Integrand size = 20, antiderivative size = 246 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx=\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}+\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d} \]
1/864*d^2*exp(-6*f*x-6*e)/a^3/f^3-3/256*d^2*exp(-4*f*x-4*e)/a^3/f^3+3/32*d ^2*exp(-2*f*x-2*e)/a^3/f^3+1/144*d*exp(-6*f*x-6*e)*(d*x+c)/a^3/f^2-3/64*d* exp(-4*f*x-4*e)*(d*x+c)/a^3/f^2+3/16*d*exp(-2*f*x-2*e)*(d*x+c)/a^3/f^2+1/4 8*exp(-6*f*x-6*e)*(d*x+c)^2/a^3/f-3/32*exp(-4*f*x-4*e)*(d*x+c)^2/a^3/f+3/1 6*exp(-2*f*x-2*e)*(d*x+c)^2/a^3/f+1/24*(d*x+c)^3/a^3/d
Time = 3.17 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.51 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx=\frac {\text {csch}^3(e+f x) \left (81 \left (8 c^2 f^2+4 c d f (3+4 f x)+d^2 \left (7+12 f x+8 f^2 x^2\right )\right ) \cosh (e+f x)+8 \left (18 c^2 f^2 (1+6 f x)+6 c d f \left (1+6 f x+18 f^2 x^2\right )+d^2 \left (1+6 f x+18 f^2 x^2+36 f^3 x^3\right )\right ) \cosh (3 (e+f x))+729 d^2 \sinh (e+f x)+1620 c d f \sinh (e+f x)+1944 c^2 f^2 \sinh (e+f x)+1620 d^2 f x \sinh (e+f x)+3888 c d f^2 x \sinh (e+f x)+1944 d^2 f^2 x^2 \sinh (e+f x)-8 d^2 \sinh (3 (e+f x))-48 c d f \sinh (3 (e+f x))-144 c^2 f^2 \sinh (3 (e+f x))-48 d^2 f x \sinh (3 (e+f x))-288 c d f^2 x \sinh (3 (e+f x))+864 c^2 f^3 x \sinh (3 (e+f x))-144 d^2 f^2 x^2 \sinh (3 (e+f x))+864 c d f^3 x^2 \sinh (3 (e+f x))+288 d^2 f^3 x^3 \sinh (3 (e+f x))\right )}{6912 a^3 f^3 (1+\coth (e+f x))^3} \]
(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] + 19 44*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)] + 288*d^2 *f^3*x^3*Sinh[3*(e + f*x)]))/(6912*a^3*f^3*(1 + Coth[e + f*x])^3)
Time = 0.52 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4212, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^2}{(a \coth (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^2}{\left (a-i a \tan \left (i e+i f x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4212 |
| \(\displaystyle \int \left (-\frac {(c+d x)^2 e^{-6 e-6 f x}}{8 a^3}+\frac {3 (c+d x)^2 e^{-4 e-4 f x}}{8 a^3}-\frac {3 (c+d x)^2 e^{-2 e-2 f x}}{8 a^3}+\frac {(c+d x)^2}{8 a^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac {3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}+\frac {3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}+\frac {(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}+\frac {3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}+\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}\) |
(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)/(4 8*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)
3.1.28.3.1 Defintions of rubi rules used
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]
Time = 0.52 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {d^{2} x^{3}}{24 a^{3}}+\frac {d c \,x^{2}}{8 a^{3}}+\frac {c^{2} x}{8 a^{3}}+\frac {c^{3}}{24 a^{3} d}+\frac {3 \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}+2 d^{2} f x +2 c d f +d^{2}\right ) {\mathrm e}^{-2 f x -2 e}}{32 a^{3} f^{3}}-\frac {3 \left (8 d^{2} x^{2} f^{2}+16 c d \,f^{2} x +8 c^{2} f^{2}+4 d^{2} f x +4 c d f +d^{2}\right ) {\mathrm e}^{-4 f x -4 e}}{256 a^{3} f^{3}}+\frac {\left (18 d^{2} x^{2} f^{2}+36 c d \,f^{2} x +18 c^{2} f^{2}+6 d^{2} f x +6 c d f +d^{2}\right ) {\mathrm e}^{-6 f x -6 e}}{864 a^{3} f^{3}}\) | \(223\) |
| parallelrisch | \(\frac {\left (\left (288 d^{2} x^{3}+864 d c \,x^{2}+864 c^{2} x \right ) f^{3}+\left (144 x^{2} d^{2}+288 c d x +216 c^{2}\right ) f^{2}+\left (48 x \,d^{2}+324 c d \right ) f +189 d^{2}\right ) \cosh \left (3 f x +3 e \right )+\left (\left (288 d^{2} x^{3}+864 d c \,x^{2}+864 c^{2} x \right ) f^{3}+\left (-144 x^{2} d^{2}-288 c d x -72 c^{2}\right ) f^{2}+\left (-48 x \,d^{2}+228 c d \right ) f +173 d^{2}\right ) \sinh \left (3 f x +3 e \right )+\left (648 \left (d x +c \right )^{2} f^{2}+972 d \left (d x +c \right ) f +567 d^{2}\right ) \cosh \left (f x +e \right )+1944 \left (\left (d x +c \right )^{2} f^{2}+\frac {5 d \left (d x +c \right ) f}{6}+\frac {3 d^{2}}{8}\right ) \sinh \left (f x +e \right )}{6912 a^{3} f^{3} \left (\cosh \left (3 f x +3 e \right )+\sinh \left (3 f x +3 e \right )\right )}\) | \(255\) |
1/24/a^3*d^2*x^3+1/8/a^3*d*c*x^2+1/8/a^3*c^2*x+1/24/a^3/d*c^3+3/32*(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^3/f^3*exp(-2*f*x-2* e)-3/256*(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^3/ f^3*exp(-4*f*x-4*e)+1/864*(18*d^2*f^2*x^2+36*c*d*f^2*x+18*c^2*f^2+6*d^2*f* x+6*c*d*f+d^2)/a^3/f^3*exp(-6*f*x-6*e)
Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (217) = 434\).
Time = 0.26 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.16 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx=\frac {8 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{3} + 24 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + 8 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \sinh \left (f x + e\right )^{3} + 81 \, {\left (8 \, d^{2} f^{2} x^{2} + 8 \, c^{2} f^{2} + 12 \, c d f + 7 \, d^{2} + 4 \, {\left (4 \, c d f^{2} + 3 \, d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) + 3 \, {\left (648 \, d^{2} f^{2} x^{2} + 648 \, c^{2} f^{2} + 540 \, c d f + 8 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{2} + 243 \, d^{2} + 108 \, {\left (12 \, c d f^{2} + 5 \, d^{2} f\right )} x\right )} \sinh \left (f x + e\right )}{6912 \, {\left (a^{3} f^{3} \cosh \left (f x + e\right )^{3} + 3 \, a^{3} f^{3} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, a^{3} f^{3} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + a^{3} f^{3} \sinh \left (f x + e\right )^{3}\right )}} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 2443 vs. \(2 (252) = 504\).
Time = 1.21 (sec) , antiderivative size = 2443, normalized size of antiderivative = 9.93 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx=\text {Too large to display} \]
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) + 1 728*a**3*f**3) + 648*c**2*f**3*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) + 1 728*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) + 1512*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) + 1944*c**2*f**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 ) + 720*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*t anh(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) + 64 8*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)...
Time = 1.15 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx=\frac {1}{96} \, c^{2} {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} + \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} - 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} + 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d e^{\left (-6 \, e\right )}}{576 \, a^{3} f^{2}} + \frac {{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} + 648 \, {\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 8 \, {\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{2} e^{\left (-6 \, e\right )}}{6912 \, a^{3} f^{3}} \]
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)
Time = 0.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx=\frac {{\left (288 \, d^{2} f^{3} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c d f^{3} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c^{2} f^{3} x e^{\left (6 \, f x + 6 \, e\right )} + 1296 \, d^{2} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 648 \, d^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 144 \, d^{2} f^{2} x^{2} + 2592 \, c d f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 1296 \, c d f^{2} x e^{\left (2 \, f x + 2 \, e\right )} + 288 \, c d f^{2} x + 1296 \, c^{2} f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 1296 \, d^{2} f x e^{\left (4 \, f x + 4 \, e\right )} - 648 \, c^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 324 \, d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} + 144 \, c^{2} f^{2} + 48 \, d^{2} f x + 1296 \, c d f e^{\left (4 \, f x + 4 \, e\right )} - 324 \, c d f e^{\left (2 \, f x + 2 \, e\right )} + 48 \, c d f + 648 \, d^{2} e^{\left (4 \, f x + 4 \, e\right )} - 81 \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} + 8 \, d^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{6912 \, a^{3} f^{3}} \]
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) + 1296*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) - 3 24*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)
Time = 2.14 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx={\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {18\,c^2\,f^2+6\,c\,d\,f+d^2}{864\,a^3\,f^3}+\frac {d^2\,x^2}{48\,a^3\,f}+\frac {d\,x\,\left (d+6\,c\,f\right )}{144\,a^3\,f^2}\right )+{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {6\,c^2\,f^2+6\,c\,d\,f+3\,d^2}{32\,a^3\,f^3}+\frac {3\,d^2\,x^2}{16\,a^3\,f}+\frac {3\,d\,x\,\left (d+2\,c\,f\right )}{16\,a^3\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {24\,c^2\,f^2+12\,c\,d\,f+3\,d^2}{256\,a^3\,f^3}+\frac {3\,d^2\,x^2}{32\,a^3\,f}+\frac {3\,d\,x\,\left (d+4\,c\,f\right )}{64\,a^3\,f^2}\right )+\frac {c^2\,x}{8\,a^3}+\frac {d^2\,x^3}{24\,a^3}+\frac {c\,d\,x^2}{8\,a^3} \]
exp(- 6*e - 6*f*x)*((d^2 + 18*c^2*f^2 + 6*c*d*f)/(864*a^3*f^3) + (d^2*x^2) /(48*a^3*f) + (d*x*(d + 6*c*f))/(144*a^3*f^2)) + exp(- 2*e - 2*f*x)*((3*d^ 2 + 6*c^2*f^2 + 6*c*d*f)/(32*a^3*f^3) + (3*d^2*x^2)/(16*a^3*f) + (3*d*x*(d + 2*c*f))/(16*a^3*f^2)) - exp(- 4*e - 4*f*x)*((3*d^2 + 24*c^2*f^2 + 12*c* d*f)/(256*a^3*f^3) + (3*d^2*x^2)/(32*a^3*f) + (3*d*x*(d + 4*c*f))/(64*a^3* f^2)) + (c^2*x)/(8*a^3) + (d^2*x^3)/(24*a^3) + (c*d*x^2)/(8*a^3)