3.156 \(\int \frac{e^{n \tanh ^{-1}(a x)}}{x^4} \, dx\)

Optimal. Leaf size=147 \[ -\frac{2 a^3 \left (n^2+2\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )}{3 (2-n)}-\frac{a n (a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{6 x^2}-\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{3 x^3} \]

[Out]

-((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(3*x^3) - (a*n*(1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(6*x^2)
- (2*a^3*(2 + n^2)*(1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x
)/(1 + a*x)])/(3*(2 - n))

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Rubi [A]  time = 0.0676119, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6126, 129, 151, 12, 131} \[ -\frac{2 a^3 \left (n^2+2\right ) (a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{a x+1}\right )}{3 (2-n)}-\frac{a n (a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{6 x^2}-\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{1-\frac{n}{2}}}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTanh[a*x])/x^4,x]

[Out]

-((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(3*x^3) - (a*n*(1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(6*x^2)
- (2*a^3*(2 + n^2)*(1 - a*x)^(1 - n/2)*(1 + a*x)^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x
)/(1 + a*x)])/(3*(2 - n))

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rule 129

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{x^4} \, dx\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{1}{3} \int \frac{(1-a x)^{-n/2} (1+a x)^{n/2} \left (-a n-a^2 x\right )}{x^3} \, dx\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{a n (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{6 x^2}+\frac{1}{6} \int \frac{a^2 \left (2+n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{x^2} \, dx\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{a n (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{6 x^2}+\frac{1}{6} \left (a^2 \left (2+n^2\right )\right ) \int \frac{(1-a x)^{-n/2} (1+a x)^{n/2}}{x^2} \, dx\\ &=-\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{a n (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{6 x^2}-\frac{2 a^3 \left (2+n^2\right ) (1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-a x}{1+a x}\right )}{3 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.0437709, size = 101, normalized size = 0.69 \[ \frac{(1-a x)^{1-\frac{n}{2}} (a x+1)^{\frac{n}{2}-1} \left (4 a^3 \left (n^2+2\right ) x^3 \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )-(n-2) (a x+1)^2 (a n x+2)\right )}{6 (n-2) x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTanh[a*x])/x^4,x]

[Out]

((1 - a*x)^(1 - n/2)*(1 + a*x)^(-1 + n/2)*(-((-2 + n)*(1 + a*x)^2*(2 + a*n*x)) + 4*a^3*(2 + n^2)*x^3*Hypergeom
etric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)]))/(6*(-2 + n)*x^3)

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctanh(a*x))/x^4,x)

[Out]

int(exp(n*arctanh(a*x))/x^4,x)

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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}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x^4,x, algorithm="fricas")

[Out]

integral(((a*x + 1)/(a*x - 1))^(1/2*n)/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{atanh}{\left (a x \right )}}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atanh(a*x))/x**4,x)

[Out]

Integral(exp(n*atanh(a*x))/x**4, x)

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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}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))/x^4,x, algorithm="giac")

[Out]

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