c*f^2 - d*f)*x - d)*sinh(f*x
+ e)^3 + 27*(4*d*f*x + 4*c*f + 3*d)*cosh(f*x + e) + 3*(108*d*f*x + 4*(18*d*f^2*x^2 - 6*c*f + 6*(6*c*f^2 - d*f
)*x - d)*cosh(f*x + e)^2 + 108*c*f + 45*d)*sinh(f*x + e))/(a^3*f^2*cosh(f*x + e)^3 + 3*a^3*f^2*cosh(f*x + e)^2
*sinh(f*x + e) + 3*a^3*f^2*cosh(f*x + e)*sinh(f*x + e)^2 + a^3*f^2*sinh(f*x + e)^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1287 vs. \(2 (170) = 340\).

Time = 1.11 (sec) , antiderivative size = 1287, normalized size of antiderivative = 7.03 \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)/(a+a*coth(f*x+e))**3,x)

[Out]

Piecewise((36*c*f**2*x*tanh(e + f*x)**3/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864
*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 108*c*f**2*x*tanh(e + f*x)**2/(288*a**3*f**2*tanh(e + f*x)**3 + 86
4*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 108*c*f**2*x*tanh(e + f*x)/(288*
a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 3
6*c*f**2*x/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 28
8*a**3*f**2) + 252*c*f*tanh(e + f*x)**2/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864
*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 324*c*f*tanh(e + f*x)/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f
**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 120*c*f/(288*a**3*f**2*tanh(e + f*x)**3
+ 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 18*d*f**2*x**2*tanh(e + f*x)
**3/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*
f**2) + 54*d*f**2*x**2*tanh(e + f*x)**2/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864
*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 54*d*f**2*x**2*tanh(e + f*x)/(288*a**3*f**2*tanh(e + f*x)**3 + 864
*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 18*d*f**2*x**2/(288*a**3*f**2*tan
h(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) - 87*d*f*x*tanh(
e + f*x)**3/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 2
88*a**3*f**2) - 9*d*f*x*tanh(e + f*x)**2/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 86
4*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 63*d*f*x*tanh(e + f*x)/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3
*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 33*d*f*x/(288*a**3*f**2*tanh(e + f*x)*
*3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) + 87*d*tanh(e + f*x)**2/(28
8*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e + f*x) + 288*a**3*f**2) +
135*d*tanh(e + f*x)/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2*tanh(e +
f*x) + 288*a**3*f**2) + 56*d/(288*a**3*f**2*tanh(e + f*x)**3 + 864*a**3*f**2*tanh(e + f*x)**2 + 864*a**3*f**2
*tanh(e + f*x) + 288*a**3*f**2), Ne(f, 0)), ((c*x + d*x**2/2)/(a*coth(e) + a)**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.68 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.75 \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx=\frac {1}{96} \, c {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} + \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} - 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} + 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d e^{\left (-6 \, e\right )}}{1152 \, a^{3} f^{2}} \]

[In]

integrate((d*x+c)/(a+a*coth(f*x+e))^3,x, algorithm="maxima")

[Out]

1/96*c*(12*(f*x + e)/(a^3*f) + (18*e^(-2*f*x - 2*e) - 9*e^(-4*f*x - 4*e) + 2*e^(-6*f*x - 6*e))/(a^3*f)) + 1/11
52*(72*f^2*x^2*e^(6*e) + 108*(2*f*x*e^(4*e) + e^(4*e))*e^(-2*f*x) - 27*(4*f*x*e^(2*e) + e^(2*e))*e^(-4*f*x) +
4*(6*f*x + 1)*e^(-6*f*x))*d*e^(-6*e)/(a^3*f^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx=\frac {{\left (72 \, d f^{2} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 144 \, c f^{2} x e^{\left (6 \, f x + 6 \, e\right )} + 216 \, d f x e^{\left (4 \, f x + 4 \, e\right )} - 108 \, d f x e^{\left (2 \, f x + 2 \, e\right )} + 24 \, d f x + 216 \, c f e^{\left (4 \, f x + 4 \, e\right )} - 108 \, c f e^{\left (2 \, f x + 2 \, e\right )} + 24 \, c f + 108 \, d e^{\left (4 \, f x + 4 \, e\right )} - 27 \, d e^{\left (2 \, f x + 2 \, e\right )} + 4 \, d\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{1152 \, a^{3} f^{2}} \]

[In]

integrate((d*x+c)/(a+a*coth(f*x+e))^3,x, algorithm="giac")

[Out]

1/1152*(72*d*f^2*x^2*e^(6*f*x + 6*e) + 144*c*f^2*x*e^(6*f*x + 6*e) + 216*d*f*x*e^(4*f*x + 4*e) - 108*d*f*x*e^(
2*f*x + 2*e) + 24*d*f*x + 216*c*f*e^(4*f*x + 4*e) - 108*c*f*e^(2*f*x + 2*e) + 24*c*f + 108*d*e^(4*f*x + 4*e) -
27*d*e^(2*f*x + 2*e) + 4*d)*e^(-6*f*x - 6*e)/(a^3*f^2)

Mupad [B] (verification not implemented)

Time = 2.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69 \[ \int \frac {c+d x}{(a+a \coth (e+f x))^3} \, dx={\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {d+6\,c\,f}{288\,a^3\,f^2}+\frac {d\,x}{48\,a^3\,f}\right )+{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {3\,d+6\,c\,f}{32\,a^3\,f^2}+\frac {3\,d\,x}{16\,a^3\,f}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {3\,d+12\,c\,f}{128\,a^3\,f^2}+\frac {3\,d\,x}{32\,a^3\,f}\right )+\frac {d\,x^2}{16\,a^3}+\frac {c\,x}{8\,a^3} \]

[In]

int((c + d*x)/(a + a*coth(e + f*x))^3,x)

[Out]

exp(- 6*e - 6*f*x)*((d + 6*c*f)/(288*a^3*f^2) + (d*x)/(48*a^3*f)) + exp(- 2*e - 2*f*x)*((3*d + 6*c*f)/(32*a^3*
f^2) + (3*d*x)/(16*a^3*f)) - exp(- 4*e - 4*f*x)*((3*d + 12*c*f)/(128*a^3*f^2) + (3*d*x)/(32*a^3*f)) + (d*x^2)/
(16*a^3) + (c*x)/(8*a^3)