\(\int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx\) [23]

Optimal result

Integrand size = 20, antiderivative size = 170 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx=-\frac {d^2 e^{-4 e-4 f x}}{128 a^2 f^3}+\frac {d^2 e^{-2 e-2 f x}}{8 a^2 f^3}-\frac {d e^{-4 e-4 f x} (c+d x)}{32 a^2 f^2}+\frac {d e^{-2 e-2 f x} (c+d x)}{4 a^2 f^2}-\frac {e^{-4 e-4 f x} (c+d x)^2}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^2}{4 a^2 f}+\frac {(c+d x)^3}{12 a^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3810, 2207, 2225} \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx=-\frac {d (c+d x) e^{-4 e-4 f x}}{32 a^2 f^2}+\frac {d (c+d x) e^{-2 e-2 f x}}{4 a^2 f^2}-\frac {(c+d x)^2 e^{-4 e-4 f x}}{16 a^2 f}+\frac {(c+d x)^2 e^{-2 e-2 f x}}{4 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}-\frac {d^2 e^{-4 e-4 f x}}{128 a^2 f^3}+\frac {d^2 e^{-2 e-2 f x}}{8 a^2 f^3} \]

[In]

Int[(c + d*x)^2/(a + a*Coth[e + f*x])^2,x]

[Out]

-1/128*(d^2*E^(-4*e - 4*f*x))/(a^2*f^3) + (d^2*E^(-2*e - 2*f*x))/(8*a^2*f^3) - (d*E^(-4*e - 4*f*x)*(c + d*x))/
(32*a^2*f^2) + (d*E^(-2*e - 2*f*x)*(c + d*x))/(4*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^2)/(16*a^2*f) + (E^(-2
*e - 2*f*x)*(c + d*x)^2)/(4*a^2*f) + (c + d*x)^3/(12*a^2*d)

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 3810

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c
+ d*x)^m, (1/(2*a) + E^(2*(a/b)*(e + f*x))/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(c+d x)^2}{4 a^2}+\frac {e^{-4 e-4 f x} (c+d x)^2}{4 a^2}-\frac {e^{-2 e-2 f x} (c+d x)^2}{2 a^2}\right ) \, dx \\ & = \frac {(c+d x)^3}{12 a^2 d}+\frac {\int e^{-4 e-4 f x} (c+d x)^2 \, dx}{4 a^2}-\frac {\int e^{-2 e-2 f x} (c+d x)^2 \, dx}{2 a^2} \\ & = -\frac {e^{-4 e-4 f x} (c+d x)^2}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^2}{4 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}+\frac {d \int e^{-4 e-4 f x} (c+d x) \, dx}{8 a^2 f}-\frac {d \int e^{-2 e-2 f x} (c+d x) \, dx}{2 a^2 f} \\ & = -\frac {d e^{-4 e-4 f x} (c+d x)}{32 a^2 f^2}+\frac {d e^{-2 e-2 f x} (c+d x)}{4 a^2 f^2}-\frac {e^{-4 e-4 f x} (c+d x)^2}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^2}{4 a^2 f}+\frac {(c+d x)^3}{12 a^2 d}+\frac {d^2 \int e^{-4 e-4 f x} \, dx}{32 a^2 f^2}-\frac {d^2 \int e^{-2 e-2 f x} \, dx}{4 a^2 f^2} \\ & = -\frac {d^2 e^{-4 e-4 f x}}{128 a^2 f^3}+\frac {d^2 e^{-2 e-2 f x}}{8 a^2 f^3}-\frac {d e^{-4 e-4 f x} (c+d x)}{32 a^2 f^2}+\frac {d e^{-2 e-2 f x} (c+d x)}{4 a^2 f^2}-\frac {e^{-4 e-4 f x} (c+d x)^2}{16 a^2 f}+\frac {e^{-2 e-2 f x} (c+d x)^2}{4 a^2 f}+\frac {(c+d x)^3}{12 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx=\frac {\text {csch}^2(e+f x) \left (48 \left (2 c^2 f^2+2 c d f (1+2 f x)+d^2 \left (1+2 f x+2 f^2 x^2\right )\right )+\left (24 c^2 f^2 (-1+4 f x)+12 c d f \left (-1-4 f x+8 f^2 x^2\right )+d^2 \left (-3-12 f x-24 f^2 x^2+32 f^3 x^3\right )\right ) \cosh (2 (e+f x))+\left (24 c^2 f^2 (1+4 f x)+12 c d f \left (1+4 f x+8 f^2 x^2\right )+d^2 \left (3+12 f x+24 f^2 x^2+32 f^3 x^3\right )\right ) \sinh (2 (e+f x))\right )}{384 a^2 f^3 (1+\coth (e+f x))^2} \]

[In]

Integrate[(c + d*x)^2/(a + a*Coth[e + f*x])^2,x]

[Out]

(Csch[e + f*x]^2*(48*(2*c^2*f^2 + 2*c*d*f*(1 + 2*f*x) + d^2*(1 + 2*f*x + 2*f^2*x^2)) + (24*c^2*f^2*(-1 + 4*f*x
) + 12*c*d*f*(-1 - 4*f*x + 8*f^2*x^2) + d^2*(-3 - 12*f*x - 24*f^2*x^2 + 32*f^3*x^3))*Cosh[2*(e + f*x)] + (24*c
^2*f^2*(1 + 4*f*x) + 12*c*d*f*(1 + 4*f*x + 8*f^2*x^2) + d^2*(3 + 12*f*x + 24*f^2*x^2 + 32*f^3*x^3))*Sinh[2*(e
+ f*x)]))/(384*a^2*f^3*(1 + Coth[e + f*x])^2)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {d^{2} x^{3}}{12 a^{2}}+\frac {d c \,x^{2}}{4 a^{2}}+\frac {c^{2} x}{4 a^{2}}+\frac {c^{3}}{12 a^{2} d}+\frac {\left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}+2 d^{2} f x +2 c d f +d^{2}\right ) {\mathrm e}^{-2 f x -2 e}}{8 a^{2} f^{3}}-\frac {\left (8 d^{2} x^{2} f^{2}+16 c d \,f^{2} x +8 c^{2} f^{2}+4 d^{2} f x +4 c d f +d^{2}\right ) {\mathrm e}^{-4 f x -4 e}}{128 a^{2} f^{3}}\)	\(163\)
parallelrisch	\(\frac {24 x f \left (\left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) f^{2}-\frac {5 d \left (\frac {d x}{2}+c \right ) f}{2}-\frac {9 d^{2}}{8}\right ) \tanh \left (f x +e \right )^{2}+\left (16 \left (d^{2} x^{3}+3 d c \,x^{2}+3 c^{2} x \right ) f^{3}+12 \left (x^{2} d^{2}+2 c d x +6 c^{2}\right ) f^{2}+6 \left (x \,d^{2}+10 c d \right ) f +27 d^{2}\right ) \tanh \left (f x +e \right )+8 \left (d^{2} x^{3}+3 d c \,x^{2}+3 c^{2} x \right ) f^{3}+6 \left (3 x^{2} d^{2}+6 c d x +8 c^{2}\right ) f^{2}+3 \left (7 x \,d^{2}+16 c d \right ) f +24 d^{2}}{96 f^{3} a^{2} \left (1+\tanh \left (f x +e \right )\right )^{2}}\)	\(212\)

[In]

int((d*x+c)^2/(a+a*coth(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/12/a^2*d^2*x^3+1/4/a^2*d*c*x^2+1/4/a^2*c^2*x+1/12/a^2/d*c^3+1/8*(2*d^2*f^2*x^2+4*c*d*f^2*x+2*c^2*f^2+2*d^2*f
*x+2*c*d*f+d^2)/a^2/f^3*exp(-2*f*x-2*e)-1/128*(8*d^2*f^2*x^2+16*c*d*f^2*x+8*c^2*f^2+4*d^2*f*x+4*c*d*f+d^2)/a^2
/f^3*exp(-4*f*x-4*e)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (150) = 300\).

Time = 0.27 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.11 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx=\frac {96 \, d^{2} f^{2} x^{2} + 96 \, c^{2} f^{2} + 96 \, c d f + {\left (32 \, d^{2} f^{3} x^{3} - 24 \, c^{2} f^{2} - 12 \, c d f + 24 \, {\left (4 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - 3 \, d^{2} + 12 \, {\left (8 \, c^{2} f^{3} - 4 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (32 \, d^{2} f^{3} x^{3} + 24 \, c^{2} f^{2} + 12 \, c d f + 24 \, {\left (4 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + 3 \, d^{2} + 12 \, {\left (8 \, c^{2} f^{3} + 4 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (32 \, d^{2} f^{3} x^{3} - 24 \, c^{2} f^{2} - 12 \, c d f + 24 \, {\left (4 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - 3 \, d^{2} + 12 \, {\left (8 \, c^{2} f^{3} - 4 \, c d f^{2} - d^{2} f\right )} x\right )} \sinh \left (f x + e\right )^{2} + 48 \, d^{2} + 96 \, {\left (2 \, c d f^{2} + d^{2} f\right )} x}{384 \, {\left (a^{2} f^{3} \cosh \left (f x + e\right )^{2} + 2 \, a^{2} f^{3} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a^{2} f^{3} \sinh \left (f x + e\right )^{2}\right )}} \]

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e))^2,x, algorithm="fricas")

[Out]

1/384*(96*d^2*f^2*x^2 + 96*c^2*f^2 + 96*c*d*f + (32*d^2*f^3*x^3 - 24*c^2*f^2 - 12*c*d*f + 24*(4*c*d*f^3 - d^2*
f^2)*x^2 - 3*d^2 + 12*(8*c^2*f^3 - 4*c*d*f^2 - d^2*f)*x)*cosh(f*x + e)^2 + 2*(32*d^2*f^3*x^3 + 24*c^2*f^2 + 12
*c*d*f + 24*(4*c*d*f^3 + d^2*f^2)*x^2 + 3*d^2 + 12*(8*c^2*f^3 + 4*c*d*f^2 + d^2*f)*x)*cosh(f*x + e)*sinh(f*x +
e) + (32*d^2*f^3*x^3 - 24*c^2*f^2 - 12*c*d*f + 24*(4*c*d*f^3 - d^2*f^2)*x^2 - 3*d^2 + 12*(8*c^2*f^3 - 4*c*d*f
^2 - d^2*f)*x)*sinh(f*x + e)^2 + 48*d^2 + 96*(2*c*d*f^2 + d^2*f)*x)/(a^2*f^3*cosh(f*x + e)^2 + 2*a^2*f^3*cosh(
f*x + e)*sinh(f*x + e) + a^2*f^3*sinh(f*x + e)^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1353 vs. \(2 (165) = 330\).

Time = 0.89 (sec) , antiderivative size = 1353, normalized size of antiderivative = 7.96 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)**2/(a+a*coth(f*x+e))**2,x)

[Out]

Piecewise((24*c**2*f**3*x*tanh(e + f*x)**2/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a
**2*f**3) + 48*c**2*f**3*x*tanh(e + f*x)/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**
2*f**3) + 24*c**2*f**3*x/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 72*c**
2*f**2*tanh(e + f*x)/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 48*c**2*f*
*2/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 24*c*d*f**3*x**2*tanh(e + f*
x)**2/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 48*c*d*f**3*x**2*tanh(e +
f*x)/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 24*c*d*f**3*x**2/(96*a**2
*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) - 60*c*d*f**2*x*tanh(e + f*x)**2/(96*a**2
*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 24*c*d*f**2*x*tanh(e + f*x)/(96*a**2*f*
*3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 36*c*d*f**2*x/(96*a**2*f**3*tanh(e + f*x)*
*2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 60*c*d*f*tanh(e + f*x)/(96*a**2*f**3*tanh(e + f*x)**2 + 192
*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 48*c*d*f/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*
x) + 96*a**2*f**3) + 8*d**2*f**3*x**3*tanh(e + f*x)**2/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e +
f*x) + 96*a**2*f**3) + 16*d**2*f**3*x**3*tanh(e + f*x)/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e
+ f*x) + 96*a**2*f**3) + 8*d**2*f**3*x**3/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a*
*2*f**3) - 30*d**2*f**2*x**2*tanh(e + f*x)**2/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 9
6*a**2*f**3) + 12*d**2*f**2*x**2*tanh(e + f*x)/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) +
96*a**2*f**3) + 18*d**2*f**2*x**2/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3)
- 27*d**2*f*x*tanh(e + f*x)**2/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) +
6*d**2*f*x*tanh(e + f*x)/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 21*d*
*2*f*x/(96*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 27*d**2*tanh(e + f*x)/(9
6*a**2*f**3*tanh(e + f*x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3) + 24*d**2/(96*a**2*f**3*tanh(e + f*
x)**2 + 192*a**2*f**3*tanh(e + f*x) + 96*a**2*f**3), Ne(f, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)/(a*coth(e)
+ a)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.12 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx=\frac {1}{16} \, c^{2} {\left (\frac {4 \, {\left (f x + e\right )}}{a^{2} f} + \frac {4 \, e^{\left (-2 \, f x - 2 \, e\right )} - e^{\left (-4 \, f x - 4 \, e\right )}}{a^{2} f}\right )} + \frac {{\left (8 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 8 \, {\left (2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - {\left (4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} c d e^{\left (-4 \, e\right )}}{32 \, a^{2} f^{2}} + \frac {{\left (32 \, f^{3} x^{3} e^{\left (4 \, e\right )} + 48 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 3 \, {\left (8 \, f^{2} x^{2} + 4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} d^{2} e^{\left (-4 \, e\right )}}{384 \, a^{2} f^{3}} \]

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e))^2,x, algorithm="maxima")

[Out]

1/16*c^2*(4*(f*x + e)/(a^2*f) + (4*e^(-2*f*x - 2*e) - e^(-4*f*x - 4*e))/(a^2*f)) + 1/32*(8*f^2*x^2*e^(4*e) + 8
*(2*f*x*e^(2*e) + e^(2*e))*e^(-2*f*x) - (4*f*x + 1)*e^(-4*f*x))*c*d*e^(-4*e)/(a^2*f^2) + 1/384*(32*f^3*x^3*e^(
4*e) + 48*(2*f^2*x^2*e^(2*e) + 2*f*x*e^(2*e) + e^(2*e))*e^(-2*f*x) - 3*(8*f^2*x^2 + 4*f*x + 1)*e^(-4*f*x))*d^2
*e^(-4*e)/(a^2*f^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx=\frac {{\left (32 \, d^{2} f^{3} x^{3} e^{\left (4 \, f x + 4 \, e\right )} + 96 \, c d f^{3} x^{2} e^{\left (4 \, f x + 4 \, e\right )} + 96 \, c^{2} f^{3} x e^{\left (4 \, f x + 4 \, e\right )} + 96 \, d^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 24 \, d^{2} f^{2} x^{2} + 192 \, c d f^{2} x e^{\left (2 \, f x + 2 \, e\right )} - 48 \, c d f^{2} x + 96 \, c^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 96 \, d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} - 24 \, c^{2} f^{2} - 12 \, d^{2} f x + 96 \, c d f e^{\left (2 \, f x + 2 \, e\right )} - 12 \, c d f + 48 \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, d^{2}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{384 \, a^{2} f^{3}} \]

[In]

integrate((d*x+c)^2/(a+a*coth(f*x+e))^2,x, algorithm="giac")

[Out]

1/384*(32*d^2*f^3*x^3*e^(4*f*x + 4*e) + 96*c*d*f^3*x^2*e^(4*f*x + 4*e) + 96*c^2*f^3*x*e^(4*f*x + 4*e) + 96*d^2
*f^2*x^2*e^(2*f*x + 2*e) - 24*d^2*f^2*x^2 + 192*c*d*f^2*x*e^(2*f*x + 2*e) - 48*c*d*f^2*x + 96*c^2*f^2*e^(2*f*x
+ 2*e) + 96*d^2*f*x*e^(2*f*x + 2*e) - 24*c^2*f^2 - 12*d^2*f*x + 96*c*d*f*e^(2*f*x + 2*e) - 12*c*d*f + 48*d^2*
e^(2*f*x + 2*e) - 3*d^2)*e^(-4*f*x - 4*e)/(a^2*f^3)

Mupad [B] (verification not implemented)

Time = 1.94 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx={\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {2\,c^2\,f^2+2\,c\,d\,f+d^2}{8\,a^2\,f^3}+\frac {d^2\,x^2}{4\,a^2\,f}+\frac {d\,x\,\left (d+2\,c\,f\right )}{4\,a^2\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {8\,c^2\,f^2+4\,c\,d\,f+d^2}{128\,a^2\,f^3}+\frac {d^2\,x^2}{16\,a^2\,f}+\frac {d\,x\,\left (d+4\,c\,f\right )}{32\,a^2\,f^2}\right )+\frac {c^2\,x}{4\,a^2}+\frac {d^2\,x^3}{12\,a^2}+\frac {c\,d\,x^2}{4\,a^2} \]

[In]

int((c + d*x)^2/(a + a*coth(e + f*x))^2,x)

[Out]

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