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

Optimal. Leaf size=336 \[ \frac {d^2 (c+d x) e^{-6 e-6 f x}}{288 a^3 f^3}-\frac {9 d^2 (c+d x) e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {9 d^2 (c+d x) e^{-2 e-2 f x}}{32 a^3 f^3}+\frac {d (c+d x)^2 e^{-6 e-6 f x}}{96 a^3 f^2}-\frac {9 d (c+d x)^2 e^{-4 e-4 f x}}{128 a^3 f^2}+\frac {9 d (c+d x)^2 e^{-2 e-2 f x}}{32 a^3 f^2}+\frac {(c+d x)^3 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^3 e^{-4 e-4 f x}}{32 a^3 f}+\frac {3 (c+d x)^3 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}+\frac {d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac {9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}+\frac {9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4} \]

[Out]

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+9/64*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

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Rubi [A]  time = 0.37, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3729, 2176, 2194} \[ \frac {d^2 (c+d x) e^{-6 e-6 f x}}{288 a^3 f^3}-\frac {9 d^2 (c+d x) e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {9 d^2 (c+d x) e^{-2 e-2 f x}}{32 a^3 f^3}+\frac {d (c+d x)^2 e^{-6 e-6 f x}}{96 a^3 f^2}-\frac {9 d (c+d x)^2 e^{-4 e-4 f x}}{128 a^3 f^2}+\frac {9 d (c+d x)^2 e^{-2 e-2 f x}}{32 a^3 f^2}+\frac {(c+d x)^3 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^3 e^{-4 e-4 f x}}{32 a^3 f}+\frac {3 (c+d x)^3 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^4}{32 a^3 d}+\frac {d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac {9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}+\frac {9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*E^(-6*e - 6*f*x))/(1728*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)

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]

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]

Rubi steps

\begin {align*} \int \frac {(c+d x)^3}{(a+a \coth (e+f x))^3} \, dx &=\int \left (\frac {(c+d x)^3}{8 a^3}-\frac {e^{-6 e-6 f x} (c+d x)^3}{8 a^3}+\frac {3 e^{-4 e-4 f x} (c+d x)^3}{8 a^3}-\frac {3 e^{-2 e-2 f x} (c+d x)^3}{8 a^3}\right ) \, dx\\ &=\frac {(c+d x)^4}{32 a^3 d}-\frac {\int e^{-6 e-6 f x} (c+d x)^3 \, dx}{8 a^3}+\frac {3 \int e^{-4 e-4 f x} (c+d x)^3 \, dx}{8 a^3}-\frac {3 \int e^{-2 e-2 f x} (c+d x)^3 \, dx}{8 a^3}\\ &=\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}-\frac {d \int e^{-6 e-6 f x} (c+d x)^2 \, dx}{16 a^3 f}+\frac {(9 d) \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{32 a^3 f}-\frac {(9 d) \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{16 a^3 f}\\ &=\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}-\frac {d^2 \int e^{-6 e-6 f x} (c+d x) \, dx}{48 a^3 f^2}+\frac {\left (9 d^2\right ) \int e^{-4 e-4 f x} (c+d x) \, dx}{64 a^3 f^2}-\frac {\left (9 d^2\right ) \int e^{-2 e-2 f x} (c+d x) \, dx}{16 a^3 f^2}\\ &=\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}-\frac {d^3 \int e^{-6 e-6 f x} \, dx}{288 a^3 f^3}+\frac {\left (9 d^3\right ) \int e^{-4 e-4 f x} \, dx}{256 a^3 f^3}-\frac {\left (9 d^3\right ) \int e^{-2 e-2 f x} \, dx}{32 a^3 f^3}\\ &=\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}\\ \end {align*}

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Mathematica [A]  time = 2.48, size = 615, normalized size = 1.83 \[ \frac {\text {csch}^3(e+f x) \left (3456 c^3 f^4 x \sinh (3 (e+f x))+7776 c^3 f^3 \sinh (e+f x)-576 c^3 f^3 \sinh (3 (e+f x))+5184 c^2 d f^4 x^2 \sinh (3 (e+f x))+23328 c^2 d f^3 x \sinh (e+f x)-1728 c^2 d f^3 x \sinh (3 (e+f x))+9720 c^2 d f^2 \sinh (e+f x)-288 c^2 d f^2 \sinh (3 (e+f x))+81 \left (32 c^3 f^3+24 c^2 d f^2 (4 f x+3)+12 c d^2 f \left (8 f^2 x^2+12 f x+7\right )+d^3 \left (32 f^3 x^3+72 f^2 x^2+84 f x+45\right )\right ) \cosh (e+f x)+16 \left (36 c^3 f^3 (6 f x+1)+18 c^2 d f^2 \left (18 f^2 x^2+6 f x+1\right )+6 c d^2 f \left (36 f^3 x^3+18 f^2 x^2+6 f x+1\right )+d^3 \left (54 f^4 x^4+36 f^3 x^3+18 f^2 x^2+6 f x+1\right )\right ) \cosh (3 (e+f x))+3456 c d^2 f^4 x^3 \sinh (3 (e+f x))+23328 c d^2 f^3 x^2 \sinh (e+f x)-1728 c d^2 f^3 x^2 \sinh (3 (e+f x))+19440 c d^2 f^2 x \sinh (e+f x)-576 c d^2 f^2 x \sinh (3 (e+f x))+8748 c d^2 f \sinh (e+f x)-96 c d^2 f \sinh (3 (e+f x))+864 d^3 f^4 x^4 \sinh (3 (e+f x))+7776 d^3 f^3 x^3 \sinh (e+f x)-576 d^3 f^3 x^3 \sinh (3 (e+f x))+9720 d^3 f^2 x^2 \sinh (e+f x)-288 d^3 f^2 x^2 \sinh (3 (e+f x))+8748 d^3 f x \sinh (e+f x)-96 d^3 f x \sinh (3 (e+f x))+4131 d^3 \sinh (e+f x)-16 d^3 \sinh (3 (e+f x))\right )}{27648 a^3 f^4 (\coth (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csch[e + f*x]^3*(81*(32*c^3*f^3 + 24*c^2*d*f^2*(3 + 4*f*x) + 12*c*d^2*f*(7 + 12*f*x + 8*f^2*x^2) + d^3*(45 +
84*f*x + 72*f^2*x^2 + 32*f^3*x^3))*Cosh[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)] + 4131*d^3*Sinh[e + f*x] + 8748*c*d^2*f*Sinh[e + f*x] + 9720*c^2*d*f^2*Sinh[e + f*x] +
7776*c^3*f^3*Sinh[e + f*x] + 8748*d^3*f*x*Sinh[e + f*x] + 19440*c*d^2*f^2*x*Sinh[e + f*x] + 23328*c^2*d*f^3*x
*Sinh[e + f*x] + 9720*d^3*f^2*x^2*Sinh[e + f*x] + 23328*c*d^2*f^3*x^2*Sinh[e + f*x] + 7776*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*Sin
h[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*Sin
h[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 + Coth[e + f*x])^3)

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fricas [B]  time = 0.40, size = 844, normalized size = 2.51 \[ \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} + 81 \, {\left (32 \, d^{3} f^{3} x^{3} + 32 \, c^{3} f^{3} + 72 \, c^{2} d f^{2} + 84 \, c d^{2} f + 45 \, d^{3} + 24 \, {\left (4 \, c d^{2} f^{3} + 3 \, d^{3} f^{2}\right )} x^{2} + 12 \, {\left (8 \, c^{2} d f^{3} + 12 \, c d^{2} f^{2} + 7 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) + 3 \, {\left (2592 \, d^{3} f^{3} x^{3} + 2592 \, c^{3} f^{3} + 3240 \, c^{2} d f^{2} + 2916 \, c d^{2} f + 1377 \, d^{3} + 648 \, {\left (12 \, c d^{2} f^{3} + 5 \, 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 (24 \, c^{2} d f^{3} + 20 \, c d^{2} f^{2} + 9 \, 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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*coth(f*x+e))^3,x, algorithm="fricas")

[Out]

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 + 1
8*(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 + 81*(32*d^3*f^3*x^3 + 32*c^3*f^3 + 72*c^2*d*f^2 + 84*c*d^2*f + 45*d^3 + 24*(4*c*d^2*f
^3 + 3*d^3*f^2)*x^2 + 12*(8*c^2*d*f^3 + 12*c*d^2*f^2 + 7*d^3*f)*x)*cosh(f*x + e) + 3*(2592*d^3*f^3*x^3 + 2592*
c^3*f^3 + 3240*c^2*d*f^2 + 2916*c*d^2*f + 1377*d^3 + 648*(12*c*d^2*f^3 + 5*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*(24*c^2*d*f^3 +
20*c*d^2*f^2 + 9*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*cosh(f*x + e)*sinh(f*x + e)^2 + a^3*f^4*sinh(f*x + e)^3)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 573, normalized size = 1.71 \[ \frac {{\left (864 \, d^{3} f^{4} x^{4} e^{\left (6 \, f x + 6 \, e\right )} + 3456 \, c d^{2} f^{4} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 5184 \, d^{3} f^{3} x^{3} e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, d^{3} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 576 \, d^{3} f^{3} x^{3} + 3456 \, c^{3} f^{4} x e^{\left (6 \, f x + 6 \, e\right )} + 15552 \, c d^{2} f^{3} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c d^{2} f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 1728 \, c d^{2} f^{3} x^{2} + 15552 \, c^{2} d f^{3} x e^{\left (4 \, f x + 4 \, e\right )} + 7776 \, d^{3} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c^{2} d f^{3} x e^{\left (2 \, f x + 2 \, e\right )} - 1944 \, d^{3} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 1728 \, c^{2} d f^{3} x + 288 \, d^{3} f^{2} x^{2} + 5184 \, c^{3} f^{3} e^{\left (4 \, f x + 4 \, e\right )} + 15552 \, c d^{2} f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, c^{3} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3888 \, c d^{2} f^{2} x e^{\left (2 \, f x + 2 \, e\right )} + 576 \, c^{3} f^{3} + 576 \, c d^{2} f^{2} x + 7776 \, c^{2} d f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 7776 \, d^{3} f x e^{\left (4 \, f x + 4 \, e\right )} - 1944 \, c^{2} d f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 972 \, d^{3} f x e^{\left (2 \, f x + 2 \, e\right )} + 288 \, c^{2} d f^{2} + 96 \, d^{3} f x + 7776 \, c d^{2} f e^{\left (4 \, f x + 4 \, e\right )} - 972 \, c d^{2} f e^{\left (2 \, f x + 2 \, e\right )} + 96 \, c d^{2} f + 3888 \, d^{3} e^{\left (4 \, f x + 4 \, e\right )} - 243 \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} + 16 \, d^{3}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{27648 \, a^{3} f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*coth(f*x+e))^3,x, algorithm="giac")

[Out]

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 + 5
76*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)

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maple [B]  time = 0.71, size = 3691, normalized size = 10.99 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+a*coth(f*x+e))^3,x)

[Out]

1/a^3/f*(1/f^3*d^3*(1/4*(f*x+e)^3*cosh(f*x+e)*sinh(f*x+e)^3-3/8*(f*x+e)^3*cosh(f*x+e)*sinh(f*x+e)+3/32*(f*x+e)
^4-3/16*(f*x+e)^2*sinh(f*x+e)^4+3/32*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)^3-45/64*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-2
7/128*(f*x+e)^2-3/128*sinh(f*x+e)^4+45/128*cosh(f*x+e)^2+9/16*(f*x+e)^2*cosh(f*x+e)^2)+4/f^3*d^3*(1/6*(f*x+e)^
3*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)^3*cosh(f*x+e)^4-1/12*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^5+1/12*(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+1/36*(f*x+e)*cosh(f*x+e)^6
-1/216*cosh(f*x+e)^5*sinh(f*x+e)+1/216*cosh(f*x+e)^3*sinh(f*x+e)+5/72*cosh(f*x+e)*sinh(f*x+e)+5/72*f*x+5/72*e-
1/24*(f*x+e)*cosh(f*x+e)^4-1/8*(f*x+e)*cosh(f*x+e)^2)-4/f^3*d^3*(1/6*(f*x+e)^3*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8
*(f*x+e)^3*sinh(f*x+e)*cosh(f*x+e)^3+1/16*(f*x+e)^3*cosh(f*x+e)*sinh(f*x+e)+1/64*(f*x+e)^4-1/12*(f*x+e)^2*sinh
(f*x+e)^2*cosh(f*x+e)^4+13/96*(f*x+e)^2*cosh(f*x+e)^4+1/36*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5-43/576*(f*x+e)*si
nh(f*x+e)*cosh(f*x+e)^3-7/384*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-7/768*(f*x+e)^2-1/216*cosh(f*x+e)^6+43/2304*cosh
(f*x+e)^4+7/768*cosh(f*x+e)^2-3/32*(f*x+e)^2*cosh(f*x+e)^2)+4*c^3*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*cosh(f
*x+e)^4)-4*c^3*(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)+1/1
6*f*x+1/16*e)+c^3*((1/4*sinh(f*x+e)^3-3/8*sinh(f*x+e))*cosh(f*x+e)+3/8*f*x+3/8*e)-3/4*c^3*sinh(f*x+e)^4-3/f^3*
d^3*(1/4*(f*x+e)^3*sinh(f*x+e)^4-3/16*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)^3+9/32*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+
e)-3/32*(f*x+e)^3+3/32*(f*x+e)*sinh(f*x+e)^4-3/128*sinh(f*x+e)^3*cosh(f*x+e)+45/256*cosh(f*x+e)*sinh(f*x+e)+27
/256*f*x+27/256*e-9/32*(f*x+e)*cosh(f*x+e)^2)+24/f^2*d^2*c*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)*sinh(f*x+e)+1/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh(f*x+
e)^4+13/288*cosh(f*x+e)^4-1/32*cosh(f*x+e)^2)+3*d^2*e^2/f^2*c*((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*d*e/f*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)-24/f^2*d^2*c*e*(1/
6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)*cosh(f*x+e)^4-1/36*cosh(f*x+e)^5*sinh(f*x+e)+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)+12*d^2*e^2/f^2*c*(1/6*sinh(f*x+e)^2*cosh(f*x+e)
^4-1/12*cosh(f*x+e)^4)-12*d*e/f*c^2*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*cosh(f*x+e)^4)-6/f^2*d^2*c*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)^4+3/16*cos
h(f*x+e)^2)-12*d^2*e^2/f^2*c*(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)*s
inh(f*x+e)+1/16*f*x+1/16*e)+12*d*e/f*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*c
osh(f*x+e)*sinh(f*x+e)+1/16*f*x+1/16*e)+3/4/f^3*d^3*e^3*sinh(f*x+e)^4+9/f^3*d^3*e*(1/4*(f*x+e)^2*sinh(f*x+e)^4
-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)^3+3/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-3/32*(f*x+e)^2+1/32*sinh(f*x+e)^4-
3/32*cosh(f*x+e)^2)-9/f^3*d^3*e^2*(1/4*(f*x+e)*sinh(f*x+e)^4-1/16*sinh(f*x+e)^3*cosh(f*x+e)+3/32*cosh(f*x+e)*s
inh(f*x+e)-3/32*f*x-3/32*e)-9/f^2*c*d^2*(1/4*(f*x+e)^2*sinh(f*x+e)^4-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)^3+3/1
6*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-3/32*(f*x+e)^2+1/32*sinh(f*x+e)^4-3/32*cosh(f*x+e)^2)-9/f*c^2*d*(1/4*(f*x+e)
*sinh(f*x+e)^4-1/16*sinh(f*x+e)^3*cosh(f*x+e)+3/32*cosh(f*x+e)*sinh(f*x+e)-3/32*f*x-3/32*e)-9/4/f^2*c*d^2*e^2*
sinh(f*x+e)^4+9/4/f*c^2*d*e*sinh(f*x+e)^4+18/f^2*c*d^2*e*(1/4*(f*x+e)*sinh(f*x+e)^4-1/16*sinh(f*x+e)^3*cosh(f*
x+e)+3/32*cosh(f*x+e)*sinh(f*x+e)-3/32*f*x-3/32*e)-4*d^3*e^3/f^3*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*cosh(f*
x+e)^4)+3/f^3*d^3*e^2*(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)^4+3/16*cosh(f*x+e)^2)+3/f^2*d^2*c*(1/4*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)^3-3/8*(f*x+e)^2*co
sh(f*x+e)*sinh(f*x+e)+1/8*(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)^4+1/32*sinh(f*x+e)^3*cosh(f*x+e)-15/64*cosh(f*x+e)
*sinh(f*x+e)-9/64*f*x-9/64*e+3/8*(f*x+e)*cosh(f*x+e)^2)-d^3*e^3/f^3*((1/4*sinh(f*x+e)^3-3/8*sinh(f*x+e))*cosh(
f*x+e)+3/8*f*x+3/8*e)+12/f*d*c^2*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)*cosh(f*x+e)^4-1/36*cosh
(f*x+e)^5*sinh(f*x+e)+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)-12/f^3*d^3*
e*(1/6*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)^2*cosh(f*x+e)^4-1/18*(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+1/12*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+1/24*(f*x+e)^2+1/108*cosh(f*x+e
)^6-1/72*cosh(f*x+e)^4-1/24*cosh(f*x+e)^2)-12/f*d*c^2*(1/6*(f*x+e)*sinh(f*x+e)^3*cosh(f*x+e)^3-1/8*(f*x+e)*sin
h(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*cosh(f*x+e)^4-1/32*cosh(f*x+e)^2)+4*d^3*e^3/f^3*(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)-3/f^3*d^3*e*(1/4*(f*x+e)^2*cosh(f*x+e)*sinh(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)^4+1/32*sinh(f*x+e)^3*cosh(f*x+e)-15/64*
cosh(f*x+e)*sinh(f*x+e)-9/64*f*x-9/64*e+3/8*(f*x+e)*cosh(f*x+e)^2)+3/f*d*c^2*(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)^4+3/16*cosh(f*x+e)^2)+12/f^3*d^3*e*(
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)*s
inh(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)*cosh(f*x+e)^4+1/108*cosh(f*x
+e)^5*sinh(f*x+e)-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-1/16*(f
*x+e)*cosh(f*x+e)^2)-12/f^3*d^3*e^2*(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*cosh(f*x+e)^4
-1/32*cosh(f*x+e)^2)-12/f^2*d^2*c*(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)*cosh(f*x+e)^4+1/108*cosh(f*x+e)^5*sinh(f*x+e)-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-1/16*(f*x+e)*cosh(f*x+e)^2)+12/f^3*d^3*e^2*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^
4-1/12*(f*x+e)*cosh(f*x+e)^4-1/36*cosh(f*x+e)^5*sinh(f*x+e)+1/36*cosh(f*x+e)^3*sinh(f*x+e)+1/24*cosh(f*x+e)*si
nh(f*x+e)+1/24*f*x+1/24*e)+12/f^2*d^2*c*(1/6*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^4-1/12*(f*x+e)^2*cosh(f*x+e)^
4-1/18*(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+1/12*(f*x+e)*cosh(f*x+e)*sinh(
f*x+e)+1/24*(f*x+e)^2+1/108*cosh(f*x+e)^6-1/72*cosh(f*x+e)^4-1/24*cosh(f*x+e)^2))

________________________________________________________________________________________

maxima [A]  time = 2.86, size = 406, normalized size = 1.21 \[ \frac {1}{96} \, c^{3} {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} + \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} - 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} + 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c^{2} d e^{\left (-6 \, e\right )}}{384 \, a^{3} f^{2}} + \frac {{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} + 648 \, {\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 8 \, {\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d^{2} e^{\left (-6 \, e\right )}}{2304 \, a^{3} f^{3}} + \frac {{\left (864 \, f^{4} x^{4} e^{\left (6 \, e\right )} + 1296 \, {\left (4 \, f^{3} x^{3} e^{\left (4 \, e\right )} + 6 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 6 \, f x e^{\left (4 \, e\right )} + 3 \, e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (32 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 24 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 12 \, f x e^{\left (2 \, e\right )} + 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 16 \, {\left (36 \, f^{3} x^{3} + 18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{3} e^{\left (-6 \, e\right )}}{27648 \, a^{3} f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*coth(f*x+e))^3,x, algorithm="maxima")

[Out]

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)

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mupad [B]  time = 1.44, size = 374, normalized size = 1.11 \[ {\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {12\,c^3\,f^3+18\,c^2\,d\,f^2+18\,c\,d^2\,f+9\,d^3}{64\,a^3\,f^4}+\frac {3\,d^3\,x^3}{16\,a^3\,f}+\frac {9\,d\,x\,\left (2\,c^2\,f^2+2\,c\,d\,f+d^2\right )}{32\,a^3\,f^3}+\frac {9\,d^2\,x^2\,\left (d+2\,c\,f\right )}{32\,a^3\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {96\,c^3\,f^3+72\,c^2\,d\,f^2+36\,c\,d^2\,f+9\,d^3}{1024\,a^3\,f^4}+\frac {3\,d^3\,x^3}{32\,a^3\,f}+\frac {9\,d\,x\,\left (8\,c^2\,f^2+4\,c\,d\,f+d^2\right )}{256\,a^3\,f^3}+\frac {9\,d^2\,x^2\,\left (d+4\,c\,f\right )}{128\,a^3\,f^2}\right )+{\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {36\,c^3\,f^3+18\,c^2\,d\,f^2+6\,c\,d^2\,f+d^3}{1728\,a^3\,f^4}+\frac {d^3\,x^3}{48\,a^3\,f}+\frac {d\,x\,\left (18\,c^2\,f^2+6\,c\,d\,f+d^2\right )}{288\,a^3\,f^3}+\frac {d^2\,x^2\,\left (d+6\,c\,f\right )}{96\,a^3\,f^2}\right )+\frac {c^3\,x}{8\,a^3}+\frac {d^3\,x^4}{32\,a^3}+\frac {3\,c^2\,d\,x^2}{16\,a^3}+\frac {c\,d^2\,x^3}{8\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + a*coth(e + f*x))^3,x)

[Out]

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)) - 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)) + (c^3*x)/(8*a^3) + (d^3*x^4)/(32*a^3) + (3*c^2*d*x^2)/(16*a^3)
+ (c*d^2*x^3)/(8*a^3)

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sympy [A]  time = 4.28, size = 3918, normalized size = 11.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+a*coth(f*x+e))**3,x)

[Out]

Piecewise((864*c**3*f**4*x*tanh(e + f*x)**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**
2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 2592*c**3*f**4*x*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e
+ f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 2592*c**3*f*
*4*x*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(
e + f*x) + 6912*a**3*f**4) + 864*c**3*f**4*x/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)*
*2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 6048*c**3*f**3*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e
+ f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 7776*c**3*f**
3*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e +
f*x) + 6912*a**3*f**4) + 2880*c**3*f**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 +
20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 1296*c**2*d*f**4*x**2*tanh(e + f*x)**3/(6912*a**3*f**4*tanh
(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 3888*c**2*
d*f**4*x**2*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*
f**4*tanh(e + f*x) + 6912*a**3*f**4) + 3888*c**2*d*f**4*x**2*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 +
20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 1296*c**2*d*f**4*x**2/(69
12*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f
**4) - 6264*c**2*d*f**3*x*tanh(e + f*x)**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2
+ 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) - 648*c**2*d*f**3*x*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e
+ f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 4536*c**2*d*
f**3*x*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tan
h(e + f*x) + 6912*a**3*f**4) + 2376*c**2*d*f**3*x/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e +
f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 6264*c**2*d*f**2*tanh(e + f*x)**2/(6912*a**3*f**4*
tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 9720*c
**2*d*f**2*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4
*tanh(e + f*x) + 6912*a**3*f**4) + 4032*c**2*d*f**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e
+ f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 864*c*d**2*f**4*x**3*tanh(e + f*x)**3/(6912*a**3
*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) +
2592*c*d**2*f**4*x**3*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 2
0736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 2592*c*d**2*f**4*x**3*tanh(e + f*x)/(6912*a**3*f**4*tanh(e +
f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 864*c*d**2*f**4
*x**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 69
12*a**3*f**4) - 6264*c*d**2*f**3*x**2*tanh(e + f*x)**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh
(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) - 648*c*d**2*f**3*x**2*tanh(e + f*x)**2/(6912*a
**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4)
+ 4536*c*d**2*f**3*x**2*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 2
0736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 2376*c*d**2*f**3*x**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 2073
6*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) - 5004*c*d**2*f**2*x*tanh(e + f
*x)**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6
912*a**3*f**4) - 2484*c*d**2*f**2*x*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e
+ f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 4428*c*d**2*f**2*x*tanh(e + f*x)/(6912*a**3*f**
4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 3060
*c*d**2*f**2*x/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e +
f*x) + 6912*a**3*f**4) + 5004*c*d**2*f*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tan
h(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 8748*c*d**2*f*tanh(e + f*x)/(6912*a**3*f**4*
tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 3936*c
*d**2*f/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) +
6912*a**3*f**4) + 216*d**3*f**4*x**4*tanh(e + f*x)**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(
e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 648*d**3*f**4*x**4*tanh(e + f*x)**2/(6912*a**3
*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) +
648*d**3*f**4*x**4*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a
**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 216*d**3*f**4*x**4/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f*
*4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) - 2088*d**3*f**3*x**3*tanh(e + f*x)**3/(
6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3
*f**4) - 216*d**3*f**3*x**3*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)*
*2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 1512*d**3*f**3*x**3*tanh(e + f*x)/(6912*a**3*f**4*tanh(
e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 792*d**3*f*
*3*x**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) +
6912*a**3*f**4) - 2502*d**3*f**2*x**2*tanh(e + f*x)**3/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh
(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) - 1242*d**3*f**2*x**2*tanh(e + f*x)**2/(6912*a*
*3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4)
+ 2214*d**3*f**2*x**2*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 2073
6*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 1530*d**3*f**2*x**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**
3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) - 2211*d**3*f*x*tanh(e + f*x)**3/(69
12*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f
**4) - 1629*d**3*f*x*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20
736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 2115*d**3*f*x*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 +
20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 1725*d**3*f*x/(6912*a**3
*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) +
2211*d**3*tanh(e + f*x)**2/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f*
*4*tanh(e + f*x) + 6912*a**3*f**4) + 4131*d**3*tanh(e + f*x)/(6912*a**3*f**4*tanh(e + f*x)**3 + 20736*a**3*f**
4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4) + 1952*d**3/(6912*a**3*f**4*tanh(e + f*x)
**3 + 20736*a**3*f**4*tanh(e + f*x)**2 + 20736*a**3*f**4*tanh(e + f*x) + 6912*a**3*f**4), Ne(f, 0)), ((c**3*x
+ 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4)/(a*coth(e) + a)**3, True))

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