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

Optimal. Leaf size=230 \[ -\frac{3 d^2 (c+d x) e^{-4 e-4 f x}}{128 a^2 f^3}+\frac{3 d^2 (c+d x) e^{-2 e-2 f x}}{8 a^2 f^3}-\frac{3 d (c+d x)^2 e^{-4 e-4 f x}}{64 a^2 f^2}+\frac{3 d (c+d x)^2 e^{-2 e-2 f x}}{8 a^2 f^2}-\frac{(c+d x)^3 e^{-4 e-4 f x}}{16 a^2 f}+\frac{(c+d x)^3 e^{-2 e-2 f x}}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}+\frac{3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4} \]

[Out]

(-3*d^3*E^(-4*e - 4*f*x))/(512*a^2*f^4) + (3*d^3*E^(-2*e - 2*f*x))/(16*a^2*f^4) - (3*d^2*E^(-4*e - 4*f*x)*(c +
d*x))/(128*a^2*f^3) + (3*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(8*a^2*f^3) - (3*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(64
*a^2*f^2) + (3*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(8*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^3)/(16*a^2*f) + (E^(-
2*e - 2*f*x)*(c + d*x)^3)/(4*a^2*f) + (c + d*x)^4/(16*a^2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.26818, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3729, 2176, 2194} \[ -\frac{3 d^2 (c+d x) e^{-4 e-4 f x}}{128 a^2 f^3}+\frac{3 d^2 (c+d x) e^{-2 e-2 f x}}{8 a^2 f^3}-\frac{3 d (c+d x)^2 e^{-4 e-4 f x}}{64 a^2 f^2}+\frac{3 d (c+d x)^2 e^{-2 e-2 f x}}{8 a^2 f^2}-\frac{(c+d x)^3 e^{-4 e-4 f x}}{16 a^2 f}+\frac{(c+d x)^3 e^{-2 e-2 f x}}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}+\frac{3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*d^3*E^(-4*e - 4*f*x))/(512*a^2*f^4) + (3*d^3*E^(-2*e - 2*f*x))/(16*a^2*f^4) - (3*d^2*E^(-4*e - 4*f*x)*(c +
d*x))/(128*a^2*f^3) + (3*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(8*a^2*f^3) - (3*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(64
*a^2*f^2) + (3*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(8*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^3)/(16*a^2*f) + (E^(-
2*e - 2*f*x)*(c + d*x)^3)/(4*a^2*f) + (c + d*x)^4/(16*a^2*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)^3}{(a+a \coth (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^3}{4 a^2}+\frac{e^{-4 e-4 f x} (c+d x)^3}{4 a^2}-\frac{e^{-2 e-2 f x} (c+d x)^3}{2 a^2}\right ) \, dx\\ &=\frac{(c+d x)^4}{16 a^2 d}+\frac{\int e^{-4 e-4 f x} (c+d x)^3 \, dx}{4 a^2}-\frac{\int e^{-2 e-2 f x} (c+d x)^3 \, dx}{2 a^2}\\ &=-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{(3 d) \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{16 a^2 f}-\frac{(3 d) \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{4 a^2 f}\\ &=-\frac{3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}+\frac{3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{\left (3 d^2\right ) \int e^{-4 e-4 f x} (c+d x) \, dx}{32 a^2 f^2}-\frac{\left (3 d^2\right ) \int e^{-2 e-2 f x} (c+d x) \, dx}{4 a^2 f^2}\\ &=-\frac{3 d^2 e^{-4 e-4 f x} (c+d x)}{128 a^2 f^3}+\frac{3 d^2 e^{-2 e-2 f x} (c+d x)}{8 a^2 f^3}-\frac{3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}+\frac{3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{\left (3 d^3\right ) \int e^{-4 e-4 f x} \, dx}{128 a^2 f^3}-\frac{\left (3 d^3\right ) \int e^{-2 e-2 f x} \, dx}{8 a^2 f^3}\\ &=-\frac{3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}+\frac{3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4}-\frac{3 d^2 e^{-4 e-4 f x} (c+d x)}{128 a^2 f^3}+\frac{3 d^2 e^{-2 e-2 f x} (c+d x)}{8 a^2 f^3}-\frac{3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}+\frac{3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}+\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}\\ \end{align*}

Mathematica [A]  time = 1.02336, size = 420, normalized size = 1.83 \[ \frac{\text{csch}^2(e+f x) (\sinh (f x)+\cosh (f x))^2 \left (f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) (\sinh (2 e)+\cosh (2 e))+\frac{1}{32} (\sinh (2 e)-\cosh (2 e)) \cosh (4 f x) \left (24 c^2 d f^2 (4 f x+1)+32 c^3 f^3+12 c d^2 f \left (8 f^2 x^2+4 f x+1\right )+d^3 \left (32 f^3 x^3+24 f^2 x^2+12 f x+3\right )\right )+\frac{1}{32} (\cosh (2 e)-\sinh (2 e)) \sinh (4 f x) \left (24 c^2 d f^2 (4 f x+1)+32 c^3 f^3+12 c d^2 f \left (8 f^2 x^2+4 f x+1\right )+d^3 \left (32 f^3 x^3+24 f^2 x^2+12 f x+3\right )\right )-\sinh (2 f x) \left (6 c^2 d f^2 (2 f x+1)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 f x+1\right )+d^3 \left (4 f^3 x^3+6 f^2 x^2+6 f x+3\right )\right )+\cosh (2 f x) \left (6 c^2 d f^2 (2 f x+1)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 f x+1\right )+d^3 \left (4 f^3 x^3+6 f^2 x^2+6 f x+3\right )\right )\right )}{16 a^2 f^4 (\coth (e+f x)+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csch[e + f*x]^2*(Cosh[f*x] + Sinh[f*x])^2*((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] + ((32*c^3*f^3 + 24*c^2*d*f^2*(1 + 4*f*x) + 12*c
*d^2*f*(1 + 4*f*x + 8*f^2*x^2) + d^3*(3 + 12*f*x + 24*f^2*x^2 + 32*f^3*x^3))*Cosh[4*f*x]*(-Cosh[2*e] + Sinh[2*
e]))/32 + f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*(Cosh[2*e] + Sinh[2*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))*Sinh[2*f*x] + ((32
*c^3*f^3 + 24*c^2*d*f^2*(1 + 4*f*x) + 12*c*d^2*f*(1 + 4*f*x + 8*f^2*x^2) + d^3*(3 + 12*f*x + 24*f^2*x^2 + 32*f
^3*x^3))*(Cosh[2*e] - Sinh[2*e])*Sinh[4*f*x])/32))/(16*a^2*f^4*(1 + Coth[e + f*x])^2)

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Maple [B]  time = 0.108, size = 2200, normalized size = 9.6 \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)^3/(a+a*coth(f*x+e))^2,x)

[Out]

1/a^2/f*(6/f^2*d^2*e*c*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^2-1/4*cosh(f*x+e)^2)-3*d^2*e^2/f^2*c*(
1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+3*d*e/f*c^2*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+6*c^2/f*d*e
*(1/4*sinh(f*x+e)^2*cosh(f*x+e)^2-1/4*cosh(f*x+e)^2)-6*c^2/f*d*e*(1/4*cosh(f*x+e)^3*sinh(f*x+e)-1/8*cosh(f*x+e
)*sinh(f*x+e)-1/8*f*x-1/8*e)+12/f^2*c*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)-12/f^2*c*d^2*e*(1/4*(f*x+e)*sinh
(f*x+e)*cosh(f*x+e)^3-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/16*(f*x+e)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e)^2)+6/f
^2*c*d^2*e^2*(1/4*cosh(f*x+e)^3*sinh(f*x+e)-1/8*cosh(f*x+e)*sinh(f*x+e)-1/8*f*x-1/8*e)-6/f^2*c*d^2*e^2*(1/4*si
nh(f*x+e)^2*cosh(f*x+e)^2-1/4*cosh(f*x+e)^2)-3/f^2*d^2*c*(1/2*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)-1/6*(f*x+e)^3-
1/2*(f*x+e)*cosh(f*x+e)^2+1/4*cosh(f*x+e)*sinh(f*x+e)+1/4*f*x+1/4*e)-3/f^3*d^3*e^2*(1/2*(f*x+e)*cosh(f*x+e)*si
nh(f*x+e)-1/4*(f*x+e)^2-1/4*cosh(f*x+e)^2)-3/f*d*c^2*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/4*(f*x+e)^2-1/4*co
sh(f*x+e)^2)+d^3*e^3/f^3*(1/2*cosh(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e)+3/f^3*d^3*e*(1/2*(f*x+e)^2*cosh(f*x+e)*si
nh(f*x+e)-1/6*(f*x+e)^3-1/2*(f*x+e)*cosh(f*x+e)^2+1/4*cosh(f*x+e)*sinh(f*x+e)+1/4*f*x+1/4*e)-1/f^3*d^3*(1/2*(f
*x+e)^3*cosh(f*x+e)*sinh(f*x+e)-1/8*(f*x+e)^4-3/4*(f*x+e)^2*cosh(f*x+e)^2+3/4*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+
3/8*(f*x+e)^2-3/8*cosh(f*x+e)^2)+2/f^3*d^3*(1/4*(f*x+e)^3*sinh(f*x+e)*cosh(f*x+e)^3-1/8*(f*x+e)^3*cosh(f*x+e)*
sinh(f*x+e)-1/32*(f*x+e)^4-3/16*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^2+3/32*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3-3
/64*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-3/128*(f*x+e)^2-3/128*sinh(f*x+e)^2*cosh(f*x+e)^2)-2/f^3*d^3*(1/4*(f*x+e)^
3*sinh(f*x+e)^2*cosh(f*x+e)^2-1/4*(f*x+e)^3*cosh(f*x+e)^2-3/16*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3+15/32*(f*x+
e)^2*cosh(f*x+e)*sinh(f*x+e)+5/32*(f*x+e)^3+3/32*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2-3/128*cosh(f*x+e)^3*sinh(
f*x+e)+51/256*cosh(f*x+e)*sinh(f*x+e)+51/256*f*x+51/256*e-3/8*(f*x+e)*cosh(f*x+e)^2)+6/f^3*d^3*e*(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)*c
osh(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)-6/f^3*d^3*e*(1/4*(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)^3-1/8*(f*x+e)*sinh(f*x+e)^
2*cosh(f*x+e)^2+1/32*cosh(f*x+e)^3*sinh(f*x+e)-1/64*cosh(f*x+e)*sinh(f*x+e)-1/64*f*x-1/64*e)+6/f^3*d^3*e^2*(1/
4*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/16*(f*x+e)^2-1/16*sinh(f*x+e)^2*cosh
(f*x+e)^2)+6/f^2*c*d^2*(1/4*(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)^3-1/8*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2+1/32*cosh(f*x+e)^3*sinh(f*x+e)-1/64*cosh(f*x+e)*sinh(f*x+e)-1/6
4*f*x-1/64*e)+6*c^2/f*d*(1/4*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3-1/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-1/16*(f*x+e
)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e)^2)-6/f^3*d^3*e^2*(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)-2/f^3*d^3*e^3*(1/4*cosh(
f*x+e)^3*sinh(f*x+e)-1/8*cosh(f*x+e)*sinh(f*x+e)-1/8*f*x-1/8*e)+2/f^3*d^3*e^3*(1/4*sinh(f*x+e)^2*cosh(f*x+e)^2
-1/4*cosh(f*x+e)^2)-6/f^2*c*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)-6*c^2/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*cos
h(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)+2*c^3*(1/4*cosh(f*x+e)^3*sinh(f*x+e)-1/8*
cosh(f*x+e)*sinh(f*x+e)-1/8*f*x-1/8*e)-2*c^3*(1/4*sinh(f*x+e)^2*cosh(f*x+e)^2-1/4*cosh(f*x+e)^2)-c^3*(1/2*cosh
(f*x+e)*sinh(f*x+e)-1/2*f*x-1/2*e))

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Maxima [A]  time = 2.49278, size = 401, normalized size = 1.74 \begin{align*} \frac{1}{16} \, c^{3}{\left (\frac{4 \,{\left (f x + e\right )}}{a^{2} f} + \frac{4 \, e^{\left (-2 \, f x - 2 \, e\right )} - e^{\left (-4 \, f x - 4 \, e\right )}}{a^{2} f}\right )} + \frac{3 \,{\left (8 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 8 \,{\left (2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} -{\left (4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} c^{2} d e^{\left (-4 \, e\right )}}{64 \, a^{2} f^{2}} + \frac{{\left (32 \, f^{3} x^{3} e^{\left (4 \, e\right )} + 48 \,{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 3 \,{\left (8 \, f^{2} x^{2} + 4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} c d^{2} e^{\left (-4 \, e\right )}}{128 \, a^{2} f^{3}} + \frac{{\left (32 \, f^{4} x^{4} e^{\left (4 \, e\right )} + 32 \,{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} + 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} -{\left (32 \, f^{3} x^{3} + 24 \, f^{2} x^{2} + 12 \, f x + 3\right )} e^{\left (-4 \, f x\right )}\right )} d^{3} e^{\left (-4 \, e\right )}}{512 \, a^{2} f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*coth(f*x+e))^2,x, algorithm="maxima")

[Out]

1/16*c^3*(4*(f*x + e)/(a^2*f) + (4*e^(-2*f*x - 2*e) - e^(-4*f*x - 4*e))/(a^2*f)) + 3/64*(8*f^2*x^2*e^(4*e) + 8
*(2*f*x*e^(2*e) + e^(2*e))*e^(-2*f*x) - (4*f*x + 1)*e^(-4*f*x))*c^2*d*e^(-4*e)/(a^2*f^2) + 1/128*(32*f^3*x^3*e
^(4*e) + 48*(2*f^2*x^2*e^(2*e) + 2*f*x*e^(2*e) + e^(2*e))*e^(-2*f*x) - 3*(8*f^2*x^2 + 4*f*x + 1)*e^(-4*f*x))*c
*d^2*e^(-4*e)/(a^2*f^3) + 1/512*(32*f^4*x^4*e^(4*e) + 32*(4*f^3*x^3*e^(2*e) + 6*f^2*x^2*e^(2*e) + 6*f*x*e^(2*e
) + 3*e^(2*e))*e^(-2*f*x) - (32*f^3*x^3 + 24*f^2*x^2 + 12*f*x + 3)*e^(-4*f*x))*d^3*e^(-4*e)/(a^2*f^4)

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Fricas [B]  time = 2.17092, size = 1257, normalized size = 5.47 \begin{align*} \frac{128 \, d^{3} f^{3} x^{3} + 128 \, c^{3} f^{3} + 192 \, c^{2} d f^{2} + 192 \, c d^{2} f + 96 \, d^{3} + 192 \,{\left (2 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} +{\left (32 \, d^{3} f^{4} x^{4} - 32 \, c^{3} f^{3} - 24 \, c^{2} d f^{2} - 12 \, c d^{2} f + 32 \,{\left (4 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 24 \,{\left (8 \, c^{2} d f^{4} - 4 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 4 \,{\left (32 \, c^{3} f^{4} - 24 \, c^{2} d f^{3} - 12 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right )^{2} + 2 \,{\left (32 \, d^{3} f^{4} x^{4} + 32 \, c^{3} f^{3} + 24 \, c^{2} d f^{2} + 12 \, c d^{2} f + 32 \,{\left (4 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + 3 \, d^{3} + 24 \,{\left (8 \, c^{2} d f^{4} + 4 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 4 \,{\left (32 \, c^{3} f^{4} + 24 \, c^{2} d f^{3} + 12 \, c d^{2} f^{2} + 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) +{\left (32 \, d^{3} f^{4} x^{4} - 32 \, c^{3} f^{3} - 24 \, c^{2} d f^{2} - 12 \, c d^{2} f + 32 \,{\left (4 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 24 \,{\left (8 \, c^{2} d f^{4} - 4 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 4 \,{\left (32 \, c^{3} f^{4} - 24 \, c^{2} d f^{3} - 12 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \sinh \left (f x + e\right )^{2} + 192 \,{\left (2 \, c^{2} d f^{3} + 2 \, c d^{2} f^{2} + d^{3} f\right )} x}{512 \,{\left (a^{2} f^{4} \cosh \left (f x + e\right )^{2} + 2 \, a^{2} f^{4} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a^{2} f^{4} \sinh \left (f x + e\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*coth(f*x+e))^2,x, algorithm="fricas")

[Out]

1/512*(128*d^3*f^3*x^3 + 128*c^3*f^3 + 192*c^2*d*f^2 + 192*c*d^2*f + 96*d^3 + 192*(2*c*d^2*f^3 + d^3*f^2)*x^2
+ (32*d^3*f^4*x^4 - 32*c^3*f^3 - 24*c^2*d*f^2 - 12*c*d^2*f + 32*(4*c*d^2*f^4 - d^3*f^3)*x^3 - 3*d^3 + 24*(8*c^
2*d*f^4 - 4*c*d^2*f^3 - d^3*f^2)*x^2 + 4*(32*c^3*f^4 - 24*c^2*d*f^3 - 12*c*d^2*f^2 - 3*d^3*f)*x)*cosh(f*x + e)
^2 + 2*(32*d^3*f^4*x^4 + 32*c^3*f^3 + 24*c^2*d*f^2 + 12*c*d^2*f + 32*(4*c*d^2*f^4 + d^3*f^3)*x^3 + 3*d^3 + 24*
(8*c^2*d*f^4 + 4*c*d^2*f^3 + d^3*f^2)*x^2 + 4*(32*c^3*f^4 + 24*c^2*d*f^3 + 12*c*d^2*f^2 + 3*d^3*f)*x)*cosh(f*x
+ e)*sinh(f*x + e) + (32*d^3*f^4*x^4 - 32*c^3*f^3 - 24*c^2*d*f^2 - 12*c*d^2*f + 32*(4*c*d^2*f^4 - d^3*f^3)*x^
3 - 3*d^3 + 24*(8*c^2*d*f^4 - 4*c*d^2*f^3 - d^3*f^2)*x^2 + 4*(32*c^3*f^4 - 24*c^2*d*f^3 - 12*c*d^2*f^2 - 3*d^3
*f)*x)*sinh(f*x + e)^2 + 192*(2*c^2*d*f^3 + 2*c*d^2*f^2 + d^3*f)*x)/(a^2*f^4*cosh(f*x + e)^2 + 2*a^2*f^4*cosh(
f*x + e)*sinh(f*x + e) + a^2*f^4*sinh(f*x + e)^2)

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Sympy [A]  time = 10.8872, size = 2200, normalized size = 9.57 \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)**3/(a+a*coth(f*x+e))**2,x)

[Out]

Piecewise((64*c**3*f**4*x*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256
*a**2*f**4) + 128*c**3*f**4*x*tanh(e + f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 25
6*a**2*f**4) + 64*c**3*f**4*x/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) -
96*c**3*f**3*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4)
+ 32*c**3*f**3/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 96*c**2*d*f**4
*x**2*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 192*c*
*2*d*f**4*x**2*tanh(e + f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) +
96*c**2*d*f**4*x**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 240*c**2*
d*f**3*x*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 96*
c**2*d*f**3*x*tanh(e + f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 1
44*c**2*d*f**3*x/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 120*c**2*d*f
**2*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 72*c**2*
d*f**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 64*c*d**2*f**4*x**3*ta
nh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 128*c*d**2*f**
4*x**3*tanh(e + f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 64*c*d**
2*f**4*x**3/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 240*c*d**2*f**3*x
**2*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 96*c*d**
2*f**3*x**2*tanh(e + f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 144
*c*d**2*f**3*x**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 216*c*d**2*
f**2*x*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 48*c*
d**2*f**2*x*tanh(e + f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 168
*c*d**2*f**2*x/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 108*c*d**2*f*t
anh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 84*c*d**2*f/(
256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 16*d**3*f**4*x**4*tanh(e + f*x
)**2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 32*d**3*f**4*x**4*tanh(e
+ f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 16*d**3*f**4*x**4/(25
6*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 80*d**3*f**3*x**3*tanh(e + f*x)*
*2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 32*d**3*f**3*x**3*tanh(e +
f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 48*d**3*f**3*x**3/(256*
a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 108*d**3*f**2*x**2*tanh(e + f*x)**
2/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 24*d**3*f**2*x**2*tanh(e +
f*x)/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 84*d**3*f**2*x**2/(256*a
**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 102*d**3*f*x*tanh(e + f*x)**2/(256*
a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 12*d**3*f*x*tanh(e + f*x)/(256*a**
2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 90*d**3*f*x/(256*a**2*f**4*tanh(e + f
*x)**2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) - 51*d**3*tanh(e + f*x)**2/(256*a**2*f**4*tanh(e + f*x)*
*2 + 512*a**2*f**4*tanh(e + f*x) + 256*a**2*f**4) + 45*d**3/(256*a**2*f**4*tanh(e + f*x)**2 + 512*a**2*f**4*ta
nh(e + f*x) + 256*a**2*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)**2, True))

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Giac [A]  time = 1.16722, size = 517, normalized size = 2.25 \begin{align*} \frac{{\left (32 \, d^{3} f^{4} x^{4} e^{\left (4 \, f x + 4 \, e\right )} + 128 \, c d^{2} f^{4} x^{3} e^{\left (4 \, f x + 4 \, e\right )} + 192 \, c^{2} d f^{4} x^{2} e^{\left (4 \, f x + 4 \, e\right )} + 128 \, d^{3} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} - 32 \, d^{3} f^{3} x^{3} + 128 \, c^{3} f^{4} x e^{\left (4 \, f x + 4 \, e\right )} + 384 \, c d^{2} f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c d^{2} f^{3} x^{2} + 384 \, c^{2} d f^{3} x e^{\left (2 \, f x + 2 \, e\right )} + 192 \, d^{3} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c^{2} d f^{3} x - 24 \, d^{3} f^{2} x^{2} + 128 \, c^{3} f^{3} e^{\left (2 \, f x + 2 \, e\right )} + 384 \, c d^{2} f^{2} x e^{\left (2 \, f x + 2 \, e\right )} - 32 \, c^{3} f^{3} - 48 \, c d^{2} f^{2} x + 192 \, c^{2} d f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 192 \, d^{3} f x e^{\left (2 \, f x + 2 \, e\right )} - 24 \, c^{2} d f^{2} - 12 \, d^{3} f x + 192 \, c d^{2} f e^{\left (2 \, f x + 2 \, e\right )} - 12 \, c d^{2} f + 96 \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, d^{3}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{512 \, a^{2} f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+a*coth(f*x+e))^2,x, algorithm="giac")

[Out]

1/512*(32*d^3*f^4*x^4*e^(4*f*x + 4*e) + 128*c*d^2*f^4*x^3*e^(4*f*x + 4*e) + 192*c^2*d*f^4*x^2*e^(4*f*x + 4*e)
+ 128*d^3*f^3*x^3*e^(2*f*x + 2*e) - 32*d^3*f^3*x^3 + 128*c^3*f^4*x*e^(4*f*x + 4*e) + 384*c*d^2*f^3*x^2*e^(2*f*
x + 2*e) - 96*c*d^2*f^3*x^2 + 384*c^2*d*f^3*x*e^(2*f*x + 2*e) + 192*d^3*f^2*x^2*e^(2*f*x + 2*e) - 96*c^2*d*f^3
*x - 24*d^3*f^2*x^2 + 128*c^3*f^3*e^(2*f*x + 2*e) + 384*c*d^2*f^2*x*e^(2*f*x + 2*e) - 32*c^3*f^3 - 48*c*d^2*f^
2*x + 192*c^2*d*f^2*e^(2*f*x + 2*e) + 192*d^3*f*x*e^(2*f*x + 2*e) - 24*c^2*d*f^2 - 12*d^3*f*x + 192*c*d^2*f*e^
(2*f*x + 2*e) - 12*c*d^2*f + 96*d^3*e^(2*f*x + 2*e) - 3*d^3)*e^(-4*f*x - 4*e)/(a^2*f^4)