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

Optimal. Leaf size=20 \[ \frac {1}{3} a^2 c^2 x^3-c^2 x \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 41} \[ \frac {1}{3} a^2 c^2 x^3-c^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*x) + (a^2*c^2*x^3)/3

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.85 \[ -c^2 \left (x-\frac {a^2 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*(x - (a^2*x^3)/3))

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fricas [A]  time = 0.54, size = 18, normalized size = 0.90 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

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giac [A]  time = 0.13, size = 18, normalized size = 0.90 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

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maple [A]  time = 0.03, size = 17, normalized size = 0.85 \[ c^{2} \left (\frac {1}{3} x^{3} a^{2}-x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 18, normalized size = 0.90 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} - c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*c^2*x^3 - c^2*x

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mupad [B]  time = 0.03, size = 15, normalized size = 0.75 \[ \frac {c^2\,x\,\left (a^2\,x^2-3\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*x*(a^2*x^2 - 3))/3

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sympy [A]  time = 0.06, size = 15, normalized size = 0.75 \[ \frac {a^{2} c^{2} x^{3}}{3} - c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**2*x**3/3 - c**2*x

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