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

Optimal. Leaf size=167 \[ \frac{a^3 2^{n/2} \left (n^2+2\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{a-\frac{1}{x}}{2 a}\right )}{3 (2-n)}+\frac{1}{6} a^3 n \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}+\frac{a^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}}{3 x} \]

[Out]

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

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Rubi [A]  time = 0.110067, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6171, 90, 80, 69} \[ \frac{a^3 2^{n/2} \left (n^2+2\right ) \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right )}{3 (2-n)}+\frac{1}{6} a^3 n \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}+\frac{a^2 \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{1-\frac{n}{2}}}{3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6171

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

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

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

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

Mathematica [A]  time = 0.589756, size = 132, normalized size = 0.79 \[ -\frac{a^3 e^{n \coth ^{-1}(a x)} \left ((n+2) \left (\left (n^2+2\right ) \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,-e^{2 \coth ^{-1}(a x)}\right )-\left (1-\frac{1}{a^2 x^2}\right ) \left (\frac{2}{a x}+n\right )+\frac{n^2+2}{a x}\right )-n \left (n^2+2\right ) e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \coth ^{-1}(a x)}\right )\right )}{6 (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(n*arccoth(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*arccoth(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*arccoth(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{acoth}{\left (a x \right )}}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(n*acoth(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*arccoth(a*x))/x^4,x, algorithm="giac")

[Out]

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