Integrand size = 20, antiderivative size = 169 \[ \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx=\frac {3 d^3 x}{8 a f^3}+\frac {3 d (c+d x)^2}{8 a f^2}+\frac {(c+d x)^3}{4 a f}+\frac {(c+d x)^4}{8 a d}-\frac {3 d^3}{8 f^4 (a+a \coth (e+f x))}-\frac {3 d^2 (c+d x)}{4 f^3 (a+a \coth (e+f x))}-\frac {3 d (c+d x)^2}{4 f^2 (a+a \coth (e+f x))}-\frac {(c+d x)^3}{2 f (a+a \coth (e+f x))} \]
3/8*d^3*x/a/f^3+3/8*d*(d*x+c)^2/a/f^2+1/4*(d*x+c)^3/a/f+1/8*(d*x+c)^4/a/d- 3/8*d^3/f^4/(a+a*coth(f*x+e))-3/4*d^2*(d*x+c)/f^3/(a+a*coth(f*x+e))-3/4*d* (d*x+c)^2/f^2/(a+a*coth(f*x+e))-1/2*(d*x+c)^3/f/(a+a*coth(f*x+e))
Time = 0.93 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.44 \[ \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx=\frac {\text {csch}(e+f x) (\cosh (f x)+\sinh (f x)) \left (\left (4 c^3 f^3+6 c^2 d f^2 (1+2 f x)+6 c d^2 f \left (1+2 f x+2 f^2 x^2\right )+d^3 \left (3+6 f x+6 f^2 x^2+4 f^3 x^3\right )\right ) \cosh (2 f x) (\cosh (e)-\sinh (e))+2 f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) (\cosh (e)+\sinh (e))+\left (4 c^3 f^3+6 c^2 d f^2 (1+2 f x)+6 c d^2 f \left (1+2 f x+2 f^2 x^2\right )+d^3 \left (3+6 f x+6 f^2 x^2+4 f^3 x^3\right )\right ) (-\cosh (e)+\sinh (e)) \sinh (2 f x)\right )}{16 a f^4 (1+\coth (e+f x))} \]
(Csch[e + f*x]*(Cosh[f*x] + Sinh[f*x])*((4*c^3*f^3 + 6*c^2*d*f^2*(1 + 2*f* x) + 6*c*d^2*f*(1 + 2*f*x + 2*f^2*x^2) + d^3*(3 + 6*f*x + 6*f^2*x^2 + 4*f^ 3*x^3))*Cosh[2*f*x]*(Cosh[e] - Sinh[e]) + 2*f^4*x*(4*c^3 + 6*c^2*d*x + 4*c *d^2*x^2 + d^3*x^3)*(Cosh[e] + Sinh[e]) + (4*c^3*f^3 + 6*c^2*d*f^2*(1 + 2* f*x) + 6*c*d^2*f*(1 + 2*f*x + 2*f^2*x^2) + d^3*(3 + 6*f*x + 6*f^2*x^2 + 4* f^3*x^3))*(-Cosh[e] + Sinh[e])*Sinh[2*f*x]))/(16*a*f^4*(1 + Coth[e + f*x]) )
Time = 0.69 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3042, 4206, 3042, 4206, 3042, 4206, 3042, 3960, 24}
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.
| \(\displaystyle \int \frac {(c+d x)^3}{a \coth (e+f x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^3}{a-i a \tan \left (i e+i f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4206 |
| \(\displaystyle \frac {3 d \int \frac {(c+d x)^2}{\coth (e+f x) a+a}dx}{2 f}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 d \int \frac {(c+d x)^2}{a-i a \tan \left (i e+i f x+\frac {\pi }{2}\right )}dx}{2 f}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^4}{8 a d}\) |
| \(\Big \downarrow \) 4206 |
| \(\displaystyle \frac {3 d \left (\frac {d \int \frac {c+d x}{\coth (e+f x) a+a}dx}{f}-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^3}{6 a d}\right )}{2 f}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 d \left (\frac {d \int \frac {c+d x}{a-i a \tan \left (i e+i f x+\frac {\pi }{2}\right )}dx}{f}-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^3}{6 a d}\right )}{2 f}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^4}{8 a d}\) |
| \(\Big \downarrow \) 4206 |
| \(\displaystyle \frac {3 d \left (\frac {d \left (\frac {d \int \frac {1}{\coth (e+f x) a+a}dx}{2 f}-\frac {c+d x}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^3}{6 a d}\right )}{2 f}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^4}{8 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 d \left (\frac {d \left (\frac {d \int \frac {1}{a-i a \tan \left (i e+i f x+\frac {\pi }{2}\right )}dx}{2 f}-\frac {c+d x}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^3}{6 a d}\right )}{2 f}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^4}{8 a d}\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {3 d \left (\frac {d \left (\frac {d \left (\frac {\int 1dx}{2 a}-\frac {1}{2 f (a \coth (e+f x)+a)}\right )}{2 f}-\frac {c+d x}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^2}{4 a d}\right )}{f}-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^3}{6 a d}\right )}{2 f}-\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^4}{8 a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {(c+d x)^3}{2 f (a \coth (e+f x)+a)}+\frac {3 d \left (-\frac {(c+d x)^2}{2 f (a \coth (e+f x)+a)}+\frac {d \left (-\frac {c+d x}{2 f (a \coth (e+f x)+a)}+\frac {(c+d x)^2}{4 a d}+\frac {d \left (\frac {x}{2 a}-\frac {1}{2 f (a \coth (e+f x)+a)}\right )}{2 f}\right )}{f}+\frac {(c+d x)^3}{6 a d}\right )}{2 f}+\frac {(c+d x)^4}{8 a d}\) |
(c + d*x)^4/(8*a*d) - (c + d*x)^3/(2*f*(a + a*Coth[e + f*x])) + (3*d*((c + d*x)^3/(6*a*d) - (c + d*x)^2/(2*f*(a + a*Coth[e + f*x])) + (d*((c + d*x)^ 2/(4*a*d) - (c + d*x)/(2*f*(a + a*Coth[e + f*x])) + (d*(x/(2*a) - 1/(2*f*( a + a*Coth[e + f*x]))))/(2*f)))/f))/(2*f)
3.1.16.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sy mbol] :> Simp[(c + d*x)^(m + 1)/(2*a*d*(m + 1)), x] + (Simp[a*d*(m/(2*b*f)) Int[(c + d*x)^(m - 1)/(a + b*Tan[e + f*x]), x], x] - Simp[a*((c + d*x)^m /(2*b*f*(a + b*Tan[e + f*x]))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[ a^2 + b^2, 0] && GtQ[m, 0]
Time = 0.38 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(\frac {d^{3} x^{4}}{8 a}+\frac {d^{2} c \,x^{3}}{2 a}+\frac {3 d \,c^{2} x^{2}}{4 a}+\frac {c^{3} x}{2 a}+\frac {c^{4}}{8 a d}+\frac {\left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{16 a \,f^{4}}\) | \(165\) |
| parallelrisch | \(\frac {\left (\left (d^{3} x^{4}+4 d^{2} c \,x^{3}+6 d \,c^{2} x^{2}+4 c^{3} x \right ) f^{4}+\left (-2 d^{3} x^{3}-6 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}\right ) f^{3}+\left (-3 d^{3} x^{2}-6 c \,d^{2} x -6 d \,c^{2}\right ) f^{2}+\left (-3 d^{3} x -6 d^{2} c \right ) f -3 d^{3}\right ) \tanh \left (f x +e \right )+4 x \left (\left (\frac {1}{4} d^{3} x^{3}+c \,d^{2} x^{2}+\frac {3}{2} c^{2} d x +c^{3}\right ) f^{3}+\frac {3 d \left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) f^{2}}{2}+\frac {3 d^{2} \left (\frac {d x}{2}+c \right ) f}{2}+\frac {3 d^{3}}{4}\right ) f}{8 f^{4} a \left (1+\tanh \left (f x +e \right )\right )}\) | \(219\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(929\) |
| default | \(\text {Expression too large to display}\) | \(929\) |
1/8/a*d^3*x^4+1/2/a*d^2*c*x^3+3/4/a*d*c^2*x^2+1/2/a*c^3*x+1/8/a/d*c^4+1/16 *(4*d^3*f^3*x^3+12*c*d^2*f^3*x^2+12*c^2*d*f^3*x+6*d^3*f^2*x^2+4*c^3*f^3+12 *c*d^2*f^2*x+6*c^2*d*f^2+6*d^3*f*x+6*c*d^2*f+3*d^3)/a/f^4*exp(-2*f*x-2*e)
Time = 0.26 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx=\frac {{\left (2 \, d^{3} f^{4} x^{4} + 4 \, c^{3} f^{3} + 6 \, c^{2} d f^{2} + 6 \, c d^{2} f + 4 \, {\left (2 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + 3 \, d^{3} + 6 \, {\left (2 \, c^{2} d f^{4} + 2 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 2 \, {\left (4 \, c^{3} f^{4} + 6 \, c^{2} d f^{3} + 6 \, c d^{2} f^{2} + 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) + {\left (2 \, d^{3} f^{4} x^{4} - 4 \, c^{3} f^{3} - 6 \, c^{2} d f^{2} - 6 \, c d^{2} f + 4 \, {\left (2 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 6 \, {\left (2 \, c^{2} d f^{4} - 2 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 2 \, {\left (4 \, c^{3} f^{4} - 6 \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \sinh \left (f x + e\right )}{16 \, {\left (a f^{4} \cosh \left (f x + e\right ) + a f^{4} \sinh \left (f x + e\right )\right )}} \]
1/16*((2*d^3*f^4*x^4 + 4*c^3*f^3 + 6*c^2*d*f^2 + 6*c*d^2*f + 4*(2*c*d^2*f^ 4 + d^3*f^3)*x^3 + 3*d^3 + 6*(2*c^2*d*f^4 + 2*c*d^2*f^3 + d^3*f^2)*x^2 + 2 *(4*c^3*f^4 + 6*c^2*d*f^3 + 6*c*d^2*f^2 + 3*d^3*f)*x)*cosh(f*x + e) + (2*d ^3*f^4*x^4 - 4*c^3*f^3 - 6*c^2*d*f^2 - 6*c*d^2*f + 4*(2*c*d^2*f^4 - d^3*f^ 3)*x^3 - 3*d^3 + 6*(2*c^2*d*f^4 - 2*c*d^2*f^3 - d^3*f^2)*x^2 + 2*(4*c^3*f^ 4 - 6*c^2*d*f^3 - 6*c*d^2*f^2 - 3*d^3*f)*x)*sinh(f*x + e))/(a*f^4*cosh(f*x + e) + a*f^4*sinh(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (144) = 288\).
Time = 0.76 (sec) , antiderivative size = 864, normalized size of antiderivative = 5.11 \[ \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx=\begin {cases} \frac {4 c^{3} f^{4} x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c^{3} f^{4} x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c^{3} f^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{4} x^{2} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{4} x^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {6 c^{2} d f^{3} x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{3} x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c^{2} d f^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c d^{2} f^{4} x^{3} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {4 c d^{2} f^{4} x^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {6 c d^{2} f^{3} x^{2} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c d^{2} f^{3} x^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {6 c d^{2} f^{2} x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c d^{2} f^{2} x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {6 c d^{2} f}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {d^{3} f^{4} x^{4} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {d^{3} f^{4} x^{4}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {2 d^{3} f^{3} x^{3} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {2 d^{3} f^{3} x^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {3 d^{3} f^{2} x^{2} \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {3 d^{3} f^{2} x^{2}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} - \frac {3 d^{3} f x \tanh {\left (e + f x \right )}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {3 d^{3} f x}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} + \frac {3 d^{3}}{8 a f^{4} \tanh {\left (e + f x \right )} + 8 a f^{4}} & \text {for}\: f \neq 0 \\\frac {c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}}{a \coth {\left (e \right )} + a} & \text {otherwise} \end {cases} \]
Piecewise((4*c**3*f**4*x*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 4*c**3*f**4*x/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 4*c**3*f**3/(8*a*f* *4*tanh(e + f*x) + 8*a*f**4) + 6*c**2*d*f**4*x**2*tanh(e + f*x)/(8*a*f**4* tanh(e + f*x) + 8*a*f**4) + 6*c**2*d*f**4*x**2/(8*a*f**4*tanh(e + f*x) + 8 *a*f**4) - 6*c**2*d*f**3*x*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f** 4) + 6*c**2*d*f**3*x/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c**2*d*f**2/( 8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 4*c*d**2*f**4*x**3*tanh(e + f*x)/(8*a *f**4*tanh(e + f*x) + 8*a*f**4) + 4*c*d**2*f**4*x**3/(8*a*f**4*tanh(e + f* x) + 8*a*f**4) - 6*c*d**2*f**3*x**2*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c*d**2*f**3*x**2/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 6*c *d**2*f**2*x*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c*d**2* f**2*x/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 6*c*d**2*f/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + d**3*f**4*x**4*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + d**3*f**4*x**4/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 2*d**3*f **3*x**3*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 2*d**3*f**3*x **3/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) - 3*d**3*f**2*x**2*tanh(e + f*x)/( 8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 3*d**3*f**2*x**2/(8*a*f**4*tanh(e + f *x) + 8*a*f**4) - 3*d**3*f*x*tanh(e + f*x)/(8*a*f**4*tanh(e + f*x) + 8*a*f **4) + 3*d**3*f*x/(8*a*f**4*tanh(e + f*x) + 8*a*f**4) + 3*d**3/(8*a*f**4*t anh(e + f*x) + 8*a*f**4), Ne(f, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**...
Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx=\frac {1}{4} \, c^{3} {\left (\frac {2 \, {\left (f x + e\right )}}{a f} + \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{a f}\right )} + \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} + {\left (2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c^{2} d e^{\left (-2 \, e\right )}}{8 \, a f^{2}} + \frac {{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c d^{2} e^{\left (-2 \, e\right )}}{8 \, a f^{3}} + \frac {{\left (2 \, f^{4} x^{4} e^{\left (2 \, e\right )} + {\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x\right )}\right )} d^{3} e^{\left (-2 \, e\right )}}{16 \, a f^{4}} \]
1/4*c^3*(2*(f*x + e)/(a*f) + e^(-2*f*x - 2*e)/(a*f)) + 3/8*(2*f^2*x^2*e^(2 *e) + (2*f*x + 1)*e^(-2*f*x))*c^2*d*e^(-2*e)/(a*f^2) + 1/8*(4*f^3*x^3*e^(2 *e) + 3*(2*f^2*x^2 + 2*f*x + 1)*e^(-2*f*x))*c*d^2*e^(-2*e)/(a*f^3) + 1/16* (2*f^4*x^4*e^(2*e) + (4*f^3*x^3 + 6*f^2*x^2 + 6*f*x + 3)*e^(-2*f*x))*d^3*e ^(-2*e)/(a*f^4)
Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx=\frac {{\left (2 \, d^{3} f^{4} x^{4} e^{\left (2 \, f x + 2 \, e\right )} + 8 \, c d^{2} f^{4} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c^{2} d f^{4} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, d^{3} f^{3} x^{3} + 8 \, c^{3} f^{4} x e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c d^{2} f^{3} x^{2} + 12 \, c^{2} d f^{3} x + 6 \, d^{3} f^{2} x^{2} + 4 \, c^{3} f^{3} + 12 \, c d^{2} f^{2} x + 6 \, c^{2} d f^{2} + 6 \, d^{3} f x + 6 \, c d^{2} f + 3 \, d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, a f^{4}} \]
1/16*(2*d^3*f^4*x^4*e^(2*f*x + 2*e) + 8*c*d^2*f^4*x^3*e^(2*f*x + 2*e) + 12 *c^2*d*f^4*x^2*e^(2*f*x + 2*e) + 4*d^3*f^3*x^3 + 8*c^3*f^4*x*e^(2*f*x + 2* e) + 12*c*d^2*f^3*x^2 + 12*c^2*d*f^3*x + 6*d^3*f^2*x^2 + 4*c^3*f^3 + 12*c* d^2*f^2*x + 6*c^2*d*f^2 + 6*d^3*f*x + 6*c*d^2*f + 3*d^3)*e^(-2*f*x - 2*e)/ (a*f^4)
Time = 2.16 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.32 \[ \int \frac {(c+d x)^3}{a+a \coth (e+f x)} \, dx=\frac {4\,c^3\,f^4\,x+6\,c^2\,d\,f^4\,x^2+6\,c^2\,d\,f^3\,x+4\,c\,d^2\,f^4\,x^3+6\,c\,d^2\,f^3\,x^2+6\,c\,d^2\,f^2\,x+d^3\,f^4\,x^4+2\,d^3\,f^3\,x^3+3\,d^3\,f^2\,x^2+3\,d^3\,f\,x}{8\,a\,f^4}-\frac {4\,c^3\,f^3+12\,c^2\,d\,f^3\,x+6\,c^2\,d\,f^2+12\,c\,d^2\,f^3\,x^2+12\,c\,d^2\,f^2\,x+6\,c\,d^2\,f+4\,d^3\,f^3\,x^3+6\,d^3\,f^2\,x^2+6\,d^3\,f\,x+3\,d^3}{8\,a\,f^4\,\left (\mathrm {coth}\left (e+f\,x\right )+1\right )} \]
(3*d^3*f*x + 4*c^3*f^4*x + 3*d^3*f^2*x^2 + 2*d^3*f^3*x^3 + d^3*f^4*x^4 + 6 *c*d^2*f^3*x^2 + 6*c^2*d*f^4*x^2 + 4*c*d^2*f^4*x^3 + 6*c*d^2*f^2*x + 6*c^2 *d*f^3*x)/(8*a*f^4) - (3*d^3 + 4*c^3*f^3 + 6*d^3*f*x + 6*c^2*d*f^2 + 6*d^3 *f^2*x^2 + 4*d^3*f^3*x^3 + 6*c*d^2*f + 12*c*d^2*f^3*x^2 + 12*c*d^2*f^2*x + 12*c^2*d*f^3*x)/(8*a*f^4*(coth(e + f*x) + 1))