Integrand size = 12, antiderivative size = 167 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx=\frac {1}{6} a^3 n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}+\frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{3 x}+\frac {2^{n/2} a^3 \left (2+n^2\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{3 (2-n)} \]
1/6*a^3*n*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(1+1/2*n)+1/3*a^2*(1-1/a/x)^(1-1/2 *n)*(1+1/a/x)^(1+1/2*n)/x+1/3*2^(1/2*n)*a^3*(n^2+2)*(1-1/a/x)^(1-1/2*n)*hy pergeom([-1/2*n, 1-1/2*n],[2-1/2*n],1/2*(a-1/x)/a)/(2-n)
Time = 0.64 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.79 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx=-\frac {a^3 e^{n \coth ^{-1}(a x)} \left (-e^{2 \coth ^{-1}(a x)} n \left (2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-\left (\left (1-\frac {1}{a^2 x^2}\right ) \left (n+\frac {2}{a x}\right )\right )+\frac {2+n^2}{a x}+\left (2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},-e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{6 (2+n)} \]
-1/6*(a^3*E^(n*ArcCoth[a*x])*(-(E^(2*ArcCoth[a*x])*n*(2 + n^2)*Hypergeomet ric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[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])])))/(2 + n)
Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6721, 101, 25, 27, 90, 79}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 6721 |
\(\displaystyle -\int \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2}}{x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {1}{3} a^2 \int -\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (a+\frac {n}{x}\right )}{a}d\frac {1}{x}+\frac {a^2 \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}}{3 x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{3 x}-\frac {1}{3} a^2 \int \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (a+\frac {n}{x}\right )}{a}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{3 x}-\frac {1}{3} a \int \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (a+\frac {n}{x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{3 x}-\frac {1}{3} a \left (\frac {1}{2} a \left (n^2+2\right ) \int \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2}d\frac {1}{x}-\frac {1}{2} a^2 n \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}\right )\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {a^2 \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{3 x}-\frac {1}{3} a \left (-\frac {a^2 2^{n/2} \left (n^2+2\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {a-\frac {1}{x}}{2 a}\right )}{2-n}-\frac {1}{2} a^2 n \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}\right )\) |
(a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(3*x) - (a*(-1/2*( a^2*n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)) - (2^(n/2)*a^2*(2 + n^2)*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2 , (a - x^(-1))/(2*a)])/(2 - n)))/3
3.2.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(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]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[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]
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]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x^{4}}d x\]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{4}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x^{4}}\, dx \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{4}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x^4} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x^4} \,d x \]