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

Optimal. Leaf size=16 \[ \frac{c (1-a x)^3}{3 a} \]

[Out]

(c*(1 - a*x)^3)/(3*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0340112, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6167, 6140, 32} \[ \frac{c (1-a x)^3}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(1 - a*x)^3)/(3*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.013573, size = 21, normalized size = 1.31 \[ -c \left (\frac{a^2 x^3}{3}-a x^2+x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*(x - a*x^2 + (a^2*x^3)/3))

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 14, normalized size = 0.9 \begin{align*} -{\frac{c \left ( ax-1 \right ) ^{3}}{3\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c*(a*x-1)^3/a

________________________________________________________________________________________

Maxima [A]  time = 1.01997, size = 27, normalized size = 1.69 \begin{align*} -\frac{1}{3} \, a^{2} c x^{3} + a c x^{2} - c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*c*x^3 + a*c*x^2 - c*x

________________________________________________________________________________________

Fricas [A]  time = 1.52522, size = 43, normalized size = 2.69 \begin{align*} -\frac{1}{3} \, a^{2} c x^{3} + a c x^{2} - c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*c*x^3 + a*c*x^2 - c*x

________________________________________________________________________________________

Sympy [A]  time = 0.071557, size = 19, normalized size = 1.19 \begin{align*} - \frac{a^{2} c x^{3}}{3} + a c x^{2} - c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2*c*x**3/3 + a*c*x**2 - c*x

________________________________________________________________________________________

Giac [A]  time = 1.14692, size = 27, normalized size = 1.69 \begin{align*} -\frac{1}{3} \, a^{2} c x^{3} + a c x^{2} - c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*c*x^3 + a*c*x^2 - c*x