### 3.568 $$\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2) \, dx$$

Optimal. Leaf size=15 $-\frac{c (a x+1)^3}{3 a}$

[Out]

-(c*(1 + a*x)^3)/(3*a)

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Rubi [A]  time = 0.033691, antiderivative size = 15, 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} $-\frac{c (a x+1)^3}{3 a}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2),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 = 0.0121771, size = 20, normalized size = 1.33 $-c \left (\frac{a^2 x^3}{3}+a x^2+x\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(c*(x + a*x^2 + (a^2*x^3)/3))

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Maple [A]  time = 0.037, 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*x+1)/(a*x-1)*(-a^2*c*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.05957, size = 28, normalized size = 1.87 \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(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

-1/3*a^2*c*x^3 - a*c*x^2 - c*x

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Fricas [A]  time = 1.47445, size = 43, normalized size = 2.87 \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(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

-1/3*a^2*c*x^3 - a*c*x^2 - c*x

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Sympy [A]  time = 0.06979, size = 20, normalized size = 1.33 \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(1/(a*x-1)*(a*x+1)*(-a**2*c*x**2+c),x)

[Out]

-a**2*c*x**3/3 - a*c*x**2 - c*x

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Giac [A]  time = 1.09652, size = 28, normalized size = 1.87 \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(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

-1/3*a^2*c*x^3 - a*c*x^2 - c*x