Integrand size = 20, antiderivative size = 336 \[ \int \frac {(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx=-\frac {d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac {9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}-\frac {9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4}-\frac {d^2 e^{-6 e-6 f x} (c+d x)}{288 a^3 f^3}-\frac {9 d^2 e^{-4 e-4 f x} (c+d x)}{256 a^3 f^3}-\frac {9 d^2 e^{-2 e-2 f x} (c+d x)}{32 a^3 f^3}-\frac {d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac {9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}-\frac {9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}-\frac {e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}-\frac {3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d} \]
-1/1728*d^3*exp(-6*f*x-6*e)/a^3/f^4-9/1024*d^3*exp(-4*f*x-4*e)/a^3/f^4-9/6 4*d^3*exp(-2*f*x-2*e)/a^3/f^4-1/288*d^2*exp(-6*f*x-6*e)*(d*x+c)/a^3/f^3-9/ 256*d^2*exp(-4*f*x-4*e)*(d*x+c)/a^3/f^3-9/32*d^2*exp(-2*f*x-2*e)*(d*x+c)/a ^3/f^3-1/96*d*exp(-6*f*x-6*e)*(d*x+c)^2/a^3/f^2-9/128*d*exp(-4*f*x-4*e)*(d *x+c)^2/a^3/f^2-9/32*d*exp(-2*f*x-2*e)*(d*x+c)^2/a^3/f^2-1/48*exp(-6*f*x-6 *e)*(d*x+c)^3/a^3/f-3/32*exp(-4*f*x-4*e)*(d*x+c)^3/a^3/f-3/16*exp(-2*f*x-2 *e)*(d*x+c)^3/a^3/f+1/32*(d*x+c)^4/a^3/d
Time = 3.46 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.83 \[ \int \frac {(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx=\frac {\text {sech}^3(e+f x) \left (-243 \left (32 c^3 f^3+8 c^2 d f^2 (5+12 f x)+4 c d^2 f \left (9+20 f x+24 f^2 x^2\right )+d^3 \left (17+36 f x+40 f^2 x^2+32 f^3 x^3\right )\right ) \cosh (e+f x)+16 \left (36 c^3 f^3 (-1+6 f x)+18 c^2 d f^2 \left (-1-6 f x+18 f^2 x^2\right )+6 c d^2 f \left (-1-6 f x-18 f^2 x^2+36 f^3 x^3\right )+d^3 \left (-1-6 f x-18 f^2 x^2-36 f^3 x^3+54 f^4 x^4\right )\right ) \cosh (3 (e+f x))-3645 d^3 \sinh (e+f x)-6804 c d^2 f \sinh (e+f x)-5832 c^2 d f^2 \sinh (e+f x)-2592 c^3 f^3 \sinh (e+f x)-6804 d^3 f x \sinh (e+f x)-11664 c d^2 f^2 x \sinh (e+f x)-7776 c^2 d f^3 x \sinh (e+f x)-5832 d^3 f^2 x^2 \sinh (e+f x)-7776 c d^2 f^3 x^2 \sinh (e+f x)-2592 d^3 f^3 x^3 \sinh (e+f x)+16 d^3 \sinh (3 (e+f x))+96 c d^2 f \sinh (3 (e+f x))+288 c^2 d f^2 \sinh (3 (e+f x))+576 c^3 f^3 \sinh (3 (e+f x))+96 d^3 f x \sinh (3 (e+f x))+576 c d^2 f^2 x \sinh (3 (e+f x))+1728 c^2 d f^3 x \sinh (3 (e+f x))+3456 c^3 f^4 x \sinh (3 (e+f x))+288 d^3 f^2 x^2 \sinh (3 (e+f x))+1728 c d^2 f^3 x^2 \sinh (3 (e+f x))+5184 c^2 d f^4 x^2 \sinh (3 (e+f x))+576 d^3 f^3 x^3 \sinh (3 (e+f x))+3456 c d^2 f^4 x^3 \sinh (3 (e+f x))+864 d^3 f^4 x^4 \sinh (3 (e+f x))\right )}{27648 a^3 f^4 (1+\tanh (e+f x))^3} \]
(Sech[e + f*x]^3*(-243*(32*c^3*f^3 + 8*c^2*d*f^2*(5 + 12*f*x) + 4*c*d^2*f* (9 + 20*f*x + 24*f^2*x^2) + d^3*(17 + 36*f*x + 40*f^2*x^2 + 32*f^3*x^3))*C osh[e + f*x] + 16*(36*c^3*f^3*(-1 + 6*f*x) + 18*c^2*d*f^2*(-1 - 6*f*x + 18 *f^2*x^2) + 6*c*d^2*f*(-1 - 6*f*x - 18*f^2*x^2 + 36*f^3*x^3) + d^3*(-1 - 6 *f*x - 18*f^2*x^2 - 36*f^3*x^3 + 54*f^4*x^4))*Cosh[3*(e + f*x)] - 3645*d^3 *Sinh[e + f*x] - 6804*c*d^2*f*Sinh[e + f*x] - 5832*c^2*d*f^2*Sinh[e + f*x] - 2592*c^3*f^3*Sinh[e + f*x] - 6804*d^3*f*x*Sinh[e + f*x] - 11664*c*d^2*f ^2*x*Sinh[e + f*x] - 7776*c^2*d*f^3*x*Sinh[e + f*x] - 5832*d^3*f^2*x^2*Sin h[e + f*x] - 7776*c*d^2*f^3*x^2*Sinh[e + f*x] - 2592*d^3*f^3*x^3*Sinh[e + f*x] + 16*d^3*Sinh[3*(e + f*x)] + 96*c*d^2*f*Sinh[3*(e + f*x)] + 288*c^2*d *f^2*Sinh[3*(e + f*x)] + 576*c^3*f^3*Sinh[3*(e + f*x)] + 96*d^3*f*x*Sinh[3 *(e + f*x)] + 576*c*d^2*f^2*x*Sinh[3*(e + f*x)] + 1728*c^2*d*f^3*x*Sinh[3* (e + f*x)] + 3456*c^3*f^4*x*Sinh[3*(e + f*x)] + 288*d^3*f^2*x^2*Sinh[3*(e + f*x)] + 1728*c*d^2*f^3*x^2*Sinh[3*(e + f*x)] + 5184*c^2*d*f^4*x^2*Sinh[3 *(e + f*x)] + 576*d^3*f^3*x^3*Sinh[3*(e + f*x)] + 3456*c*d^2*f^4*x^3*Sinh[ 3*(e + f*x)] + 864*d^3*f^4*x^4*Sinh[3*(e + f*x)]))/(27648*a^3*f^4*(1 + Tan h[e + f*x])^3)
Time = 0.68 (sec) , antiderivative size = 336, 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.
| \(\displaystyle \int \frac {(c+d x)^3}{(a \tanh (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^3}{(a-i a \tan (i e+i f x))^3}dx\) |
| \(\Big \downarrow \) 4212 |
| \(\displaystyle \int \left (\frac {(c+d x)^3 e^{-6 e-6 f x}}{8 a^3}+\frac {3 (c+d x)^3 e^{-4 e-4 f x}}{8 a^3}+\frac {3 (c+d x)^3 e^{-2 e-2 f x}}{8 a^3}+\frac {(c+d x)^3}{8 a^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^2 (c+d x) e^{-6 e-6 f x}}{288 a^3 f^3}-\frac {9 d^2 (c+d x) e^{-4 e-4 f x}}{256 a^3 f^3}-\frac {9 d^2 (c+d x) e^{-2 e-2 f x}}{32 a^3 f^3}-\frac {d (c+d x)^2 e^{-6 e-6 f x}}{96 a^3 f^2}-\frac {9 d (c+d x)^2 e^{-4 e-4 f x}}{128 a^3 f^2}-\frac {9 d (c+d x)^2 e^{-2 e-2 f x}}{32 a^3 f^2}-\frac {(c+d x)^3 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^3 e^{-4 e-4 f x}}{32 a^3 f}-\frac {3 (c+d x)^3 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}-\frac {d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac {9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}-\frac {9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4}\) |
-1/1728*(d^3*E^(-6*e - 6*f*x))/(a^3*f^4) - (9*d^3*E^(-4*e - 4*f*x))/(1024* a^3*f^4) - (9*d^3*E^(-2*e - 2*f*x))/(64*a^3*f^4) - (d^2*E^(-6*e - 6*f*x)*( c + d*x))/(288*a^3*f^3) - (9*d^2*E^(-4*e - 4*f*x)*(c + d*x))/(256*a^3*f^3) - (9*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(32*a^3*f^3) - (d*E^(-6*e - 6*f*x)*( c + d*x)^2)/(96*a^3*f^2) - (9*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(128*a^3*f^2 ) - (9*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(32*a^3*f^2) - (E^(-6*e - 6*f*x)*(c + d*x)^3)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c + d*x)^3)/(32*a^3*f) - (3*E ^(-2*e - 2*f*x)*(c + d*x)^3)/(16*a^3*f) + (c + d*x)^4/(32*a^3*d)
3.1.43.3.1 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 = 0.33 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\frac {d^{3} x^{4}}{32 a^{3}}+\frac {d^{2} c \,x^{3}}{8 a^{3}}+\frac {3 d \,c^{2} x^{2}}{16 a^{3}}+\frac {c^{3} x}{8 a^{3}}+\frac {c^{4}}{32 a^{3} d}-\frac {3 \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{64 a^{3} f^{4}}-\frac {3 \left (32 d^{3} x^{3} f^{3}+96 c \,d^{2} f^{3} x^{2}+96 c^{2} d \,f^{3} x +24 d^{3} f^{2} x^{2}+32 c^{3} f^{3}+48 c \,d^{2} f^{2} x +24 c^{2} d \,f^{2}+12 d^{3} f x +12 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-4 f x -4 e}}{1024 a^{3} f^{4}}-\frac {\left (36 d^{3} x^{3} f^{3}+108 c \,d^{2} f^{3} x^{2}+108 c^{2} d \,f^{3} x +18 d^{3} f^{2} x^{2}+36 c^{3} f^{3}+36 c \,d^{2} f^{2} x +18 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +d^{3}\right ) {\mathrm e}^{-6 f x -6 e}}{1728 a^{3} f^{4}}\) | \(379\) |
| parallelrisch | \(\frac {-1952 d^{3}-1725 \tanh \left (f x +e \right )^{2} d^{3}-6264 c^{2} d \,f^{3} x -3645 \tanh \left (f x +e \right ) d^{3}-864 \tanh \left (f x +e \right )^{2} c^{3} f^{3}-2484 x \tanh \left (f x +e \right ) c \,d^{2} f^{2}+2592 x^{3} \tanh \left (f x +e \right ) c \,d^{2} f^{4}+3888 x^{2} \tanh \left (f x +e \right ) c^{2} d \,f^{4}-648 x^{2} \tanh \left (f x +e \right ) c \,d^{2} f^{3}-648 x \tanh \left (f x +e \right ) c^{2} d \,f^{3}-2880 c^{3} f^{3}+648 d^{3} \tanh \left (f x +e \right )^{2} x^{4} f^{4}-5832 \tanh \left (f x +e \right ) c^{2} d \,f^{2}-6804 \tanh \left (f x +e \right ) c \,d^{2} f +864 x \tanh \left (f x +e \right )^{3} c^{3} f^{4}+1725 x \tanh \left (f x +e \right )^{3} d^{3} f +1530 x^{2} \tanh \left (f x +e \right )^{3} d^{3} f^{2}+792 x^{3} \tanh \left (f x +e \right )^{3} d^{3} f^{3}+216 d^{3} \tanh \left (f x +e \right )^{3} x^{4} f^{4}-2376 \tanh \left (f x +e \right )^{2} c^{2} d \,f^{2}-3060 \tanh \left (f x +e \right )^{2} c \,d^{2} f +864 x \,c^{3} f^{4}-5004 c \,d^{2} f^{2} x -2502 d^{3} f^{2} x^{2}-4032 c^{2} d \,f^{2}+4536 x \tanh \left (f x +e \right )^{2} c^{2} d \,f^{3}+4428 x \tanh \left (f x +e \right )^{2} c \,d^{2} f^{2}+3888 x^{2} \tanh \left (f x +e \right )^{2} c^{2} d \,f^{4}+4536 x^{2} \tanh \left (f x +e \right )^{2} c \,d^{2} f^{3}+2592 x^{3} \tanh \left (f x +e \right )^{2} c \,d^{2} f^{4}+216 d^{3} x^{4} f^{4}+2376 x^{2} \tanh \left (f x +e \right )^{3} c \,d^{2} f^{3}+864 x^{3} \tanh \left (f x +e \right )^{3} c \,d^{2} f^{4}-1242 x^{2} \tanh \left (f x +e \right ) d^{3} f^{2}-216 x^{3} \tanh \left (f x +e \right ) d^{3} f^{3}+2592 x \tanh \left (f x +e \right ) c^{3} f^{4}-1629 x \tanh \left (f x +e \right ) d^{3} f +648 d^{3} \tanh \left (f x +e \right ) x^{4} f^{4}+1296 x^{2} c^{2} d \,f^{4}+864 x^{3} c \,d^{2} f^{4}-2211 d^{3} f x -3936 c \,d^{2} f +2592 x \tanh \left (f x +e \right )^{2} c^{3} f^{4}+2115 x \tanh \left (f x +e \right )^{2} d^{3} f +2214 x^{2} \tanh \left (f x +e \right )^{2} d^{3} f^{2}+1512 x^{3} \tanh \left (f x +e \right )^{2} d^{3} f^{3}+2376 x \tanh \left (f x +e \right )^{3} c^{2} d \,f^{3}+3060 x \tanh \left (f x +e \right )^{3} c \,d^{2} f^{2}+1296 x^{2} \tanh \left (f x +e \right )^{3} c^{2} d \,f^{4}-2088 d^{3} x^{3} f^{3}-6264 c \,d^{2} f^{3} x^{2}-2592 \tanh \left (f x +e \right ) c^{3} f^{3}}{6912 f^{4} a^{3} \left (1+\tanh \left (f x +e \right )\right )^{3}}\) | \(803\) |
1/32/a^3*d^3*x^4+1/8/a^3*d^2*c*x^3+3/16/a^3*d*c^2*x^2+1/8/a^3*c^3*x+1/32/a ^3/d*c^4-3/64*(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^3/f^4*e xp(-2*f*x-2*e)-3/1024*(32*d^3*f^3*x^3+96*c*d^2*f^3*x^2+96*c^2*d*f^3*x+24*d ^3*f^2*x^2+32*c^3*f^3+48*c*d^2*f^2*x+24*c^2*d*f^2+12*d^3*f*x+12*c*d^2*f+3* d^3)/a^3/f^4*exp(-4*f*x-4*e)-1/1728*(36*d^3*f^3*x^3+108*c*d^2*f^3*x^2+108* c^2*d*f^3*x+18*d^3*f^2*x^2+36*c^3*f^3+36*c*d^2*f^2*x+18*c^2*d*f^2+6*d^3*f* x+6*c*d^2*f+d^3)/a^3/f^4*exp(-6*f*x-6*e)
Leaf count of result is larger than twice the leaf count of optimal. 844 vs. \(2 (298) = 596\).
Time = 0.24 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.51 \[ \int \frac {(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx=\frac {16 \, {\left (54 \, d^{3} f^{4} x^{4} - 36 \, c^{3} f^{3} - 18 \, c^{2} d f^{2} - 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} - 6 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} - 18 \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} - d^{3} f\right )} x\right )} \cosh \left (f x + e\right )^{3} + 48 \, {\left (54 \, d^{3} f^{4} x^{4} - 36 \, c^{3} f^{3} - 18 \, c^{2} d f^{2} - 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} - 6 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} - 18 \, c^{2} d f^{3} - 6 \, c d^{2} f^{2} - d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + 16 \, {\left (54 \, d^{3} f^{4} x^{4} + 36 \, c^{3} f^{3} + 18 \, c^{2} d f^{2} + 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} + 6 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} + 18 \, c^{2} d f^{3} + 6 \, c d^{2} f^{2} + d^{3} f\right )} x\right )} \sinh \left (f x + e\right )^{3} - 243 \, {\left (32 \, d^{3} f^{3} x^{3} + 32 \, c^{3} f^{3} + 40 \, c^{2} d f^{2} + 36 \, c d^{2} f + 17 \, d^{3} + 8 \, {\left (12 \, c d^{2} f^{3} + 5 \, d^{3} f^{2}\right )} x^{2} + 4 \, {\left (24 \, c^{2} d f^{3} + 20 \, c d^{2} f^{2} + 9 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) - 3 \, {\left (864 \, d^{3} f^{3} x^{3} + 864 \, c^{3} f^{3} + 1944 \, c^{2} d f^{2} + 2268 \, c d^{2} f + 1215 \, d^{3} + 648 \, {\left (4 \, c d^{2} f^{3} + 3 \, d^{3} f^{2}\right )} x^{2} - 16 \, {\left (54 \, d^{3} f^{4} x^{4} + 36 \, c^{3} f^{3} + 18 \, c^{2} d f^{2} + 6 \, c d^{2} f + 36 \, {\left (6 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + d^{3} + 18 \, {\left (18 \, c^{2} d f^{4} + 6 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 6 \, {\left (36 \, c^{3} f^{4} + 18 \, c^{2} d f^{3} + 6 \, c d^{2} f^{2} + d^{3} f\right )} x\right )} \cosh \left (f x + e\right )^{2} + 324 \, {\left (8 \, c^{2} d f^{3} + 12 \, c d^{2} f^{2} + 7 \, d^{3} f\right )} x\right )} \sinh \left (f x + e\right )}{27648 \, {\left (a^{3} f^{4} \cosh \left (f x + e\right )^{3} + 3 \, a^{3} f^{4} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, a^{3} f^{4} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + a^{3} f^{4} \sinh \left (f x + e\right )^{3}\right )}} \]
1/27648*(16*(54*d^3*f^4*x^4 - 36*c^3*f^3 - 18*c^2*d*f^2 - 6*c*d^2*f + 36*( 6*c*d^2*f^4 - d^3*f^3)*x^3 - d^3 + 18*(18*c^2*d*f^4 - 6*c*d^2*f^3 - d^3*f^ 2)*x^2 + 6*(36*c^3*f^4 - 18*c^2*d*f^3 - 6*c*d^2*f^2 - d^3*f)*x)*cosh(f*x + e)^3 + 48*(54*d^3*f^4*x^4 - 36*c^3*f^3 - 18*c^2*d*f^2 - 6*c*d^2*f + 36*(6 *c*d^2*f^4 - d^3*f^3)*x^3 - d^3 + 18*(18*c^2*d*f^4 - 6*c*d^2*f^3 - d^3*f^2 )*x^2 + 6*(36*c^3*f^4 - 18*c^2*d*f^3 - 6*c*d^2*f^2 - d^3*f)*x)*cosh(f*x + e)*sinh(f*x + e)^2 + 16*(54*d^3*f^4*x^4 + 36*c^3*f^3 + 18*c^2*d*f^2 + 6*c* d^2*f + 36*(6*c*d^2*f^4 + d^3*f^3)*x^3 + d^3 + 18*(18*c^2*d*f^4 + 6*c*d^2* f^3 + d^3*f^2)*x^2 + 6*(36*c^3*f^4 + 18*c^2*d*f^3 + 6*c*d^2*f^2 + d^3*f)*x )*sinh(f*x + e)^3 - 243*(32*d^3*f^3*x^3 + 32*c^3*f^3 + 40*c^2*d*f^2 + 36*c *d^2*f + 17*d^3 + 8*(12*c*d^2*f^3 + 5*d^3*f^2)*x^2 + 4*(24*c^2*d*f^3 + 20* c*d^2*f^2 + 9*d^3*f)*x)*cosh(f*x + e) - 3*(864*d^3*f^3*x^3 + 864*c^3*f^3 + 1944*c^2*d*f^2 + 2268*c*d^2*f + 1215*d^3 + 648*(4*c*d^2*f^3 + 3*d^3*f^2)* x^2 - 16*(54*d^3*f^4*x^4 + 36*c^3*f^3 + 18*c^2*d*f^2 + 6*c*d^2*f + 36*(6*c *d^2*f^4 + d^3*f^3)*x^3 + d^3 + 18*(18*c^2*d*f^4 + 6*c*d^2*f^3 + d^3*f^2)* x^2 + 6*(36*c^3*f^4 + 18*c^2*d*f^3 + 6*c*d^2*f^2 + d^3*f)*x)*cosh(f*x + e) ^2 + 324*(8*c^2*d*f^3 + 12*c*d^2*f^2 + 7*d^3*f)*x)*sinh(f*x + e))/(a^3*f^4 *cosh(f*x + e)^3 + 3*a^3*f^4*cosh(f*x + e)^2*sinh(f*x + e) + 3*a^3*f^4*cos h(f*x + e)*sinh(f*x + e)^2 + a^3*f^4*sinh(f*x + e)^3)
\[ \int \frac {(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx=\frac {\int \frac {c^{3}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
(Integral(c**3/(tanh(e + f*x)**3 + 3*tanh(e + f*x)**2 + 3*tanh(e + f*x) + 1), x) + Integral(d**3*x**3/(tanh(e + f*x)**3 + 3*tanh(e + f*x)**2 + 3*tan h(e + f*x) + 1), x) + Integral(3*c*d**2*x**2/(tanh(e + f*x)**3 + 3*tanh(e + f*x)**2 + 3*tanh(e + f*x) + 1), x) + Integral(3*c**2*d*x/(tanh(e + f*x)* *3 + 3*tanh(e + f*x)**2 + 3*tanh(e + f*x) + 1), x))/a**3
Time = 1.67 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx=\frac {1}{96} \, c^{3} {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} - \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} + 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} - 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c^{2} d e^{\left (-6 \, e\right )}}{384 \, a^{3} f^{2}} + \frac {{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} - 648 \, {\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 8 \, {\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d^{2} e^{\left (-6 \, e\right )}}{2304 \, a^{3} f^{3}} + \frac {{\left (864 \, f^{4} x^{4} e^{\left (6 \, e\right )} - 1296 \, {\left (4 \, f^{3} x^{3} e^{\left (4 \, e\right )} + 6 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 6 \, f x e^{\left (4 \, e\right )} + 3 \, e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (32 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 24 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 12 \, f x e^{\left (2 \, e\right )} + 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 16 \, {\left (36 \, f^{3} x^{3} + 18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{3} e^{\left (-6 \, e\right )}}{27648 \, a^{3} f^{4}} \]
1/96*c^3*(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)) + 1/384*(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^2*d*e^(-6*e)/(a^3*f^2) + 1/2304*(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))*c*d^2*e^(-6*e)/(a^3*f^3) + 1/27648*(864*f^4*x^4*e^ (6*e) - 1296*(4*f^3*x^3*e^(4*e) + 6*f^2*x^2*e^(4*e) + 6*f*x*e^(4*e) + 3*e^ (4*e))*e^(-2*f*x) - 81*(32*f^3*x^3*e^(2*e) + 24*f^2*x^2*e^(2*e) + 12*f*x*e ^(2*e) + 3*e^(2*e))*e^(-4*f*x) - 16*(36*f^3*x^3 + 18*f^2*x^2 + 6*f*x + 1)* e^(-6*f*x))*d^3*e^(-6*e)/(a^3*f^4)
Time = 0.28 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.63 \[ \int \frac {(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx=\frac {{\left (864 \, d^{3} f^{4} x^{4} e^{\left (6 \, f x + 6 \, e\right )} + 3456 \, c d^{2} f^{4} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} e^{\left (6 \, f x + 6 \, e\right )} - 5184 \, d^{3} f^{3} x^{3} e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, d^{3} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} - 576 \, d^{3} f^{3} x^{3} + 3456 \, c^{3} f^{4} x e^{\left (6 \, f x + 6 \, e\right )} - 15552 \, c d^{2} f^{3} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c d^{2} f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 1728 \, c d^{2} f^{3} x^{2} - 15552 \, c^{2} d f^{3} x e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, d^{3} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c^{2} d f^{3} x e^{\left (2 \, f x + 2 \, e\right )} - 1944 \, d^{3} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 1728 \, c^{2} d f^{3} x - 288 \, d^{3} f^{2} x^{2} - 5184 \, c^{3} f^{3} e^{\left (4 \, f x + 4 \, e\right )} - 15552 \, c d^{2} f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, c^{3} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3888 \, c d^{2} f^{2} x e^{\left (2 \, f x + 2 \, e\right )} - 576 \, c^{3} f^{3} - 576 \, c d^{2} f^{2} x - 7776 \, c^{2} d f^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, d^{3} f x e^{\left (4 \, f x + 4 \, e\right )} - 1944 \, c^{2} d f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 972 \, d^{3} f x e^{\left (2 \, f x + 2 \, e\right )} - 288 \, c^{2} d f^{2} - 96 \, d^{3} f x - 7776 \, c d^{2} f e^{\left (4 \, f x + 4 \, e\right )} - 972 \, c d^{2} f e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c d^{2} f - 3888 \, d^{3} e^{\left (4 \, f x + 4 \, e\right )} - 243 \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} - 16 \, d^{3}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{27648 \, a^{3} f^{4}} \]
1/27648*(864*d^3*f^4*x^4*e^(6*f*x + 6*e) + 3456*c*d^2*f^4*x^3*e^(6*f*x + 6 *e) + 5184*c^2*d*f^4*x^2*e^(6*f*x + 6*e) - 5184*d^3*f^3*x^3*e^(4*f*x + 4*e ) - 2592*d^3*f^3*x^3*e^(2*f*x + 2*e) - 576*d^3*f^3*x^3 + 3456*c^3*f^4*x*e^ (6*f*x + 6*e) - 15552*c*d^2*f^3*x^2*e^(4*f*x + 4*e) - 7776*c*d^2*f^3*x^2*e ^(2*f*x + 2*e) - 1728*c*d^2*f^3*x^2 - 15552*c^2*d*f^3*x*e^(4*f*x + 4*e) - 7776*d^3*f^2*x^2*e^(4*f*x + 4*e) - 7776*c^2*d*f^3*x*e^(2*f*x + 2*e) - 1944 *d^3*f^2*x^2*e^(2*f*x + 2*e) - 1728*c^2*d*f^3*x - 288*d^3*f^2*x^2 - 5184*c ^3*f^3*e^(4*f*x + 4*e) - 15552*c*d^2*f^2*x*e^(4*f*x + 4*e) - 2592*c^3*f^3* e^(2*f*x + 2*e) - 3888*c*d^2*f^2*x*e^(2*f*x + 2*e) - 576*c^3*f^3 - 576*c*d ^2*f^2*x - 7776*c^2*d*f^2*e^(4*f*x + 4*e) - 7776*d^3*f*x*e^(4*f*x + 4*e) - 1944*c^2*d*f^2*e^(2*f*x + 2*e) - 972*d^3*f*x*e^(2*f*x + 2*e) - 288*c^2*d* f^2 - 96*d^3*f*x - 7776*c*d^2*f*e^(4*f*x + 4*e) - 972*c*d^2*f*e^(2*f*x + 2 *e) - 96*c*d^2*f - 3888*d^3*e^(4*f*x + 4*e) - 243*d^3*e^(2*f*x + 2*e) - 16 *d^3)*e^(-6*f*x - 6*e)/(a^3*f^4)
Time = 2.03 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.12 \[ \int \frac {(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx=\frac {c^3\,x}{8\,a^3}-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {96\,c^3\,f^3+72\,c^2\,d\,f^2+36\,c\,d^2\,f+9\,d^3}{1024\,a^3\,f^4}+\frac {3\,d^3\,x^3}{32\,a^3\,f}+\frac {9\,d\,x\,\left (8\,c^2\,f^2+4\,c\,d\,f+d^2\right )}{256\,a^3\,f^3}+\frac {9\,d^2\,x^2\,\left (d+4\,c\,f\right )}{128\,a^3\,f^2}\right )-{\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {36\,c^3\,f^3+18\,c^2\,d\,f^2+6\,c\,d^2\,f+d^3}{1728\,a^3\,f^4}+\frac {d^3\,x^3}{48\,a^3\,f}+\frac {d\,x\,\left (18\,c^2\,f^2+6\,c\,d\,f+d^2\right )}{288\,a^3\,f^3}+\frac {d^2\,x^2\,\left (d+6\,c\,f\right )}{96\,a^3\,f^2}\right )-{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {12\,c^3\,f^3+18\,c^2\,d\,f^2+18\,c\,d^2\,f+9\,d^3}{64\,a^3\,f^4}+\frac {3\,d^3\,x^3}{16\,a^3\,f}+\frac {9\,d\,x\,\left (2\,c^2\,f^2+2\,c\,d\,f+d^2\right )}{32\,a^3\,f^3}+\frac {9\,d^2\,x^2\,\left (d+2\,c\,f\right )}{32\,a^3\,f^2}\right )+\frac {d^3\,x^4}{32\,a^3}+\frac {3\,c^2\,d\,x^2}{16\,a^3}+\frac {c\,d^2\,x^3}{8\,a^3} \]
(c^3*x)/(8*a^3) - exp(- 4*e - 4*f*x)*((9*d^3 + 96*c^3*f^3 + 72*c^2*d*f^2 + 36*c*d^2*f)/(1024*a^3*f^4) + (3*d^3*x^3)/(32*a^3*f) + (9*d*x*(d^2 + 8*c^2 *f^2 + 4*c*d*f))/(256*a^3*f^3) + (9*d^2*x^2*(d + 4*c*f))/(128*a^3*f^2)) - exp(- 6*e - 6*f*x)*((d^3 + 36*c^3*f^3 + 18*c^2*d*f^2 + 6*c*d^2*f)/(1728*a^ 3*f^4) + (d^3*x^3)/(48*a^3*f) + (d*x*(d^2 + 18*c^2*f^2 + 6*c*d*f))/(288*a^ 3*f^3) + (d^2*x^2*(d + 6*c*f))/(96*a^3*f^2)) - exp(- 2*e - 2*f*x)*((9*d^3 + 12*c^3*f^3 + 18*c^2*d*f^2 + 18*c*d^2*f)/(64*a^3*f^4) + (3*d^3*x^3)/(16*a ^3*f) + (9*d*x*(d^2 + 2*c^2*f^2 + 2*c*d*f))/(32*a^3*f^3) + (9*d^2*x^2*(d + 2*c*f))/(32*a^3*f^2)) + (d^3*x^4)/(32*a^3) + (3*c^2*d*x^2)/(16*a^3) + (c* d^2*x^3)/(8*a^3)