3.600 \(\int e^{-2 \coth ^{-1}(a x)} (c-a^2 c x^2)^2 \, dx\)

Optimal. Leaf size=37 \[ \frac{c^2 (1-a x)^4}{2 a}-\frac{c^2 (1-a x)^5}{5 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.063194, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 43} \[ \frac{c^2 (1-a x)^4}{2 a}-\frac{c^2 (1-a x)^5}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^2/E^(2*ArcCoth[a*x]),x]

[Out]

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

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 6140

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0165207, size = 30, normalized size = 0.81 \[ \frac{1}{10} c^2 x \left (2 a^4 x^4-5 a^3 x^3+10 a x-10\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)^2/E^(2*ArcCoth[a*x]),x]

[Out]

(c^2*x*(-10 + 10*a*x - 5*a^3*x^3 + 2*a^4*x^4))/10

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 30, normalized size = 0.8 \begin{align*}{c}^{2} \left ({\frac{{x}^{5}{a}^{4}}{5}}-{\frac{{x}^{4}{a}^{3}}{2}}+a{x}^{2}-x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.13044, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} - \frac{1}{2} \, a^{3} c^{2} x^{4} + a c^{2} x^{2} - c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.5016, size = 74, normalized size = 2. \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} - \frac{1}{2} \, a^{3} c^{2} x^{4} + a c^{2} x^{2} - c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.083979, size = 36, normalized size = 0.97 \begin{align*} \frac{a^{4} c^{2} x^{5}}{5} - \frac{a^{3} c^{2} x^{4}}{2} + a c^{2} x^{2} - c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13428, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} - \frac{1}{2} \, a^{3} c^{2} x^{4} + a c^{2} x^{2} - c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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