3.2.75 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\) [175]

Optimal. Leaf size=37 \[ \frac {2}{3 a c^3 (1-a x)^3}-\frac {1}{2 a c^3 (1-a x)^2} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 37, 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 = {6264, 45} \begin {gather*} \frac {2}{3 a c^3 (1-a x)^3}-\frac {1}{2 a c^3 (1-a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - a*c*x)^3,x]

[Out]

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

Rule 45

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 6264

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^{2 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx &=\frac {\int \frac {1+a x}{(1-a x)^4} \, dx}{c^3}\\ &=\frac {\int \left (\frac {2}{(-1+a x)^4}+\frac {1}{(-1+a x)^3}\right ) \, dx}{c^3}\\ &=\frac {2}{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 = 0.01, size = 23, normalized size = 0.62 \begin {gather*} -\frac {1+3 a x}{6 a c^3 (-1+a x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - a*c*x)^3,x]

[Out]

-1/6*(1 + 3*a*x)/(a*c^3*(-1 + a*x)^3)

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Maple [A]
time = 1.10, size = 30, normalized size = 0.81

method	result	size
risch	\(\frac {-\frac {x}{2}-\frac {1}{6 a}}{c^{3} \left (a x -1\right )^{3}}\)	\(21\)
gosper	\(-\frac {3 a x +1}{6 a \,c^{3} \left (a x -1\right )^{3}}\)	\(22\)
default	\(\frac {-\frac {1}{2 \left (a x -1\right )^{2} a}-\frac {2}{3 a \left (a x -1\right )^{3}}}{c^{3}}\)	\(30\)
norman	\(\frac {-\frac {x}{c}+\frac {a \,x^{2}}{2 c}-\frac {a^{2} x^{3}}{6 c}}{\left (a x -1\right )^{3} c^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 47, normalized size = 1.27 \begin {gather*} -\frac {3 \, a x + 1}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 47, normalized size = 1.27 \begin {gather*} -\frac {3 \, a x + 1}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.14, size = 48, normalized size = 1.30 \begin {gather*} \frac {- 3 a x - 1}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**3,x)

[Out]

(-3*a*x - 1)/(6*a**4*c**3*x**3 - 18*a**3*c**3*x**2 + 18*a**2*c**3*x - 6*a*c**3)

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Giac [A]
time = 0.43, size = 21, normalized size = 0.57 \begin {gather*} -\frac {3 \, a x + 1}{6 \, {\left (a x - 1\right )}^{3} a c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a*c*x+c)^3,x, algorithm="giac")

[Out]

-1/6*(3*a*x + 1)/((a*x - 1)^3*a*c^3)

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Mupad [B]
time = 0.08, size = 21, normalized size = 0.57 \begin {gather*} -\frac {3\,a\,x+1}{6\,a\,c^3\,{\left (a\,x-1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a*x + 1)/(6*a*c^3*(a*x - 1)^3)

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