3.189 \(\int e^{4 \coth ^{-1}(a x)} (c-a c x)^4 \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{5} a^4 c^4 x^5-\frac{2}{3} a^2 c^4 x^3+c^4 x \]

[Out]

c^4*x - (2*a^2*c^4*x^3)/3 + (a^4*c^4*x^5)/5

________________________________________________________________________________________

Rubi [A]  time = 0.0550847, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6129, 41, 194} \[ \frac{1}{5} a^4 c^4 x^5-\frac{2}{3} a^2 c^4 x^3+c^4 x \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcCoth[a*x])*(c - a*c*x)^4,x]

[Out]

c^4*x - (2*a^2*c^4*x^3)/3 + (a^4*c^4*x^5)/5

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\\ &=c^4 \int (1-a x)^2 (1+a x)^2 \, dx\\ &=c^4 \int \left (1-a^2 x^2\right )^2 \, dx\\ &=c^4 \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx\\ &=c^4 x-\frac{2}{3} a^2 c^4 x^3+\frac{1}{5} a^4 c^4 x^5\\ \end{align*}

Mathematica [A]  time = 0.0161992, size = 26, normalized size = 0.81 \[ c^4 \left (\frac{a^4 x^5}{5}-\frac{2 a^2 x^3}{3}+x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x)^4,x]

[Out]

c^4*(x - (2*a^2*x^3)/3 + (a^4*x^5)/5)

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 23, normalized size = 0.7 \begin{align*}{c}^{4} \left ({\frac{{x}^{5}{a}^{4}}{5}}-{\frac{2\,{x}^{3}{a}^{2}}{3}}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^4,x)

[Out]

c^4*(1/5*x^5*a^4-2/3*x^3*a^2+x)

________________________________________________________________________________________

Maxima [A]  time = 1.03821, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{2}{3} \, a^{2} c^{4} x^{3} + c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

1/5*a^4*c^4*x^5 - 2/3*a^2*c^4*x^3 + c^4*x

________________________________________________________________________________________

Fricas [A]  time = 1.35828, size = 58, normalized size = 1.81 \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{2}{3} \, a^{2} c^{4} x^{3} + c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/5*a^4*c^4*x^5 - 2/3*a^2*c^4*x^3 + c^4*x

________________________________________________________________________________________

Sympy [A]  time = 0.172421, size = 29, normalized size = 0.91 \begin{align*} \frac{a^{4} c^{4} x^{5}}{5} - \frac{2 a^{2} c^{4} x^{3}}{3} + c^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)**2*(a*x+1)**2*(-a*c*x+c)**4,x)

[Out]

a**4*c**4*x**5/5 - 2*a**2*c**4*x**3/3 + c**4*x

________________________________________________________________________________________

Giac [A]  time = 1.16814, size = 57, normalized size = 1.78 \begin{align*} \frac{{\left (3 \, c^{4} + \frac{15 \, c^{4}}{a x - 1} + \frac{20 \, c^{4}}{{\left (a x - 1\right )}^{2}}\right )}{\left (a x - 1\right )}^{5}}{15 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a*c*x+c)^4,x, algorithm="giac")

[Out]

1/15*(3*c^4 + 15*c^4/(a*x - 1) + 20*c^4/(a*x - 1)^2)*(a*x - 1)^5/a