Integrand size = 24, antiderivative size = 239 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac {42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {5040 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt {c-a^2 c x^2}} \]
-exp(n*arccoth(a*x))*(-7*a*x+n)/a/c/(-n^2+49)/(-a^2*c*x^2+c)^(7/2)-42*exp( n*arccoth(a*x))*(-5*a*x+n)/a/c^2/(n^4-74*n^2+1225)/(-a^2*c*x^2+c)^(5/2)-84 0*exp(n*arccoth(a*x))*(-3*a*x+n)/a/c^3/(-n^2+49)/(n^4-34*n^2+225)/(-a^2*c* x^2+c)^(3/2)-5040*exp(n*arccoth(a*x))*(-a*x+n)/a/c^4/(n^4-74*n^2+1225)/(n^ 4-10*n^2+9)/(-a^2*c*x^2+c)^(1/2)
Time = 1.95 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.09 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {a e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right ) x^2 \left (-\frac {35 n}{-1+n^2}+\frac {35 a x}{-1+n^2}-\frac {63 a \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )}{-9+n^2}+\frac {35 a \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (5 \coth ^{-1}(a x)\right )}{-25+n^2}-\frac {7 a \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (7 \coth ^{-1}(a x)\right )}{-49+n^2}+\frac {21 a n \sqrt {1-\frac {1}{a^2 x^2}} x \sinh \left (3 \coth ^{-1}(a x)\right )}{-9+n^2}-\frac {7 a n \sqrt {1-\frac {1}{a^2 x^2}} x \sinh \left (5 \coth ^{-1}(a x)\right )}{-25+n^2}+\frac {a n \sqrt {1-\frac {1}{a^2 x^2}} x \sinh \left (7 \coth ^{-1}(a x)\right )}{-49+n^2}\right )}{64 c^3 \left (c-a^2 c x^2\right )^{3/2}} \]
(a*E^(n*ArcCoth[a*x])*(1 - 1/(a^2*x^2))*x^2*((-35*n)/(-1 + n^2) + (35*a*x) /(-1 + n^2) - (63*a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]])/(-9 + n^ 2) + (35*a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[5*ArcCoth[a*x]])/(-25 + n^2) - (7* a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[7*ArcCoth[a*x]])/(-49 + n^2) + (21*a*n*Sqrt [1 - 1/(a^2*x^2)]*x*Sinh[3*ArcCoth[a*x]])/(-9 + n^2) - (7*a*n*Sqrt[1 - 1/( a^2*x^2)]*x*Sinh[5*ArcCoth[a*x]])/(-25 + n^2) + (a*n*Sqrt[1 - 1/(a^2*x^2)] *x*Sinh[7*ArcCoth[a*x]])/(-49 + n^2)))/(64*c^3*(c - a^2*c*x^2)^(3/2))
Time = 0.79 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6739, 6739, 6739, 6738}
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.
| \(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 6739 |
| \(\displaystyle \frac {42 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}}dx}{c \left (49-n^2\right )}-\frac {(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 6739 |
| \(\displaystyle \frac {42 \left (\frac {20 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}}dx}{c \left (25-n^2\right )}-\frac {(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}\right )}{c \left (49-n^2\right )}-\frac {(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 6739 |
| \(\displaystyle \frac {42 \left (\frac {20 \left (\frac {6 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}}dx}{c \left (9-n^2\right )}-\frac {(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}\right )}{c \left (25-n^2\right )}-\frac {(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}\right )}{c \left (49-n^2\right )}-\frac {(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 6738 |
| \(\displaystyle \frac {42 \left (\frac {20 \left (-\frac {6 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (1-n^2\right ) \left (9-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left (9-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}\right )}{c \left (25-n^2\right )}-\frac {(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left (25-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}\right )}{c \left (49-n^2\right )}-\frac {(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}\) |
-((E^(n*ArcCoth[a*x])*(n - 7*a*x))/(a*c*(49 - n^2)*(c - a^2*c*x^2)^(7/2))) + (42*(-((E^(n*ArcCoth[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2) ^(5/2))) + (20*(-((E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2 *c*x^2)^(3/2))) - (6*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^2*(1 - n^2)*(9 - n ^2)*Sqrt[c - a^2*c*x^2])))/(c*(25 - n^2))))/(c*(49 - n^2))
3.8.50.3.1 Defintions of rubi rules used
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(n - a*x)*(E^(n*ArcCoth[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2])), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2))), x] - Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 - 4*(p + 1)^2))) Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] || !IntegerQ[n])
Time = 0.52 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.91
| method | result | size |
| gosper | \(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (5040 a^{7} x^{7}-5040 n \,a^{6} x^{6}+2520 a^{5} n^{2} x^{5}-840 a^{4} n^{3} x^{4}-17640 a^{5} x^{5}+210 a^{3} n^{4} x^{3}+15960 n \,a^{4} x^{4}-42 a^{2} n^{5} x^{2}-7140 a^{3} n^{2} x^{3}+7 a \,n^{6} x +2100 a^{2} n^{3} x^{2}-n^{7}+22050 a^{3} x^{3}-455 a \,n^{4} x -17178 n \,x^{2} a^{2}+77 n^{5}+6433 n^{2} x a -1519 n^{3}-11025 a x +6483 n \right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a \left (n^{8}-84 n^{6}+1974 n^{4}-12916 n^{2}+11025\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\) | \(218\) |
(a*x-1)*(a*x+1)*(5040*a^7*x^7-5040*a^6*n*x^6+2520*a^5*n^2*x^5-840*a^4*n^3* x^4-17640*a^5*x^5+210*a^3*n^4*x^3+15960*a^4*n*x^4-42*a^2*n^5*x^2-7140*a^3* n^2*x^3+7*a*n^6*x+2100*a^2*n^3*x^2-n^7+22050*a^3*x^3-455*a*n^4*x-17178*a^2 *n*x^2+77*n^5+6433*a*n^2*x-1519*n^3-11025*a*x+6483*n)*exp(n*arccoth(a*x))/ a/(n^8-84*n^6+1974*n^4-12916*n^2+11025)/(-a^2*c*x^2+c)^(9/2)
Time = 0.27 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.90 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {{\left (5040 \, a^{7} x^{7} - 5040 \, a^{6} n x^{6} - n^{7} + 2520 \, {\left (a^{5} n^{2} - 7 \, a^{5}\right )} x^{5} + 77 \, n^{5} - 840 \, {\left (a^{4} n^{3} - 19 \, a^{4} n\right )} x^{4} + 210 \, {\left (a^{3} n^{4} - 34 \, a^{3} n^{2} + 105 \, a^{3}\right )} x^{3} - 1519 \, n^{3} - 42 \, {\left (a^{2} n^{5} - 50 \, a^{2} n^{3} + 409 \, a^{2} n\right )} x^{2} + 7 \, {\left (a n^{6} - 65 \, a n^{4} + 919 \, a n^{2} - 1575 \, a\right )} x + 6483 \, n\right )} \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{5} n^{8} - 84 \, a c^{5} n^{6} + 1974 \, a c^{5} n^{4} + {\left (a^{9} c^{5} n^{8} - 84 \, a^{9} c^{5} n^{6} + 1974 \, a^{9} c^{5} n^{4} - 12916 \, a^{9} c^{5} n^{2} + 11025 \, a^{9} c^{5}\right )} x^{8} - 12916 \, a c^{5} n^{2} - 4 \, {\left (a^{7} c^{5} n^{8} - 84 \, a^{7} c^{5} n^{6} + 1974 \, a^{7} c^{5} n^{4} - 12916 \, a^{7} c^{5} n^{2} + 11025 \, a^{7} c^{5}\right )} x^{6} + 11025 \, a c^{5} + 6 \, {\left (a^{5} c^{5} n^{8} - 84 \, a^{5} c^{5} n^{6} + 1974 \, a^{5} c^{5} n^{4} - 12916 \, a^{5} c^{5} n^{2} + 11025 \, a^{5} c^{5}\right )} x^{4} - 4 \, {\left (a^{3} c^{5} n^{8} - 84 \, a^{3} c^{5} n^{6} + 1974 \, a^{3} c^{5} n^{4} - 12916 \, a^{3} c^{5} n^{2} + 11025 \, a^{3} c^{5}\right )} x^{2}} \]
-(5040*a^7*x^7 - 5040*a^6*n*x^6 - n^7 + 2520*(a^5*n^2 - 7*a^5)*x^5 + 77*n^ 5 - 840*(a^4*n^3 - 19*a^4*n)*x^4 + 210*(a^3*n^4 - 34*a^3*n^2 + 105*a^3)*x^ 3 - 1519*n^3 - 42*(a^2*n^5 - 50*a^2*n^3 + 409*a^2*n)*x^2 + 7*(a*n^6 - 65*a *n^4 + 919*a*n^2 - 1575*a)*x + 6483*n)*sqrt(-a^2*c*x^2 + c)*((a*x + 1)/(a* x - 1))^(1/2*n)/(a*c^5*n^8 - 84*a*c^5*n^6 + 1974*a*c^5*n^4 + (a^9*c^5*n^8 - 84*a^9*c^5*n^6 + 1974*a^9*c^5*n^4 - 12916*a^9*c^5*n^2 + 11025*a^9*c^5)*x ^8 - 12916*a*c^5*n^2 - 4*(a^7*c^5*n^8 - 84*a^7*c^5*n^6 + 1974*a^7*c^5*n^4 - 12916*a^7*c^5*n^2 + 11025*a^7*c^5)*x^6 + 11025*a*c^5 + 6*(a^5*c^5*n^8 - 84*a^5*c^5*n^6 + 1974*a^5*c^5*n^4 - 12916*a^5*c^5*n^2 + 11025*a^5*c^5)*x^4 - 4*(a^3*c^5*n^8 - 84*a^3*c^5*n^6 + 1974*a^3*c^5*n^4 - 12916*a^3*c^5*n^2 + 11025*a^3*c^5)*x^2)
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \]
Time = 4.67 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.85 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {5040\,x^7}{c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {-n^7+77\,n^5-1519\,n^3+6483\,n}{a^7\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {5040\,n\,x^6}{a\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {x^5\,\left (2520\,n^2-17640\right )}{a^2\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {x^3\,\left (210\,n^4-7140\,n^2+22050\right )}{a^4\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {7\,x\,\left (n^6-65\,n^4+919\,n^2-1575\right )}{a^6\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {840\,n\,x^4\,\left (n^2-19\right )}{a^3\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {42\,n\,x^2\,\left (n^4-50\,n^2+409\right )}{a^5\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^6}-x^6\,\sqrt {c-a^2\,c\,x^2}+\frac {3\,x^4\,\sqrt {c-a^2\,c\,x^2}}{a^2}-\frac {3\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^4}\right )} \]
-(((a*x + 1)/(a*x))^(n/2)*((5040*x^7)/(c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) + (6483*n - 1519*n^3 + 77*n^5 - n^7)/(a^7*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) - (5040*n*x^6)/(a*c^4*(1974*n^4 - 12916 *n^2 - 84*n^6 + n^8 + 11025)) + (x^5*(2520*n^2 - 17640))/(a^2*c^4*(1974*n^ 4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) + (x^3*(210*n^4 - 7140*n^2 + 22050) )/(a^4*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11025)) + (7*x*(919*n^2 - 65*n^4 + n^6 - 1575))/(a^6*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n^8 + 11 025)) - (840*n*x^4*(n^2 - 19))/(a^3*c^4*(1974*n^4 - 12916*n^2 - 84*n^6 + n ^8 + 11025)) - (42*n*x^2*(n^4 - 50*n^2 + 409))/(a^5*c^4*(1974*n^4 - 12916* n^2 - 84*n^6 + n^8 + 11025))))/(((a*x - 1)/(a*x))^(n/2)*((c - a^2*c*x^2)^( 1/2)/a^6 - x^6*(c - a^2*c*x^2)^(1/2) + (3*x^4*(c - a^2*c*x^2)^(1/2))/a^2 - (3*x^2*(c - a^2*c*x^2)^(1/2))/a^4))