3.28 \(\int \frac{(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx\)

Optimal. Leaf size=246 \[ \frac{d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac{3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}+\frac{3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}+\frac{(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac{3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}+\frac{3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}+\frac{d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac{3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac{3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \]

[Out]

(d^2*E^(-6*e - 6*f*x))/(864*a^3*f^3) - (3*d^2*E^(-4*e - 4*f*x))/(256*a^3*f^3) + (3*d^2*E^(-2*e - 2*f*x))/(32*a
^3*f^3) + (d*E^(-6*e - 6*f*x)*(c + d*x))/(144*a^3*f^2) - (3*d*E^(-4*e - 4*f*x)*(c + d*x))/(64*a^3*f^2) + (3*d*
E^(-2*e - 2*f*x)*(c + d*x))/(16*a^3*f^2) + (E^(-6*e - 6*f*x)*(c + d*x)^2)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c
+ d*x)^2)/(32*a^3*f) + (3*E^(-2*e - 2*f*x)*(c + d*x)^2)/(16*a^3*f) + (c + d*x)^3/(24*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.257589, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3729, 2176, 2194} \[ \frac{d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac{3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}+\frac{3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}+\frac{(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac{3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}+\frac{3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}+\frac{d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac{3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac{3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + a*Coth[e + f*x])^3,x]

[Out]

(d^2*E^(-6*e - 6*f*x))/(864*a^3*f^3) - (3*d^2*E^(-4*e - 4*f*x))/(256*a^3*f^3) + (3*d^2*E^(-2*e - 2*f*x))/(32*a
^3*f^3) + (d*E^(-6*e - 6*f*x)*(c + d*x))/(144*a^3*f^2) - (3*d*E^(-4*e - 4*f*x)*(c + d*x))/(64*a^3*f^2) + (3*d*
E^(-2*e - 2*f*x)*(c + d*x))/(16*a^3*f^2) + (E^(-6*e - 6*f*x)*(c + d*x)^2)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c
+ d*x)^2)/(32*a^3*f) + (3*E^(-2*e - 2*f*x)*(c + d*x)^2)/(16*a^3*f) + (c + d*x)^3/(24*a^3*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))^3} \, dx &=\int \left (\frac{(c+d x)^2}{8 a^3}-\frac{e^{-6 e-6 f x} (c+d x)^2}{8 a^3}+\frac{3 e^{-4 e-4 f x} (c+d x)^2}{8 a^3}-\frac{3 e^{-2 e-2 f x} (c+d x)^2}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^3}{24 a^3 d}-\frac{\int e^{-6 e-6 f x} (c+d x)^2 \, dx}{8 a^3}+\frac{3 \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{8 a^3}-\frac{3 \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{8 a^3}\\ &=\frac{e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac{3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac{3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{d \int e^{-6 e-6 f x} (c+d x) \, dx}{24 a^3 f}+\frac{(3 d) \int e^{-4 e-4 f x} (c+d x) \, dx}{16 a^3 f}-\frac{(3 d) \int e^{-2 e-2 f x} (c+d x) \, dx}{8 a^3 f}\\ &=\frac{d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac{3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac{3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac{e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac{3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac{3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{d^2 \int e^{-6 e-6 f x} \, dx}{144 a^3 f^2}+\frac{\left (3 d^2\right ) \int e^{-4 e-4 f x} \, dx}{64 a^3 f^2}-\frac{\left (3 d^2\right ) \int e^{-2 e-2 f x} \, dx}{16 a^3 f^2}\\ &=\frac{d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac{3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac{3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}+\frac{d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac{3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac{3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac{e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac{3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac{3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}\\ \end{align*}

Mathematica [A]  time = 1.37168, size = 371, normalized size = 1.51 \[ \frac{\text{csch}^3(e+f x) \left (81 \left (8 c^2 f^2+4 c d f (4 f x+3)+d^2 \left (8 f^2 x^2+12 f x+7\right )\right ) \cosh (e+f x)+8 \left (18 c^2 f^2 (6 f x+1)+6 c d f \left (18 f^2 x^2+6 f x+1\right )+d^2 \left (36 f^3 x^3+18 f^2 x^2+6 f x+1\right )\right ) \cosh (3 (e+f x))+864 c^2 f^3 x \sinh (3 (e+f x))+1944 c^2 f^2 \sinh (e+f x)-144 c^2 f^2 \sinh (3 (e+f x))+864 c d f^3 x^2 \sinh (3 (e+f x))+3888 c d f^2 x \sinh (e+f x)-288 c d f^2 x \sinh (3 (e+f x))+1620 c d f \sinh (e+f x)-48 c d f \sinh (3 (e+f x))+288 d^2 f^3 x^3 \sinh (3 (e+f x))+1944 d^2 f^2 x^2 \sinh (e+f x)-144 d^2 f^2 x^2 \sinh (3 (e+f x))+1620 d^2 f x \sinh (e+f x)-48 d^2 f x \sinh (3 (e+f x))+729 d^2 \sinh (e+f x)-8 d^2 \sinh (3 (e+f x))\right )}{6912 a^3 f^3 (\coth (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + a*Coth[e + f*x])^3,x]

[Out]

(Csch[e + f*x]^3*(81*(8*c^2*f^2 + 4*c*d*f*(3 + 4*f*x) + d^2*(7 + 12*f*x + 8*f^2*x^2))*Cosh[e + f*x] + 8*(18*c^
2*f^2*(1 + 6*f*x) + 6*c*d*f*(1 + 6*f*x + 18*f^2*x^2) + d^2*(1 + 6*f*x + 18*f^2*x^2 + 36*f^3*x^3))*Cosh[3*(e +
f*x)] + 729*d^2*Sinh[e + f*x] + 1620*c*d*f*Sinh[e + f*x] + 1944*c^2*f^2*Sinh[e + f*x] + 1620*d^2*f*x*Sinh[e +
f*x] + 3888*c*d*f^2*x*Sinh[e + f*x] + 1944*d^2*f^2*x^2*Sinh[e + f*x] - 8*d^2*Sinh[3*(e + f*x)] - 48*c*d*f*Sinh
[3*(e + f*x)] - 144*c^2*f^2*Sinh[3*(e + f*x)] - 48*d^2*f*x*Sinh[3*(e + f*x)] - 288*c*d*f^2*x*Sinh[3*(e + f*x)]
+ 864*c^2*f^3*x*Sinh[3*(e + f*x)] - 144*d^2*f^2*x^2*Sinh[3*(e + f*x)] + 864*c*d*f^3*x^2*Sinh[3*(e + f*x)] + 2
88*d^2*f^3*x^3*Sinh[3*(e + f*x)]))/(6912*a^3*f^3*(1 + Coth[e + f*x])^3)

________________________________________________________________________________________

Maple [B]  time = 0.125, size = 2098, normalized size = 8.5 \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))^3,x)

[Out]

1/a^3/f*(-4/f^2*d^2*(1/6*(f*x+e)^2*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3+1/16*(f
*x+e)^2*cosh(f*x+e)*sinh(f*x+e)+1/48*(f*x+e)^3-1/18*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4+13/144*(f*x+e)*sinh(f*
x+e)^2*cosh(f*x+e)^2+1/36*(f*x+e)*cosh(f*x+e)^2+1/108*sinh(f*x+e)*cosh(f*x+e)^5-43/1728*cosh(f*x+e)^3*sinh(f*x
+e)-7/1152*cosh(f*x+e)*sinh(f*x+e)-7/1152*f*x-7/1152*e)+4/f^2*d^2*(1/6*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^4-1
/12*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^2-1/12*(f*x+e)^2*cosh(f*x+e)^2-1/18*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5+
1/18*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3+1/108*sinh(f*x+e)^2*cosh(f*x+e)^4-1/216*sinh(f*x+e)^2*cosh(f*x+e)^2-5/1
08*cosh(f*x+e)^2+1/12*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+1/24*(f*x+e)^2)+1/f^2*d^2*(1/4*(f*x+e)^2*cosh(f*x+e)*sin
h(f*x+e)^3-3/8*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)+1/8*(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2+1/2*(f*
x+e)*cosh(f*x+e)^2+1/32*cosh(f*x+e)^3*sinh(f*x+e)-17/64*cosh(f*x+e)*sinh(f*x+e)-17/64*f*x-17/64*e)-3/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*cosh(f*x+e)^2)+8/f^2*d^
2*e*(1/6*(f*x+e)*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3+1/16*(f*x+e)*cosh(f*x+e)*si
nh(f*x+e)+1/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh(f*x+e)^4+13/288*sinh(f*x+e)^2*cosh(f*x+e)^2+1/72*cosh(f*x+e)^
2)-8/f^2*d^2*e*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2-1/12*(f*x+e)*
cosh(f*x+e)^2-1/36*sinh(f*x+e)*cosh(f*x+e)^5+1/36*cosh(f*x+e)^3*sinh(f*x+e)+1/24*cosh(f*x+e)*sinh(f*x+e)+1/24*
f*x+1/24*e)-2/f^2*d^2*e*(1/4*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)^3-3/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+3/16*(f*x+e
)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e)^2+1/4*cosh(f*x+e)^2)+6/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^2*(1/6*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*cosh(f*x+e)^3*sinh(f*x+e)+1/16*cosh(f*x+e)*sinh(f*x+e)+1/16*f*x+1/1
6*e)+4/f^2*d^2*e^2*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*sinh(f*x+e)^2*cosh(f*x+e)^2-1/12*cosh(f*x+e)^2)+1/f^2
*d^2*e^2*((1/4*sinh(f*x+e)^3-3/8*sinh(f*x+e))*cosh(f*x+e)+3/8*f*x+3/8*e)-3/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)-8*c/f*d*(1/6*(f*x+e)*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)*cosh(f*x
+e)^3+1/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+1/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh(f*x+e)^4+13/288*sinh(f*x+e)^
2*cosh(f*x+e)^2+1/72*cosh(f*x+e)^2)+8*c/f*d*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)*sinh(f*x+e)^
2*cosh(f*x+e)^2-1/12*(f*x+e)*cosh(f*x+e)^2-1/36*sinh(f*x+e)*cosh(f*x+e)^5+1/36*cosh(f*x+e)^3*sinh(f*x+e)+1/24*
cosh(f*x+e)*sinh(f*x+e)+1/24*f*x+1/24*e)+2*c/f*d*(1/4*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)^3-3/8*(f*x+e)*cosh(f*x+e
)*sinh(f*x+e)+3/16*(f*x+e)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e)^2+1/4*cosh(f*x+e)^2)-6*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)+8*c/f*d*e*(1/6*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*cosh(f*x+e)^3*sinh(f*x+e)+1/16*cosh(f*x+e)*sinh(f*
x+e)+1/16*f*x+1/16*e)-8*c/f*d*e*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*sinh(f*x+e)^2*cosh(f*x+e)^2-1/12*cosh(f*
x+e)^2)-2*c/f*d*e*((1/4*sinh(f*x+e)^3-3/8*sinh(f*x+e))*cosh(f*x+e)+3/8*f*x+3/8*e)+6*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^2*(1/6*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*cosh(f*x+e)^3*sinh(f*x+e)+1/16*co
sh(f*x+e)*sinh(f*x+e)+1/16*f*x+1/16*e)+4*c^2*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*sinh(f*x+e)^2*cosh(f*x+e)^2
-1/12*cosh(f*x+e)^2)+c^2*((1/4*sinh(f*x+e)^3-3/8*sinh(f*x+e))*cosh(f*x+e)+3/8*f*x+3/8*e)-3*c^2*(1/4*sinh(f*x+e
)^2*cosh(f*x+e)^2-1/4*cosh(f*x+e)^2))

________________________________________________________________________________________

Maxima [A]  time = 5.60223, size = 343, normalized size = 1.39 \begin{align*} \frac{1}{96} \, c^{2}{\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 )} c d e^{\left (-6 \, e\right )}}{576 \, a^{3} f^{2}} + \frac{{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} + 648 \,{\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \,{\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 8 \,{\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{2} e^{\left (-6 \, e\right )}}{6912 \, a^{3} 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))^3,x, algorithm="maxima")

[Out]

1/96*c^2*(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/
576*(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))*c*d*e^(-6*e)/(a^3*f^2) + 1/6912*(288*f^3*x^3*e^(6*e) + 648*(2*f^2*x^2*e^(4*e) + 2*f
*x*e^(4*e) + e^(4*e))*e^(-2*f*x) - 81*(8*f^2*x^2*e^(2*e) + 4*f*x*e^(2*e) + e^(2*e))*e^(-4*f*x) + 8*(18*f^2*x^2
+ 6*f*x + 1)*e^(-6*f*x))*d^2*e^(-6*e)/(a^3*f^3)

________________________________________________________________________________________

Fricas [B]  time = 2.13223, size = 1215, normalized size = 4.94 \begin{align*} \frac{8 \,{\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \,{\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \,{\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{3} + 24 \,{\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \,{\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \,{\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + 8 \,{\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \,{\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \,{\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \sinh \left (f x + e\right )^{3} + 81 \,{\left (8 \, d^{2} f^{2} x^{2} + 8 \, c^{2} f^{2} + 12 \, c d f + 7 \, d^{2} + 4 \,{\left (4 \, c d f^{2} + 3 \, d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) + 3 \,{\left (648 \, d^{2} f^{2} x^{2} + 648 \, c^{2} f^{2} + 540 \, c d f + 8 \,{\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \,{\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \,{\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{2} + 243 \, d^{2} + 108 \,{\left (12 \, c d f^{2} + 5 \, d^{2} f\right )} x\right )} \sinh \left (f x + e\right )}{6912 \,{\left (a^{3} f^{3} \cosh \left (f x + e\right )^{3} + 3 \, a^{3} f^{3} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, a^{3} f^{3} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + a^{3} f^{3} \sinh \left (f x + e\right )^{3}\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))^3,x, algorithm="fricas")

[Out]

1/6912*(8*(36*d^2*f^3*x^3 + 18*c^2*f^2 + 6*c*d*f + 18*(6*c*d*f^3 + d^2*f^2)*x^2 + d^2 + 6*(18*c^2*f^3 + 6*c*d*
f^2 + d^2*f)*x)*cosh(f*x + e)^3 + 24*(36*d^2*f^3*x^3 + 18*c^2*f^2 + 6*c*d*f + 18*(6*c*d*f^3 + d^2*f^2)*x^2 + d
^2 + 6*(18*c^2*f^3 + 6*c*d*f^2 + d^2*f)*x)*cosh(f*x + e)*sinh(f*x + e)^2 + 8*(36*d^2*f^3*x^3 - 18*c^2*f^2 - 6*
c*d*f + 18*(6*c*d*f^3 - d^2*f^2)*x^2 - d^2 + 6*(18*c^2*f^3 - 6*c*d*f^2 - d^2*f)*x)*sinh(f*x + e)^3 + 81*(8*d^2
*f^2*x^2 + 8*c^2*f^2 + 12*c*d*f + 7*d^2 + 4*(4*c*d*f^2 + 3*d^2*f)*x)*cosh(f*x + e) + 3*(648*d^2*f^2*x^2 + 648*
c^2*f^2 + 540*c*d*f + 8*(36*d^2*f^3*x^3 - 18*c^2*f^2 - 6*c*d*f + 18*(6*c*d*f^3 - d^2*f^2)*x^2 - d^2 + 6*(18*c^
2*f^3 - 6*c*d*f^2 - d^2*f)*x)*cosh(f*x + e)^2 + 243*d^2 + 108*(12*c*d*f^2 + 5*d^2*f)*x)*sinh(f*x + e))/(a^3*f^
3*cosh(f*x + e)^3 + 3*a^3*f^3*cosh(f*x + e)^2*sinh(f*x + e) + 3*a^3*f^3*cosh(f*x + e)*sinh(f*x + e)^2 + a^3*f^
3*sinh(f*x + e)^3)

________________________________________________________________________________________

Sympy [A]  time = 5.25176, size = 2448, normalized size = 9.95 \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))**3,x)

[Out]

Piecewise((216*c**2*f**3*x*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2
+ 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 648*c**2*f**3*x*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e +
f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 648*c**2*f**3*x*t
anh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x)
+ 1728*a**3*f**3) + 216*c**2*f**3*x/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184
*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 648*c**2*f**2*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 +
5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 432*c**2*f**2*tanh(e + f*x
)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*
a**3*f**3) + 72*c**2*f**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*
tanh(e + f*x) + 1728*a**3*f**3) + 216*c*d*f**3*x**2*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a
**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 648*c*d*f**3*x**2*tanh(e + f*x)**
2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**
3*f**3) + 648*c*d*f**3*x**2*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 +
5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 216*c*d*f**3*x**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a*
*3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 1044*c*d*f**2*x*tanh(e + f*x)**3/(
1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f
**3) - 108*c*d*f**2*x*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 51
84*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 756*c*d*f**2*x*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 +
5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 396*c*d*f**2*x/(1728*a**3*
f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 540
*c*d*f*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*ta
nh(e + f*x) + 1728*a**3*f**3) - 576*c*d*f*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*t
anh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 132*c*d*f/(1728*a**3*f**3*tanh(e + f*x)**3
+ 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 72*d**2*f**3*x**3*tanh(e
+ f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) +
1728*a**3*f**3) + 216*d**2*f**3*x**3*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e
+ f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 216*d**2*f**3*x**3*tanh(e + f*x)/(1728*a**3*f**3
*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 72*d**2
*f**3*x**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) +
1728*a**3*f**3) - 522*d**2*f**2*x**2*tanh(e + f*x)**3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(
e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 54*d**2*f**2*x**2*tanh(e + f*x)**2/(1728*a**3*f
**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 378*
d**2*f**2*x**2*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f*
*3*tanh(e + f*x) + 1728*a**3*f**3) + 198*d**2*f**2*x**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh
(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 417*d**2*f*x*tanh(e + f*x)**3/(1728*a**3*f**3*
tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 207*d**2
*f*x*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh
(e + f*x) + 1728*a**3*f**3) + 369*d**2*f*x*tanh(e + f*x)/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tan
h(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 255*d**2*f*x/(1728*a**3*f**3*tanh(e + f*x)**3
+ 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) - 243*d**2*tanh(e + f*x)**
3/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**
3*f**3) - 312*d**2*tanh(e + f*x)**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e + f*x)**2 + 5184*
a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3) + 85*d**2/(1728*a**3*f**3*tanh(e + f*x)**3 + 5184*a**3*f**3*tanh(e +
f*x)**2 + 5184*a**3*f**3*tanh(e + f*x) + 1728*a**3*f**3), Ne(f, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)/(a*co
th(e) + a)**3, True))

________________________________________________________________________________________

Giac [A]  time = 1.25946, size = 447, normalized size = 1.82 \begin{align*} \frac{{\left (288 \, d^{2} f^{3} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c d f^{3} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c^{2} f^{3} x e^{\left (6 \, f x + 6 \, e\right )} + 1296 \, d^{2} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 648 \, d^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 144 \, d^{2} f^{2} x^{2} + 2592 \, c d f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 1296 \, c d f^{2} x e^{\left (2 \, f x + 2 \, e\right )} + 288 \, c d f^{2} x + 1296 \, c^{2} f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 1296 \, d^{2} f x e^{\left (4 \, f x + 4 \, e\right )} - 648 \, c^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 324 \, d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} + 144 \, c^{2} f^{2} + 48 \, d^{2} f x + 1296 \, c d f e^{\left (4 \, f x + 4 \, e\right )} - 324 \, c d f e^{\left (2 \, f x + 2 \, e\right )} + 48 \, c d f + 648 \, d^{2} e^{\left (4 \, f x + 4 \, e\right )} - 81 \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} + 8 \, d^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{6912 \, a^{3} 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))^3,x, algorithm="giac")

[Out]

1/6912*(288*d^2*f^3*x^3*e^(6*f*x + 6*e) + 864*c*d*f^3*x^2*e^(6*f*x + 6*e) + 864*c^2*f^3*x*e^(6*f*x + 6*e) + 12
96*d^2*f^2*x^2*e^(4*f*x + 4*e) - 648*d^2*f^2*x^2*e^(2*f*x + 2*e) + 144*d^2*f^2*x^2 + 2592*c*d*f^2*x*e^(4*f*x +
4*e) - 1296*c*d*f^2*x*e^(2*f*x + 2*e) + 288*c*d*f^2*x + 1296*c^2*f^2*e^(4*f*x + 4*e) + 1296*d^2*f*x*e^(4*f*x
+ 4*e) - 648*c^2*f^2*e^(2*f*x + 2*e) - 324*d^2*f*x*e^(2*f*x + 2*e) + 144*c^2*f^2 + 48*d^2*f*x + 1296*c*d*f*e^(
4*f*x + 4*e) - 324*c*d*f*e^(2*f*x + 2*e) + 48*c*d*f + 648*d^2*e^(4*f*x + 4*e) - 81*d^2*e^(2*f*x + 2*e) + 8*d^2
)*e^(-6*f*x - 6*e)/(a^3*f^3)