3.191 \(\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx\)

Optimal. Leaf size=17 \[ \frac {c^2 (a x+1)^3}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 32} \[ \frac {c^2 (a x+1)^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])*(c - a*c*x)^2,x]

[Out]

(c^2*(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 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])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.24 \[ c^2 \left (\frac {a^2 x^3}{3}+a x^2+x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])*(c - a*c*x)^2,x]

[Out]

c^2*(x + a*x^2 + (a^2*x^3)/3)

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fricas [A]  time = 0.42, size = 25, normalized size = 1.47 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^2,x, algorithm="fricas")

[Out]

1/3*a^2*c^2*x^3 + a*c^2*x^2 + c^2*x

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giac [A]  time = 0.16, size = 25, normalized size = 1.47 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^2,x, algorithm="giac")

[Out]

1/3*a^2*c^2*x^3 + a*c^2*x^2 + c^2*x

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maple [A]  time = 0.02, size = 16, normalized size = 0.94 \[ \frac {c^{2} \left (a x +1\right )^{3}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^2,x)

[Out]

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

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maxima [A]  time = 0.32, size = 25, normalized size = 1.47 \[ \frac {1}{3} \, a^{2} c^{2} x^{3} + a c^{2} x^{2} + c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a*c*x+c)^2,x, algorithm="maxima")

[Out]

1/3*a^2*c^2*x^3 + a*c^2*x^2 + c^2*x

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mupad [B]  time = 0.04, size = 19, normalized size = 1.12 \[ \frac {c^2\,x\,\left (a^2\,x^2+3\,a\,x+3\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^2*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)

[Out]

(c^2*x*(3*a*x + a^2*x^2 + 3))/3

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sympy [A]  time = 0.08, size = 24, normalized size = 1.41 \[ \frac {a^{2} c^{2} x^{3}}{3} + a c^{2} x^{2} + c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**2,x)

[Out]

a**2*c**2*x**3/3 + a*c**2*x**2 + c**2*x

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