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

Optimal. Leaf size=84 \[ \frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^3 \left (n^2+6 n+8\right )}+\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-2}}{a c^3 (n+4)} \]

[Out]

((1 - a*x)^(-2 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^3*(4 + n)) + ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a
*c^3*(8 + 6*n + n^2))

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Rubi [A]  time = 0.0558024, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6129, 45, 37} \[ \frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^3 \left (n^2+6 n+8\right )}+\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-2}}{a c^3 (n+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - a*x)^(-2 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^3*(4 + n)) + ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a
*c^3*(8 + 6*n + n^2))

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0311324, size = 51, normalized size = 0.61 \[ \frac{(1-a x)^{-\frac{n}{2}-2} (-a x+n+3) (a x+1)^{\frac{n}{2}+1}}{a c^3 (n+2) (n+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.032, size = 46, normalized size = 0.6 \begin{align*} -{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( ax-n-3 \right ) \left ( ax+1 \right ) }{ \left ( ax-1 \right ) ^{2}{c}^{3} \left ({n}^{2}+6\,n+8 \right ) a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x)

[Out]

-exp(n*arctanh(a*x))*(a*x-n-3)*(a*x+1)/(a*x-1)^2/c^3/(n^2+6*n+8)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a c x - c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.25484, size = 259, normalized size = 3.08 \begin{align*} -\frac{{\left (a^{2} x^{2} -{\left (a n + 2 \, a\right )} x - n - 3\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a c^{3} n^{2} + 6 \, a c^{3} n + 8 \, a c^{3} +{\left (a^{3} c^{3} n^{2} + 6 \, a^{3} c^{3} n + 8 \, a^{3} c^{3}\right )} x^{2} - 2 \,{\left (a^{2} c^{3} n^{2} + 6 \, a^{2} c^{3} n + 8 \, a^{2} c^{3}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x, algorithm="fricas")

[Out]

-(a^2*x^2 - (a*n + 2*a)*x - n - 3)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^3*n^2 + 6*a*c^3*n + 8*a*c^3 + (a^3*c^3*n
^2 + 6*a^3*c^3*n + 8*a^3*c^3)*x^2 - 2*(a^2*c^3*n^2 + 6*a^2*c^3*n + 8*a^2*c^3)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/(-a*c*x+c)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a c x - c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x, algorithm="giac")

[Out]

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