Integrand size = 27, antiderivative size = 944 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{(3+n) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^5}{\left (9-10 n^2+n^4\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}+\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^5}{(3+n) \left (1-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {6 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1+n}{2}} x^5}{\left (3+4 n+n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} \left (1+\frac {1}{a x}\right )^{\frac {3+n}{2}} x^5}{(3+n) \left (c-a^2 c x^2\right )^{5/2}}-\frac {2^{\frac {5+n}{2}} a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-3-n)} x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-n),\frac {1}{2} (-3-n),\frac {1}{2} (-1-n),\frac {a-\frac {1}{x}}{2 a}\right )}{(3+n) \left (c-a^2 c x^2\right )^{5/2}} \]
-a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5 /(3+n)/(-a^2*c*x^2+c)^(5/2)-3*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-1/2-1/2* n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c)^(5/2)+6*a^5*(1-1/ a^2/x^2)^(5/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*x^5/(-n^3-3*n^ 2+n+3)/(-a^2*c*x^2+c)^(5/2)-6*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(3/2-1/2*n )*(1+1/a/x)^(-3/2+1/2*n)*x^5/(n^4-10*n^2+9)/(-a^2*c*x^2+c)^(5/2)+4*a^5*(1- 1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x^5/(3+n)/( -a^2*c*x^2+c)^(5/2)+8*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-1/2-1/2*n)*(1+1/ a/x)^(-1/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c)^(5/2)-8*a^5*(1-1/a^2/x^2) ^(5/2)*(1-1/a/x)^(1/2-1/2*n)*(1+1/a/x)^(-1/2+1/2*n)*x^5/(-n^3-3*n^2+n+3)/( -a^2*c*x^2+c)^(5/2)-6*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/ a/x)^(1/2+1/2*n)*x^5/(3+n)/(-a^2*c*x^2+c)^(5/2)-6*a^5*(1-1/a^2/x^2)^(5/2)* (1-1/a/x)^(-1/2-1/2*n)*(1+1/a/x)^(1/2+1/2*n)*x^5/(n^2+4*n+3)/(-a^2*c*x^2+c )^(5/2)+4*a^5*(1-1/a^2/x^2)^(5/2)*(1-1/a/x)^(-3/2-1/2*n)*(1+1/a/x)^(3/2+1/ 2*n)*x^5/(3+n)/(-a^2*c*x^2+c)^(5/2)-2^(5/2+1/2*n)*a^5*(1-1/a^2/x^2)^(5/2)* (1-1/a/x)^(-3/2-1/2*n)*x^5*hypergeom([-3/2-1/2*n, -3/2-1/2*n],[-1/2-1/2*n] ,1/2*(a-1/x)/a)/(3+n)/(-a^2*c*x^2+c)^(5/2)
Time = 2.22 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.23 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (42-2 n^2-45 a n x+5 a n^3 x\right )+6 a \left (-1+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (2 \coth ^{-1}(a x)\right )-n \left (-1+n^2\right ) \left (-1+a^2 x^2\right ) \cosh \left (3 \coth ^{-1}(a x)\right )\right )-8 e^{(1+n) \coth ^{-1}(a x)} \left (9-9 n-n^2+n^3\right ) \left (-1+a^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-e^{2 \coth ^{-1}(a x)}\right )}{4 a c^2 (-1+n) (1+n) \left (-9+n^2\right ) \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a^2 c x^2}} \]
(E^(n*ArcCoth[a*x])*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(42 - 2*n^2 - 45*a*n*x + 5* a*n^3*x) + 6*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[2*ArcCoth[a*x]] - n *(-1 + n^2)*(-1 + a^2*x^2)*Cosh[3*ArcCoth[a*x]]) - 8*E^((1 + n)*ArcCoth[a* x])*(9 - 9*n - n^2 + n^3)*(-1 + a^2*x^2)*Hypergeometric2F1[1, (1 + n)/2, ( 3 + n)/2, -E^(2*ArcCoth[a*x])])/(4*a*c^2*(-1 + n)*(1 + n)*(-9 + n^2)*Sqrt[ 1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2])
Time = 0.97 (sec) , antiderivative size = 624, normalized size of antiderivative = 0.66, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6746, 6749, 137, 2009}
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.
\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^6}dx}{\left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 6749 |
\(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {\left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}}{x^4}d\frac {1}{x}}{\left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 137 |
\(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \left (-4 a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+1}+6 a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+2}-4 a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+3}+a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}+4}+a^4 \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-5)} \left (1+\frac {1}{a x}\right )^{\frac {n-5}{2}}\right )d\frac {1}{x}}{\left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {a^5 2^{\frac {n+5}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-3),\frac {1}{2} (-n-3),\frac {1}{2} (-n-1),\frac {a-\frac {1}{x}}{2 a}\right )}{n+3}+\frac {3 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n^2+4 n+3}-\frac {8 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n^2+4 n+3}+\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{n^2+4 n+3}-\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(n+3) \left (1-n^2\right )}+\frac {8 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{(n+3) \left (1-n^2\right )}+\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}}}{n^4-10 n^2+9}+\frac {a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}-\frac {4 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}+\frac {6 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}-\frac {4 a^5 \left (\frac {1}{a x}+1\right )^{\frac {n+3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-3)}}{n+3}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
-(((1 - 1/(a^2*x^2))^(5/2)*x^5*((a^5*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a* x))^((-3 + n)/2))/(3 + n) + (3*a^5*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x) )^((-3 + n)/2))/(3 + 4*n + n^2) - (6*a^5*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/ (a*x))^((-3 + n)/2))/((3 + n)*(1 - n^2)) + (6*a^5*(1 - 1/(a*x))^((3 - n)/2 )*(1 + 1/(a*x))^((-3 + n)/2))/(9 - 10*n^2 + n^4) - (4*a^5*(1 - 1/(a*x))^(( -3 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2))/(3 + n) - (8*a^5*(1 - 1/(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2))/(3 + 4*n + n^2) + (8*a^5*(1 - 1/(a*x) )^((1 - n)/2)*(1 + 1/(a*x))^((-1 + n)/2))/((3 + n)*(1 - n^2)) + (6*a^5*(1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((1 + n)/2))/(3 + n) + (6*a^5*(1 - 1 /(a*x))^((-1 - n)/2)*(1 + 1/(a*x))^((1 + n)/2))/(3 + 4*n + n^2) - (4*a^5*( 1 - 1/(a*x))^((-3 - n)/2)*(1 + 1/(a*x))^((3 + n)/2))/(3 + n) + (2^((5 + n) /2)*a^5*(1 - 1/(a*x))^((-3 - n)/2)*Hypergeometric2F1[(-3 - n)/2, (-3 - n)/ 2, (-1 - n)/2, (a - x^(-1))/(2*a)])/(3 + n)))/(c - a^2*c*x^2)^(5/2))
3.8.61.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (ILtQ[m, 0] && ILtQ[n, 0]))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_.), x _Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/ x^(m + 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] && IntegerQ[m]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
integral(-sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a^6*c^3*x^7 - 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 - c^3*x), x)
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{x \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{x\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]