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

Optimal. Leaf size=170 \[ -\frac{d (c+d x) e^{-4 e-4 f x}}{32 a^2 f^2}+\frac{d (c+d x) e^{-2 e-2 f x}}{4 a^2 f^2}-\frac{(c+d x)^2 e^{-4 e-4 f x}}{16 a^2 f}+\frac{(c+d x)^2 e^{-2 e-2 f x}}{4 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}-\frac{d^2 e^{-4 e-4 f x}}{128 a^2 f^3}+\frac{d^2 e^{-2 e-2 f x}}{8 a^2 f^3} \]

[Out]

-(d^2*E^(-4*e - 4*f*x))/(128*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))/(3
2*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)

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Rubi [A]  time = 0.191968, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3729, 2176, 2194} \[ -\frac{d (c+d x) e^{-4 e-4 f x}}{32 a^2 f^2}+\frac{d (c+d x) e^{-2 e-2 f x}}{4 a^2 f^2}-\frac{(c+d x)^2 e^{-4 e-4 f x}}{16 a^2 f}+\frac{(c+d x)^2 e^{-2 e-2 f x}}{4 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}-\frac{d^2 e^{-4 e-4 f x}}{128 a^2 f^3}+\frac{d^2 e^{-2 e-2 f x}}{8 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^2*E^(-4*e - 4*f*x))/(128*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))/(3
2*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 3729

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*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rule 2176

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] &&  !$UseGamma === True

Rule 2194

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]

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+a \coth (e+f x))^2} \, dx &=\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]  time = 0.946135, size = 207, normalized size = 1.22 \[ \frac{\text{csch}^2(e+f x) \left (\left (24 c^2 f^2 (4 f x+1)+12 c d f \left (8 f^2 x^2+4 f x+1\right )+d^2 \left (32 f^3 x^3+24 f^2 x^2+12 f x+3\right )\right ) \sinh (2 (e+f x))+\left (24 c^2 f^2 (4 f x-1)+12 c d f \left (8 f^2 x^2-4 f x-1\right )+d^2 \left (32 f^3 x^3-24 f^2 x^2-12 f x-3\right )\right ) \cosh (2 (e+f x))+48 \left (2 c^2 f^2+2 c d f (2 f x+1)+d^2 \left (2 f^2 x^2+2 f x+1\right )\right )\right )}{384 a^2 f^3 (\coth (e+f x)+1)^2} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.097, size = 1087, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2/f*(-2/f^2*d^2*(1/4*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^2-1/4*(f*x+e)^2*cosh(f*x+e)^2-1/8*(f*x+e)*sinh(f*
x+e)*cosh(f*x+e)^3+5/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+5/32*(f*x+e)^2+1/32*sinh(f*x+e)^2*cosh(f*x+e)^2-1/8*co
sh(f*x+e)^2)+2/f^2*d^2*(1/4*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3-1/8*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)-1/24*(f*
x+e)^3-1/8*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2+1/32*cosh(f*x+e)^3*sinh(f*x+e)-1/64*cosh(f*x+e)*sinh(f*x+e)-1/6
4*f*x-1/64*e)-1/f^2*d^2*(1/2*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)-1/6*(f*x+e)^3-1/2*(f*x+e)*cosh(f*x+e)^2+1/4*cos
h(f*x+e)*sinh(f*x+e)+1/4*f*x+1/4*e)+4/f^2*d^2*e*(1/4*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2-1/4*(f*x+e)*cosh(f*x+
e)^2-1/16*cosh(f*x+e)^3*sinh(f*x+e)+5/32*cosh(f*x+e)*sinh(f*x+e)+5/32*f*x+5/32*e)-4/f^2*d^2*e*(1/4*(f*x+e)*sin
h(f*x+e)*cosh(f*x+e)^3-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/16*(f*x+e)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e)^2)+2/
f^2*d^2*e*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^2-1/4*cosh(f*x+e)^2)-2/f^2*d^2*e^2*(1/4*sinh(f*x+e)
^2*cosh(f*x+e)^2-1/4*cosh(f*x+e)^2)+2/f^2*d^2*e^2*(1/4*cosh(f*x+e)^3*sinh(f*x+e)-1/8*cosh(f*x+e)*sinh(f*x+e)-1
/8*f*x-1/8*e)-d^2*e^2/f^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)-4*c/f*d*(1/4*(f*x+e)*sinh(f*x+e)^2*cosh(
f*x+e)^2-1/4*(f*x+e)*cosh(f*x+e)^2-1/16*cosh(f*x+e)^3*sinh(f*x+e)+5/32*cosh(f*x+e)*sinh(f*x+e)+5/32*f*x+5/32*e
)+4*c/f*d*(1/4*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/16*(f*x+e)^2-1/16*sinh(
f*x+e)^2*cosh(f*x+e)^2)-2/f*d*c*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^2-1/4*cosh(f*x+e)^2)+4*c/f*d*
e*(1/4*sinh(f*x+e)^2*cosh(f*x+e)^2-1/4*cosh(f*x+e)^2)-4*c/f*d*e*(1/4*cosh(f*x+e)^3*sinh(f*x+e)-1/8*cosh(f*x+e)
*sinh(f*x+e)-1/8*f*x-1/8*e)+2*d*e/f*c*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)-2*c^2*(1/4*sinh(f*x+e)^2*cos
h(f*x+e)^2-1/4*cosh(f*x+e)^2)+2*c^2*(1/4*cosh(f*x+e)^3*sinh(f*x+e)-1/8*cosh(f*x+e)*sinh(f*x+e)-1/8*f*x-1/8*e)-
c^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e))

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Maxima [A]  time = 2.02766, size = 258, normalized size = 1.52 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.26552, size = 811, normalized size = 4.77 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 3.46332, size = 1358, normalized size = 7.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((48*c**2*f**3*x*tanh(e + f*x)**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192
*a**2*f**3) + 96*c**2*f**3*x*tanh(e + f*x)/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192
*a**2*f**3) + 48*c**2*f**3*x/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) -
72*c**2*f**2*tanh(e + f*x)**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) +
 24*c**2*f**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 48*c*d*f**3*x**
2*tanh(e + f*x)**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 96*c*d*f**
3*x**2*tanh(e + f*x)/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 48*c*d*f
**3*x**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) - 120*c*d*f**2*x*tanh(
e + f*x)**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 48*c*d*f**2*x*tan
h(e + f*x)/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 72*c*d*f**2*x/(192
*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) - 60*c*d*f*tanh(e + f*x)**2/(192*a*
*2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 36*c*d*f/(192*a**2*f**3*tanh(e + f*x
)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 16*d**2*f**3*x**3*tanh(e + f*x)**2/(192*a**2*f**3*tanh(e
 + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 32*d**2*f**3*x**3*tanh(e + f*x)/(192*a**2*f**3*tan
h(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 16*d**2*f**3*x**3/(192*a**2*f**3*tanh(e + f*x)*
*2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) - 60*d**2*f**2*x**2*tanh(e + f*x)**2/(192*a**2*f**3*tanh(e +
 f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 24*d**2*f**2*x**2*tanh(e + f*x)/(192*a**2*f**3*tanh(
e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 36*d**2*f**2*x**2/(192*a**2*f**3*tanh(e + f*x)**2
 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) - 54*d**2*f*x*tanh(e + f*x)**2/(192*a**2*f**3*tanh(e + f*x)**2
 + 384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 12*d**2*f*x*tanh(e + f*x)/(192*a**2*f**3*tanh(e + f*x)**2 +
384*a**2*f**3*tanh(e + f*x) + 192*a**2*f**3) + 42*d**2*f*x/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tan
h(e + f*x) + 192*a**2*f**3) - 27*d**2*tanh(e + f*x)**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e
+ f*x) + 192*a**2*f**3) + 21*d**2/(192*a**2*f**3*tanh(e + f*x)**2 + 384*a**2*f**3*tanh(e + f*x) + 192*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))

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Giac [A]  time = 1.1428, size = 306, normalized size = 1.8 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)