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]

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

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.52, size = 21, normalized size = 1.40 \[ -\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]

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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giac [A]  time = 0.12, size = 21, normalized size = 1.40 \[ -\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]

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

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maple [A]  time = 0.03, size = 14, normalized size = 0.93 \[ -\frac {c \left (a x +1\right )^{3}}{3 a} \]

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 = 0.31, size = 21, normalized size = 1.40 \[ -\frac {1}{3} \, a^{2} c x^{3} - a c x^{2} - c x \]

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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mupad [B]  time = 0.03, size = 17, normalized size = 1.13 \[ -\frac {c\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*x*(3*a*x + a^2*x^2 + 3))/3

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sympy [A]  time = 0.06, size = 20, normalized size = 1.33 \[ - \frac {a^{2} c x^{3}}{3} - a c x^{2} - c x \]

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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