Integrand size = 22, antiderivative size = 197 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left (36-n^2\right ) \left (64-20 n^2+n^4\right )}-\frac {e^{n \coth ^{-1}(a x)} (n-6 a x)}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}-\frac {30 e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^4 \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {360 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^4 \left (4-n^2\right ) \left (16-n^2\right ) \left (36-n^2\right ) \left (1-a^2 x^2\right )} \]
720*exp(n*arccoth(a*x))/a/c^4/n/(-n^2+36)/(n^4-20*n^2+64)-exp(n*arccoth(a* x))*(-6*a*x+n)/a/c^4/(-n^2+36)/(-a^2*x^2+1)^3-30*exp(n*arccoth(a*x))*(-4*a *x+n)/a/c^4/(n^4-52*n^2+576)/(-a^2*x^2+1)^2-360*exp(n*arccoth(a*x))*(-2*a* x+n)/a/c^4/(-n^2+36)/(n^4-20*n^2+64)/(-a^2*x^2+1)
Time = 0.52 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.77 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (n^6-6 a n^5 x-120 a n^3 x \left (-2+a^2 x^2\right )+720 \left (-1+a^2 x^2\right )^3+10 n^4 \left (-5+3 a^2 x^2\right )-48 a n x \left (33-40 a^2 x^2+15 a^4 x^4\right )+8 n^2 \left (68-105 a^2 x^2+45 a^4 x^4\right )\right )}{a c^4 n \left (-36+n^2\right ) \left (-16+n^2\right ) \left (-4+n^2\right ) \left (-1+a^2 x^2\right )^3} \]
-((E^(n*ArcCoth[a*x])*(n^6 - 6*a*n^5*x - 120*a*n^3*x*(-2 + a^2*x^2) + 720* (-1 + a^2*x^2)^3 + 10*n^4*(-5 + 3*a^2*x^2) - 48*a*n*x*(33 - 40*a^2*x^2 + 1 5*a^4*x^4) + 8*n^2*(68 - 105*a^2*x^2 + 45*a^4*x^4)))/(a*c^4*n*(-36 + n^2)* (-16 + n^2)*(-4 + n^2)*(-1 + a^2*x^2)^3))
Time = 0.70 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6739, 27, 6739, 6739, 6737}
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.
| \(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 6739 |
| \(\displaystyle \frac {30 \int \frac {e^{n \coth ^{-1}(a x)}}{c^3 \left (1-a^2 x^2\right )^3}dx}{c \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {30 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-a^2 x^2\right )^3}dx}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6739 |
| \(\displaystyle \frac {30 \left (\frac {12 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-a^2 x^2\right )^2}dx}{16-n^2}-\frac {(n-4 a x) e^{n \coth ^{-1}(a x)}}{a \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\right )}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6739 |
| \(\displaystyle \frac {30 \left (\frac {12 \left (\frac {2 \int \frac {e^{n \coth ^{-1}(a x)}}{1-a^2 x^2}dx}{4-n^2}-\frac {(n-2 a x) e^{n \coth ^{-1}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{16-n^2}-\frac {(n-4 a x) e^{n \coth ^{-1}(a x)}}{a \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\right )}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\) |
| \(\Big \downarrow \) 6737 |
| \(\displaystyle \frac {30 \left (\frac {12 \left (\frac {2 e^{n \coth ^{-1}(a x)}}{a n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \coth ^{-1}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{16-n^2}-\frac {(n-4 a x) e^{n \coth ^{-1}(a x)}}{a \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}\right )}{c^4 \left (36-n^2\right )}-\frac {(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (36-n^2\right ) \left (1-a^2 x^2\right )^3}\) |
-((E^(n*ArcCoth[a*x])*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3)) + ( 30*(-((E^(n*ArcCoth[a*x])*(n - 4*a*x))/(a*(16 - n^2)*(1 - a^2*x^2)^2)) + ( 12*((2*E^(n*ArcCoth[a*x]))/(a*n*(4 - n^2)) - (E^(n*ArcCoth[a*x])*(n - 2*a* x))/(a*(4 - n^2)*(1 - a^2*x^2))))/(16 - n^2)))/(c^4*(36 - n^2))
3.8.43.3.1 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[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[ E^(n*ArcCoth[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2]
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 = 59.38 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.85
| method | result | size |
| gosper | \(-\frac {\left (720 a^{6} x^{6}-720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 a n x +544 n^{2}-720\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (a^{2} x^{2}-1\right )^{3} c^{4} a n \left (n^{6}-56 n^{4}+784 n^{2}-2304\right )}\) | \(167\) |
| risch | \(-\frac {\left (720 a^{6} x^{6}-720 a^{5} x^{5} n +360 a^{4} n^{2} x^{4}-120 a^{3} n^{3} x^{3}-2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} x^{3} n -6 a \,n^{5} x -840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}-50 n^{4}-1584 a n x +544 n^{2}-720\right ) \left (a x -1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{c^{4} \left (n^{2}-36\right ) \left (n^{2}-16\right ) \left (n^{2}-4\right ) a n \left (a^{2} x^{2}-1\right )^{3}}\) | \(182\) |
| parallelrisch | \(\frac {-720 x^{6} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{6}+2160 x^{4} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{4}+720 a^{5} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x^{5} n -360 x^{4} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{4} n^{2}+120 x^{3} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{3} n^{3}-30 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2} n^{4}+6 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a \,n^{5}+1584 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a n -2160 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2}-1920 a^{3} x^{3} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n +840 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2} n^{2}-240 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a \,n^{3}+720 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}-{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{6}+50 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{4}-544 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{2}}{c^{4} \left (a^{2} x^{2}-1\right )^{3} a n \left (n^{6}-56 n^{4}+784 n^{2}-2304\right )}\) | \(274\) |
-(720*a^6*x^6-720*a^5*n*x^5+360*a^4*n^2*x^4-120*a^3*n^3*x^3-2160*a^4*x^4+3 0*a^2*n^4*x^2+1920*a^3*n*x^3-6*a*n^5*x-840*a^2*n^2*x^2+n^6+240*a*n^3*x+216 0*a^2*x^2-50*n^4-1584*a*n*x+544*n^2-720)*exp(n*arccoth(a*x))/(a^2*x^2-1)^3 /c^4/a/n/(n^6-56*n^4+784*n^2-2304)
Time = 0.26 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.57 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (720 \, a^{6} x^{6} - 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left (a^{4} n^{2} - 6 \, a^{4}\right )} x^{4} - 50 \, n^{4} - 120 \, {\left (a^{3} n^{3} - 16 \, a^{3} n\right )} x^{3} + 30 \, {\left (a^{2} n^{4} - 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} - 6 \, {\left (a n^{5} - 40 \, a n^{3} + 264 \, a n\right )} x - 720\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{7} - 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} - {\left (a^{7} c^{4} n^{7} - 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} - 2304 \, a^{7} c^{4} n\right )} x^{6} - 2304 \, a c^{4} n + 3 \, {\left (a^{5} c^{4} n^{7} - 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} - 2304 \, a^{5} c^{4} n\right )} x^{4} - 3 \, {\left (a^{3} c^{4} n^{7} - 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} - 2304 \, a^{3} c^{4} n\right )} x^{2}} \]
(720*a^6*x^6 - 720*a^5*n*x^5 + n^6 + 360*(a^4*n^2 - 6*a^4)*x^4 - 50*n^4 - 120*(a^3*n^3 - 16*a^3*n)*x^3 + 30*(a^2*n^4 - 28*a^2*n^2 + 72*a^2)*x^2 + 54 4*n^2 - 6*(a*n^5 - 40*a*n^3 + 264*a*n)*x - 720)*((a*x + 1)/(a*x - 1))^(1/2 *n)/(a*c^4*n^7 - 56*a*c^4*n^5 + 784*a*c^4*n^3 - (a^7*c^4*n^7 - 56*a^7*c^4* n^5 + 784*a^7*c^4*n^3 - 2304*a^7*c^4*n)*x^6 - 2304*a*c^4*n + 3*(a^5*c^4*n^ 7 - 56*a^5*c^4*n^5 + 784*a^5*c^4*n^3 - 2304*a^5*c^4*n)*x^4 - 3*(a^3*c^4*n^ 7 - 56*a^3*c^4*n^5 + 784*a^3*c^4*n^3 - 2304*a^3*c^4*n)*x^2)
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\text {Timed out} \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{4}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{4}} \,d x } \]
Time = 4.83 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.59 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {n^6-50\,n^4+544\,n^2-720}{a^7\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {720\,x^5}{a^2\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {x^3\,\left (120\,n^2-1920\right )}{a^4\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {720\,x^6}{a\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}-\frac {6\,x\,\left (n^4-40\,n^2+264\right )}{a^6\,c^4\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^2\,\left (30\,n^4-840\,n^2+2160\right )}{a^5\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}+\frac {x^4\,\left (360\,n^2-2160\right )}{a^3\,c^4\,n\,\left (n^6-56\,n^4+784\,n^2-2304\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^6}-x^6+\frac {3\,x^4}{a^2}-\frac {3\,x^2}{a^4}\right )} \]
(((a*x + 1)/(a*x))^(n/2)*((544*n^2 - 50*n^4 + n^6 - 720)/(a^7*c^4*n*(784*n ^2 - 56*n^4 + n^6 - 2304)) - (720*x^5)/(a^2*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)) - (x^3*(120*n^2 - 1920))/(a^4*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)) + (720*x^6)/(a*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304)) - (6*x*(n^4 - 40*n^2 + 264))/(a^6*c^4*(784*n^2 - 56*n^4 + n^6 - 2304)) + (x^2*(30*n^4 - 840*n^ 2 + 2160))/(a^5*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304)) + (x^4*(360*n^2 - 2 160))/(a^3*c^4*n*(784*n^2 - 56*n^4 + n^6 - 2304))))/(((a*x - 1)/(a*x))^(n/ 2)*(1/a^6 - x^6 + (3*x^4)/a^2 - (3*x^2)/a^4))