Integrand size = 27, antiderivative size = 60 \[ \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {(1+5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{10} (1+a x)^{15} \sqrt {c-a^2 c x^2}} \]
Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {(1+5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (-1+a x)^{10} (1+a x)^{15} \sqrt {c-a^2 c x^2}} \]
((1 + 5*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3*c^13*(-1 + a*x)^10*(1 + a*x)^15*S qrt[c - a^2*c*x^2])
Time = 0.47 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6703, 6700, 91}
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.
\(\displaystyle \int \frac {x^2 e^{-5 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{27/2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (1-a^2 x^2\right )^{27/2}}dx}{c^{13} \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {x^2}{(1-a x)^{11} (a x+1)^{16}}dx}{c^{13} \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 91 |
\(\displaystyle \frac {(5 a x+1) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{10} (a x+1)^{15} \sqrt {c-a^2 c x^2}}\) |
((1 + 5*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3*c^13*(1 - a*x)^10*(1 + a*x)^15*Sq rt[c - a^2*c*x^2])
3.14.78.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(c + d*x)^(n + 1)*(e + f*x)^(p + 1)*((2*a*d*f*(n + p + 3 ) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x)/(d^2*f^2*(n + p + 2)*(n + p + 3))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2 , 0] && NeQ[n + p + 3, 0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*( b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1) + c*f*(p + 1))*(a* d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (5 a x +1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{120 \left (a x +1\right )^{4} a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\) | \(49\) |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (5 a x +1\right )}{120 \left (a^{2} x^{2}-1\right ) c^{14} a^{3} \left (a x +1\right )^{15} \left (a x -1\right )^{10}}\) | \(66\) |
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (53) = 106\).
Time = 1.54 (sec) , antiderivative size = 497, normalized size of antiderivative = 8.28 \[ \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {{\left (a^{22} x^{25} + 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} - 50 \, a^{18} x^{21} + 126 \, a^{17} x^{20} + 280 \, a^{16} x^{19} - 160 \, a^{15} x^{18} - 765 \, a^{14} x^{17} - 105 \, a^{13} x^{16} + 1248 \, a^{12} x^{15} + 720 \, a^{11} x^{14} - 1260 \, a^{10} x^{13} - 1260 \, a^{9} x^{12} + 720 \, a^{8} x^{11} + 1248 \, a^{7} x^{10} - 105 \, a^{6} x^{9} - 765 \, a^{5} x^{8} - 160 \, a^{4} x^{7} + 280 \, a^{3} x^{6} + 126 \, a^{2} x^{5} - 50 \, a x^{4} - 40 \, x^{3}\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{120 \, {\left (a^{27} c^{14} x^{27} + 5 \, a^{26} c^{14} x^{26} - a^{25} c^{14} x^{25} - 45 \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} + 166 \, a^{22} c^{14} x^{22} + 330 \, a^{21} c^{14} x^{21} - 286 \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} + 55 \, a^{18} c^{14} x^{18} + 2013 \, a^{17} c^{14} x^{17} + 825 \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} - 1980 \, a^{14} c^{14} x^{14} + 1980 \, a^{13} c^{14} x^{13} + 2508 \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} - 2013 \, a^{10} c^{14} x^{10} - 55 \, a^{9} c^{14} x^{9} + 1045 \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} - 330 \, a^{6} c^{14} x^{6} - 166 \, a^{5} c^{14} x^{5} + 50 \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} + a^{2} c^{14} x^{2} - 5 \, a c^{14} x - c^{14}\right )}} \]
1/120*(a^22*x^25 + 5*a^21*x^24 - 40*a^19*x^22 - 50*a^18*x^21 + 126*a^17*x^ 20 + 280*a^16*x^19 - 160*a^15*x^18 - 765*a^14*x^17 - 105*a^13*x^16 + 1248* a^12*x^15 + 720*a^11*x^14 - 1260*a^10*x^13 - 1260*a^9*x^12 + 720*a^8*x^11 + 1248*a^7*x^10 - 105*a^6*x^9 - 765*a^5*x^8 - 160*a^4*x^7 + 280*a^3*x^6 + 126*a^2*x^5 - 50*a*x^4 - 40*x^3)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/( a^27*c^14*x^27 + 5*a^26*c^14*x^26 - a^25*c^14*x^25 - 45*a^24*c^14*x^24 - 5 0*a^23*c^14*x^23 + 166*a^22*c^14*x^22 + 330*a^21*c^14*x^21 - 286*a^20*c^14 *x^20 - 1045*a^19*c^14*x^19 + 55*a^18*c^14*x^18 + 2013*a^17*c^14*x^17 + 82 5*a^16*c^14*x^16 - 2508*a^15*c^14*x^15 - 1980*a^14*c^14*x^14 + 1980*a^13*c ^14*x^13 + 2508*a^12*c^14*x^12 - 825*a^11*c^14*x^11 - 2013*a^10*c^14*x^10 - 55*a^9*c^14*x^9 + 1045*a^8*c^14*x^8 + 286*a^7*c^14*x^7 - 330*a^6*c^14*x^ 6 - 166*a^5*c^14*x^5 + 50*a^4*c^14*x^4 + 45*a^3*c^14*x^3 + a^2*c^14*x^2 - 5*a*c^14*x - c^14)
Timed out. \[ \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (53) = 106\).
Time = 0.65 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.55 \[ \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {5 \, a \sqrt {c} x + \sqrt {c}}{120 \, {\left (a^{28} c^{14} x^{25} + 5 \, a^{27} c^{14} x^{24} - 40 \, a^{25} c^{14} x^{22} - 50 \, a^{24} c^{14} x^{21} + 126 \, a^{23} c^{14} x^{20} + 280 \, a^{22} c^{14} x^{19} - 160 \, a^{21} c^{14} x^{18} - 765 \, a^{20} c^{14} x^{17} - 105 \, a^{19} c^{14} x^{16} + 1248 \, a^{18} c^{14} x^{15} + 720 \, a^{17} c^{14} x^{14} - 1260 \, a^{16} c^{14} x^{13} - 1260 \, a^{15} c^{14} x^{12} + 720 \, a^{14} c^{14} x^{11} + 1248 \, a^{13} c^{14} x^{10} - 105 \, a^{12} c^{14} x^{9} - 765 \, a^{11} c^{14} x^{8} - 160 \, a^{10} c^{14} x^{7} + 280 \, a^{9} c^{14} x^{6} + 126 \, a^{8} c^{14} x^{5} - 50 \, a^{7} c^{14} x^{4} - 40 \, a^{6} c^{14} x^{3} + 5 \, a^{4} c^{14} x + a^{3} c^{14}\right )}} \]
1/120*(5*a*sqrt(c)*x + sqrt(c))/(a^28*c^14*x^25 + 5*a^27*c^14*x^24 - 40*a^ 25*c^14*x^22 - 50*a^24*c^14*x^21 + 126*a^23*c^14*x^20 + 280*a^22*c^14*x^19 - 160*a^21*c^14*x^18 - 765*a^20*c^14*x^17 - 105*a^19*c^14*x^16 + 1248*a^1 8*c^14*x^15 + 720*a^17*c^14*x^14 - 1260*a^16*c^14*x^13 - 1260*a^15*c^14*x^ 12 + 720*a^14*c^14*x^11 + 1248*a^13*c^14*x^10 - 105*a^12*c^14*x^9 - 765*a^ 11*c^14*x^8 - 160*a^10*c^14*x^7 + 280*a^9*c^14*x^6 + 126*a^8*c^14*x^5 - 50 *a^7*c^14*x^4 - 40*a^6*c^14*x^3 + 5*a^4*c^14*x + a^3*c^14)
Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (53) = 106\).
Time = 0.40 (sec) , antiderivative size = 458, normalized size of antiderivative = 7.63 \[ \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {\frac {2451570 \, {\left (\frac {2}{a x + 1} - 1\right )}^{9} + 1514205 \, {\left (\frac {2}{a x + 1} - 1\right )}^{8} + 769120 \, {\left (\frac {2}{a x + 1} - 1\right )}^{7} + 318780 \, {\left (\frac {2}{a x + 1} - 1\right )}^{6} + 106260 \, {\left (\frac {2}{a x + 1} - 1\right )}^{5} + 27830 \, {\left (\frac {2}{a x + 1} - 1\right )}^{4} + 5520 \, {\left (\frac {2}{a x + 1} - 1\right )}^{3} + 780 \, {\left (\frac {2}{a x + 1} - 1\right )}^{2} + \frac {140}{a x + 1} - 67}{a^{2} c^{\frac {27}{2}} {\left (\frac {2}{a x + 1} - 1\right )}^{10}} - \frac {2 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{15} + 45 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{14} + 480 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{13} + 3220 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{12} + 15180 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{11} + 53130 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{10} + 141680 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{9} + 288420 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{8} + 432630 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{7} + 408595 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{6} - 891480 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{4} - 2080120 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{3} - 3120180 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}^{2} - 3565920 \, a^{28} c^{189} {\left (\frac {2}{a x + 1} - 1\right )}}{a^{30} c^{\frac {405}{2}}}}{2013265920 \, a} \]
1/2013265920*((2451570*(2/(a*x + 1) - 1)^9 + 1514205*(2/(a*x + 1) - 1)^8 + 769120*(2/(a*x + 1) - 1)^7 + 318780*(2/(a*x + 1) - 1)^6 + 106260*(2/(a*x + 1) - 1)^5 + 27830*(2/(a*x + 1) - 1)^4 + 5520*(2/(a*x + 1) - 1)^3 + 780*( 2/(a*x + 1) - 1)^2 + 140/(a*x + 1) - 67)/(a^2*c^(27/2)*(2/(a*x + 1) - 1)^1 0) - (2*a^28*c^189*(2/(a*x + 1) - 1)^15 + 45*a^28*c^189*(2/(a*x + 1) - 1)^ 14 + 480*a^28*c^189*(2/(a*x + 1) - 1)^13 + 3220*a^28*c^189*(2/(a*x + 1) - 1)^12 + 15180*a^28*c^189*(2/(a*x + 1) - 1)^11 + 53130*a^28*c^189*(2/(a*x + 1) - 1)^10 + 141680*a^28*c^189*(2/(a*x + 1) - 1)^9 + 288420*a^28*c^189*(2 /(a*x + 1) - 1)^8 + 432630*a^28*c^189*(2/(a*x + 1) - 1)^7 + 408595*a^28*c^ 189*(2/(a*x + 1) - 1)^6 - 891480*a^28*c^189*(2/(a*x + 1) - 1)^4 - 2080120* a^28*c^189*(2/(a*x + 1) - 1)^3 - 3120180*a^28*c^189*(2/(a*x + 1) - 1)^2 - 3565920*a^28*c^189*(2/(a*x + 1) - 1))/(a^30*c^(405/2)))/a
Time = 4.71 (sec) , antiderivative size = 363, normalized size of antiderivative = 6.05 \[ \int \frac {e^{-5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}+5\,a\,x\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}}{-120\,a^{30}\,c^{14}\,x^{27}-600\,a^{29}\,c^{14}\,x^{26}+120\,a^{28}\,c^{14}\,x^{25}+5400\,a^{27}\,c^{14}\,x^{24}+6000\,a^{26}\,c^{14}\,x^{23}-19920\,a^{25}\,c^{14}\,x^{22}-39600\,a^{24}\,c^{14}\,x^{21}+34320\,a^{23}\,c^{14}\,x^{20}+125400\,a^{22}\,c^{14}\,x^{19}-6600\,a^{21}\,c^{14}\,x^{18}-241560\,a^{20}\,c^{14}\,x^{17}-99000\,a^{19}\,c^{14}\,x^{16}+300960\,a^{18}\,c^{14}\,x^{15}+237600\,a^{17}\,c^{14}\,x^{14}-237600\,a^{16}\,c^{14}\,x^{13}-300960\,a^{15}\,c^{14}\,x^{12}+99000\,a^{14}\,c^{14}\,x^{11}+241560\,a^{13}\,c^{14}\,x^{10}+6600\,a^{12}\,c^{14}\,x^9-125400\,a^{11}\,c^{14}\,x^8-34320\,a^{10}\,c^{14}\,x^7+39600\,a^9\,c^{14}\,x^6+19920\,a^8\,c^{14}\,x^5-6000\,a^7\,c^{14}\,x^4-5400\,a^6\,c^{14}\,x^3-120\,a^5\,c^{14}\,x^2+600\,a^4\,c^{14}\,x+120\,a^3\,c^{14}} \]
((c - a^2*c*x^2)^(1/2)*(1 - a^2*x^2)^(1/2) + 5*a*x*(c - a^2*c*x^2)^(1/2)*( 1 - a^2*x^2)^(1/2))/(120*a^3*c^14 + 600*a^4*c^14*x - 120*a^5*c^14*x^2 - 54 00*a^6*c^14*x^3 - 6000*a^7*c^14*x^4 + 19920*a^8*c^14*x^5 + 39600*a^9*c^14* x^6 - 34320*a^10*c^14*x^7 - 125400*a^11*c^14*x^8 + 6600*a^12*c^14*x^9 + 24 1560*a^13*c^14*x^10 + 99000*a^14*c^14*x^11 - 300960*a^15*c^14*x^12 - 23760 0*a^16*c^14*x^13 + 237600*a^17*c^14*x^14 + 300960*a^18*c^14*x^15 - 99000*a ^19*c^14*x^16 - 241560*a^20*c^14*x^17 - 6600*a^21*c^14*x^18 + 125400*a^22* c^14*x^19 + 34320*a^23*c^14*x^20 - 39600*a^24*c^14*x^21 - 19920*a^25*c^14* x^22 + 6000*a^26*c^14*x^23 + 5400*a^27*c^14*x^24 + 120*a^28*c^14*x^25 - 60 0*a^29*c^14*x^26 - 120*a^30*c^14*x^27)