Integrand size = 22, antiderivative size = 289 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2+n)}+\frac {\left (6+4 n-n^2-n^3\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 (2-n) n (2+n)}-\frac {\left (6+4 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)}}{a c^2 n (2+n)}+\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x}{c^2}+\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a c^2} \]
-(3+n)*(1-1/a/x)^(-1-1/2*n)*(1+1/a/x)^(-1+1/2*n)/a/c^2/(2+n)+(-n^3-n^2+4*n +6)*(1-1/a/x)^(1-1/2*n)*(1+1/a/x)^(-1+1/2*n)/a/c^2/n/(-n^2+4)-(n^2+4*n+6)* (1+1/a/x)^(-1+1/2*n)/a/c^2/n/(2+n)/((1-1/a/x)^(1/2*n))+(1-1/a/x)^(-1-1/2*n )*(1+1/a/x)^(-1+1/2*n)*x/c^2+2*(1+1/a/x)^(1/2*n)*hypergeom([1, 1/2*n],[1+1 /2*n],(a+1/x)/(a-1/x))/a/c^2/((1-1/a/x)^(1/2*n))
Time = 1.36 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.62 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (-6+n^2+6 a n x-a n^3 x+6 a^2 x^2-2 a^2 n^2 x^2-4 a^3 n x^3+a^3 n^3 x^3+e^{2 \coth ^{-1}(a x)} (-2+n) n^2 \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+n \left (-4+n^2\right ) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{a c^2 (-2+n) n (2+n) \left (-1+a^2 x^2\right )} \]
(E^(n*ArcCoth[a*x])*(-6 + n^2 + 6*a*n*x - a*n^3*x + 6*a^2*x^2 - 2*a^2*n^2* x^2 - 4*a^3*n*x^3 + a^3*n^3*x^3 + E^(2*ArcCoth[a*x])*(-2 + n)*n^2*(-1 + a^ 2*x^2)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + n*(-4 + n^2)*(-1 + a^2*x^2)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x] )]))/(a*c^2*(-2 + n)*n*(2 + n)*(-1 + a^2*x^2))
Time = 0.48 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6748, 144, 25, 27, 172, 25, 27, 172, 25, 27, 172, 27, 141}
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.
| \(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} x^2d\frac {1}{x}}{c^2}\) |
| \(\Big \downarrow \) 144 |
| \(\displaystyle -\frac {x \left (-\left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}\right ) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}-\int -\frac {\left (a n+\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} x}{a^2}d\frac {1}{x}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\left (a n+\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} x}{a^2}d\frac {1}{x}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \left (a n+\frac {3}{x}\right ) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} xd\frac {1}{x}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {\frac {\frac {a (n+3) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{n+2}-\frac {a \int -\frac {\left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} \left (a n (n+2)+\frac {2 (n+3)}{x}\right ) x}{a}d\frac {1}{x}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {a \int \frac {\left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} \left (a n (n+2)+\frac {2 (n+3)}{x}\right ) x}{a}d\frac {1}{x}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} \left (a n (n+2)+\frac {2 (n+3)}{x}\right ) xd\frac {1}{x}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {\frac {\frac {\frac {a \left (n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{n}-\frac {a \int -\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} \left (a (n+2) n^2+\frac {n^2+4 n+6}{x}\right ) x}{a}d\frac {1}{x}}{n}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {\frac {a \int \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} \left (a (n+2) n^2+\frac {n^2+4 n+6}{x}\right ) x}{a}d\frac {1}{x}}{n}+\frac {a \left (n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{n}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {n-4}{2}} \left (a (n+2) n^2+\frac {n^2+4 n+6}{x}\right ) xd\frac {1}{x}}{n}+\frac {a \left (n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{n}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 172 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {a \int n^2 \left (4-n^2\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {n-2}{2}} xd\frac {1}{x}}{2-n}-\frac {a \left (-n^3-n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{2-n}}{n}+\frac {a \left (n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{n}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\frac {a n^2 \left (4-n^2\right ) \int \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {n-2}{2}} xd\frac {1}{x}}{2-n}-\frac {a \left (-n^3-n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{2-n}}{n}+\frac {a \left (n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{n}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
| \(\Big \downarrow \) 141 |
| \(\displaystyle -\frac {\frac {\frac {\frac {-\frac {2 a n \left (4-n^2\right ) \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {n+2}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{2-n}-\frac {a \left (-n^3-n^2+4 n+6\right ) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{1-\frac {n}{2}}}{2-n}}{n}+\frac {a \left (n^2+4 n+6\right ) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{n}}{n+2}+\frac {a (n+3) \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}}{n+2}}{a^2}-x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}}}{c^2}\) |
-((-((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*x) + ((a*(3 + n)* (1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2))/(2 + n) + ((a*(6 + 4* n + n^2)*(1 + 1/(a*x))^((-2 + n)/2))/(n*(1 - 1/(a*x))^(n/2)) + (-((a*(6 + 4*n - n^2 - n^3)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((-2 + n)/2))/(2 - n)) - (2*a*n*(4 - n^2)*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (a + x^(-1))/(a - x^(-1))])/((2 - n)*(1 - 1/(a*x))^(n/2)))/n)/(2 + n ))/a^2)/c^2)
3.10.30.3.1 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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f ))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + Simp[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*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))] /; FreeQ[{a, b, c, d, e, f , m, n, p}, x] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, 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] + Simp[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)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | | ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1 ])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^2), x], x , 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[ n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (c -\frac {c}{a^{2} x^{2}}\right )^{2}}d x\]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\frac {a^{4} \int \frac {x^{4} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^2} \,d x \]