3.195 \(\int \frac {e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\)

Optimal. Leaf size=52 \[ \frac {1}{2 a c^3 (1-a x)^2}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{a c^3 (1-a x)^4} \]

[Out]

1/a/c^3/(-a*x+1)^4-4/3/a/c^3/(-a*x+1)^3+1/2/a/c^3/(-a*x+1)^2

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Rubi [A]  time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac {1}{2 a c^3 (1-a x)^2}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{a c^3 (1-a x)^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])/(c - a*c*x)^3,x]

[Out]

1/(a*c^3*(1 - a*x)^4) - 4/(3*a*c^3*(1 - a*x)^3) + 1/(2*a*c^3*(1 - a*x)^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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])

Rubi steps

\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx &=\frac {\int \frac {(1+a x)^2}{(1-a x)^5} \, dx}{c^3}\\ &=\frac {\int \left (-\frac {4}{(-1+a x)^5}-\frac {4}{(-1+a x)^4}-\frac {1}{(-1+a x)^3}\right ) \, dx}{c^3}\\ &=\frac {1}{a c^3 (1-a x)^4}-\frac {4}{3 a c^3 (1-a x)^3}+\frac {1}{2 a c^3 (1-a x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 0.60 \[ \frac {3 a^2 x^2+2 a x+1}{6 a c^3 (a x-1)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])/(c - a*c*x)^3,x]

[Out]

(1 + 2*a*x + 3*a^2*x^2)/(6*a*c^3*(-1 + a*x)^4)

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fricas [A]  time = 0.70, size = 65, normalized size = 1.25 \[ \frac {3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/6*(3*a^2*x^2 + 2*a*x + 1)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)

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giac [A]  time = 0.16, size = 29, normalized size = 0.56 \[ \frac {3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left (a x - 1\right )}^{4} a c^{3}} \]

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)^3,x, algorithm="giac")

[Out]

1/6*(3*a^2*x^2 + 2*a*x + 1)/((a*x - 1)^4*a*c^3)

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maple [A]  time = 0.03, size = 41, normalized size = 0.79 \[ \frac {\frac {1}{2 a \left (a x -1\right )^{2}}+\frac {1}{a \left (a x -1\right )^{4}}+\frac {4}{3 a \left (a x -1\right )^{3}}}{c^{3}} \]

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)^3,x)

[Out]

1/c^3*(1/2/a/(a*x-1)^2+1/a/(a*x-1)^4+4/3/a/(a*x-1)^3)

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maxima [A]  time = 0.33, size = 65, normalized size = 1.25 \[ \frac {3 \, a^{2} x^{2} + 2 \, a x + 1}{6 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]

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)^3,x, algorithm="maxima")

[Out]

1/6*(3*a^2*x^2 + 2*a*x + 1)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)

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mupad [B]  time = 0.85, size = 29, normalized size = 0.56 \[ \frac {3\,a^2\,x^2+2\,a\,x+1}{6\,a\,c^3\,{\left (a\,x-1\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)^4/((a^2*x^2 - 1)^2*(c - a*c*x)^3),x)

[Out]

(2*a*x + 3*a^2*x^2 + 1)/(6*a*c^3*(a*x - 1)^4)

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sympy [A]  time = 0.33, size = 70, normalized size = 1.35 \[ - \frac {- 3 a^{2} x^{2} - 2 a x - 1}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}} \]

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)**3,x)

[Out]

-(-3*a**2*x**2 - 2*a*x - 1)/(6*a**5*c**3*x**4 - 24*a**4*c**3*x**3 + 36*a**3*c**3*x**2 - 24*a**2*c**3*x + 6*a*c
**3)

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