Integrand size = 22, antiderivative size = 154 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {2 (14-11 a x)}{35 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{7 a c^4 (1+a x) \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (105-61 a x)}{105 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 (105-44 a x)}{105 a c^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^4}+\frac {\arcsin (a x)}{a c^4} \] Output:
2/35*(-11*a*x+14)/a/c^4/(-a^2*x^2+1)^(5/2)-1/7/a/c^4/(a*x+1)/(-a^2*x^2+1)^ (5/2)-2/105*(-61*a*x+105)/a/c^4/(-a^2*x^2+1)^(3/2)+4/105*(-44*a*x+105)/a/c ^4/(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a/c^4+arcsin(a*x)/a/c^4
Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {384+279 a x-1065 a^2 x^2-715 a^3 x^3+965 a^4 x^4+559 a^5 x^5-281 a^6 x^6-105 a^7 x^7+105 (-1+a x)^2 (1+a x)^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{105 a c^4 (-1+a x)^2 (1+a x)^3 \sqrt {1-a^2 x^2}} \] Input:
Integrate[1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^4),x]
Output:
(384 + 279*a*x - 1065*a^2*x^2 - 715*a^3*x^3 + 965*a^4*x^4 + 559*a^5*x^5 - 281*a^6*x^6 - 105*a^7*x^7 + 105*(-1 + a*x)^2*(1 + a*x)^3*Sqrt[1 - a^2*x^2] *ArcSin[a*x])/(105*a*c^4*(-1 + a*x)^2*(1 + a*x)^3*Sqrt[1 - a^2*x^2])
Time = 1.17 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6707, 6699, 529, 2345, 2345, 2345, 27, 455, 223}
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.
| \(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx\) |
| \(\Big \downarrow \) 6707 |
| \(\displaystyle \frac {a^8 \int \frac {e^{-\text {arctanh}(a x)} x^8}{\left (1-a^2 x^2\right )^4}dx}{c^4}\) |
| \(\Big \downarrow \) 6699 |
| \(\displaystyle \frac {a^8 \int \frac {x^8 (1-a x)}{\left (1-a^2 x^2\right )^{9/2}}dx}{c^4}\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle \frac {a^8 \left (-\frac {1}{7} \int \frac {-\frac {7 x^7}{a}+\frac {7 x^6}{a^2}-\frac {7 x^5}{a^3}+\frac {7 x^4}{a^4}-\frac {7 x^3}{a^5}+\frac {7 x^2}{a^6}-\frac {7 x}{a^7}+\frac {1}{a^8}}{\left (1-a^2 x^2\right )^{7/2}}dx-\frac {1-a x}{7 a^9 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^4}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {a^8 \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {-\frac {35 x^5}{a^3}+\frac {35 x^4}{a^4}-\frac {70 x^3}{a^5}+\frac {70 x^2}{a^6}-\frac {105 x}{a^7}+\frac {17}{a^8}}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {2 (14-11 a x)}{5 a^9 \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1-a x}{7 a^9 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^4}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {a^8 \left (\frac {1}{7} \left (\frac {1}{5} \left (-\frac {1}{3} \int \frac {-\frac {105 x^3}{a^5}+\frac {105 x^2}{a^6}-\frac {315 x}{a^7}+\frac {71}{a^8}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {2 (105-61 a x)}{3 a^9 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2 (14-11 a x)}{5 a^9 \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1-a x}{7 a^9 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^4}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {a^8 \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {105 (1-a x)}{a^8 \sqrt {1-a^2 x^2}}dx+\frac {4 (105-44 a x)}{a^9 \sqrt {1-a^2 x^2}}\right )-\frac {2 (105-61 a x)}{3 a^9 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2 (14-11 a x)}{5 a^9 \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1-a x}{7 a^9 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^8 \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 \int \frac {1-a x}{\sqrt {1-a^2 x^2}}dx}{a^8}+\frac {4 (105-44 a x)}{a^9 \sqrt {1-a^2 x^2}}\right )-\frac {2 (105-61 a x)}{3 a^9 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2 (14-11 a x)}{5 a^9 \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1-a x}{7 a^9 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^4}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {a^8 \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {105 \left (\int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^8}+\frac {4 (105-44 a x)}{a^9 \sqrt {1-a^2 x^2}}\right )-\frac {2 (105-61 a x)}{3 a^9 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2 (14-11 a x)}{5 a^9 \left (1-a^2 x^2\right )^{5/2}}\right )-\frac {1-a x}{7 a^9 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^4}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {a^8 \left (\frac {1}{7} \left (\frac {2 (14-11 a x)}{5 a^9 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (\frac {1}{3} \left (\frac {4 (105-44 a x)}{a^9 \sqrt {1-a^2 x^2}}+\frac {105 \left (\frac {\sqrt {1-a^2 x^2}}{a}+\frac {\arcsin (a x)}{a}\right )}{a^8}\right )-\frac {2 (105-61 a x)}{3 a^9 \left (1-a^2 x^2\right )^{3/2}}\right )\right )-\frac {1-a x}{7 a^9 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^4}\) |
Input:
Int[1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^4),x]
Output:
(a^8*(-1/7*(1 - a*x)/(a^9*(1 - a^2*x^2)^(7/2)) + ((2*(14 - 11*a*x))/(5*a^9 *(1 - a^2*x^2)^(5/2)) + ((-2*(105 - 61*a*x))/(3*a^9*(1 - a^2*x^2)^(3/2)) + ((4*(105 - 44*a*x))/(a^9*Sqrt[1 - a^2*x^2]) + (105*(Sqrt[1 - a^2*x^2]/a + ArcSin[a*x]/a))/a^8)/3)/5)/7))/c^4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^p Int[x^m*((1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n), x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c , 0]) && ILtQ[(n - 1)/2, 0] && !IntegerQ[p - n/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(330\) vs. \(2(136)=272\).
Time = 0.36 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.15
| method | result | size |
| risch | \(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{4}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{8} \sqrt {a^{2}}}+\frac {1657 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{672 a^{10} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{80 a^{12} \left (x -\frac {1}{a}\right )^{3}}-\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{60 a^{11} \left (x -\frac {1}{a}\right )^{2}}-\frac {379 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{480 a^{10} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{56 a^{13} \left (x +\frac {1}{a}\right )^{4}}+\frac {17 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{112 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {211 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{336 a^{11} \left (x +\frac {1}{a}\right )^{2}}\right ) a^{8}}{c^{4}}\) | \(331\) |
| default | \(\frac {a^{8} \left (\frac {\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}}{32 a^{12}}+\frac {11 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{192 a^{12} \left (x -\frac {1}{a}\right )^{3}}+\frac {\frac {47 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{128 a \left (x -\frac {1}{a}\right )^{2}}+\frac {47 a \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{128}}{a^{10}}+\frac {\frac {93 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{256}-\frac {93 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{256 \sqrt {a^{2}}}}{a^{9}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{7 a \left (x +\frac {1}{a}\right )^{5}}+\frac {2 a \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}\right )}{7}}{16 a^{13}}-\frac {3 \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}\right )}{8 a^{12}}-\frac {29 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{96 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {35 \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}-a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )\right )}{32 a^{10}}+\frac {\frac {163 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{256}+\frac {163 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{256 \sqrt {a^{2}}}}{a^{9}}\right )}{c^{4}}\) | \(723\) |
Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x,method=_RETURNVERBOSE)
Output:
-1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^4+(1/a^8/(a^2)^(1/2)*arctan((a^2)^(1 /2)*x/(-a^2*x^2+1)^(1/2))+1657/672/a^10/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a ))^(1/2)-1/80/a^12/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-7/60/a^11/ (x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-379/480/a^10/(x-1/a)*(-(x-1/a )^2*a^2-2*a*(x-1/a))^(1/2)-1/56/a^13/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a) )^(1/2)+17/112/a^12/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-211/336/a ^11/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))*a^8/c^4
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (136) = 272\).
Time = 0.11 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.82 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {384 \, a^{7} x^{7} + 384 \, a^{6} x^{6} - 1152 \, a^{5} x^{5} - 1152 \, a^{4} x^{4} + 1152 \, a^{3} x^{3} + 1152 \, a^{2} x^{2} - 384 \, a x - 210 \, {\left (a^{7} x^{7} + a^{6} x^{6} - 3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (105 \, a^{7} x^{7} + 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} - 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} + 1065 \, a^{2} x^{2} - 279 \, a x - 384\right )} \sqrt {-a^{2} x^{2} + 1} - 384}{105 \, {\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right )}} \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="frica s")
Output:
1/105*(384*a^7*x^7 + 384*a^6*x^6 - 1152*a^5*x^5 - 1152*a^4*x^4 + 1152*a^3* x^3 + 1152*a^2*x^2 - 384*a*x - 210*(a^7*x^7 + a^6*x^6 - 3*a^5*x^5 - 3*a^4* x^4 + 3*a^3*x^3 + 3*a^2*x^2 - a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a* x)) + (105*a^7*x^7 + 281*a^6*x^6 - 559*a^5*x^5 - 965*a^4*x^4 + 715*a^3*x^3 + 1065*a^2*x^2 - 279*a*x - 384)*sqrt(-a^2*x^2 + 1) - 384)/(a^8*c^4*x^7 + a^7*c^4*x^6 - 3*a^6*c^4*x^5 - 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 + 3*a^3*c^4*x^ 2 - a^2*c^4*x - a*c^4)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {a^{8} \int \frac {x^{8} \sqrt {- a^{2} x^{2} + 1}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\, dx}{c^{4}} \] Input:
integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**4,x)
Output:
a**8*Integral(x**8*sqrt(-a**2*x**2 + 1)/(a**9*x**9 + a**8*x**8 - 4*a**7*x* *7 - 4*a**6*x**6 + 6*a**5*x**5 + 6*a**4*x**4 - 4*a**3*x**3 - 4*a**2*x**2 + a*x + 1), x)/c**4
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="maxim a")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a^2*x^2))^4), x)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="giac" )
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a^2*x^2))^4), x)
Time = 23.28 (sec) , antiderivative size = 614, normalized size of antiderivative = 3.99 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}-\frac {35\,a\,\sqrt {1-a^2\,x^2}}{48\,\left (a^4\,c^4\,x^2+2\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{8\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{140\,\left (a^6\,c^4\,x^2+2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {a^8\,\sqrt {1-a^2\,x^2}}{120\,\left (a^{11}\,c^4\,x^2-2\,a^{10}\,c^4\,x+a^9\,c^4\right )}+\frac {13\,a^8\,\sqrt {1-a^2\,x^2}}{120\,\left (a^{11}\,c^4\,x^2+2\,a^{10}\,c^4\,x+a^9\,c^4\right )}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}-\frac {a\,\sqrt {1-a^2\,x^2}}{56\,\left (a^6\,c^4\,x^4+4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2+4\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {1657\,\sqrt {1-a^2\,x^2}}{672\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {379\,\sqrt {1-a^2\,x^2}}{480\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}-\frac {17\,\sqrt {1-a^2\,x^2}}{112\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}+3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {\sqrt {1-a^2\,x^2}}{80\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )} \] Input:
int((1 - a^2*x^2)^(1/2)/((c - c/(a^2*x^2))^4*(a*x + 1)),x)
Output:
asinh(x*(-a^2)^(1/2))/(c^4*(-a^2)^(1/2)) - (35*a*(1 - a^2*x^2)^(1/2))/(48* (a^2*c^4 + 2*a^3*c^4*x + a^4*c^4*x^2)) - (a*(1 - a^2*x^2)^(1/2))/(8*(a^2*c ^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) - (a^3*(1 - a^2*x^2)^(1/2))/(140*(a^4*c^4 + 2*a^5*c^4*x + a^6*c^4*x^2)) + (a^8*(1 - a^2*x^2)^(1/2))/(120*(a^9*c^4 - 2*a^10*c^4*x + a^11*c^4*x^2)) + (13*a^8*(1 - a^2*x^2)^(1/2))/(120*(a^9*c^ 4 + 2*a^10*c^4*x + a^11*c^4*x^2)) + (1 - a^2*x^2)^(1/2)/(a*c^4) - (a*(1 - a^2*x^2)^(1/2))/(56*(a^2*c^4 + 4*a^3*c^4*x + 6*a^4*c^4*x^2 + 4*a^5*c^4*x^3 + a^6*c^4*x^4)) - (1657*(1 - a^2*x^2)^(1/2))/(672*(-a^2)^(1/2)*(c^4*x*(-a ^2)^(1/2) + (c^4*(-a^2)^(1/2))/a)) + (379*(1 - a^2*x^2)^(1/2))/(480*(-a^2) ^(1/2)*(c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a)) - (17*(1 - a^2*x^2)^(1 /2))/(112*(-a^2)^(1/2)*(3*c^4*x*(-a^2)^(1/2) + (c^4*(-a^2)^(1/2))/a + a^2* c^4*x^3*(-a^2)^(1/2) + 3*a*c^4*x^2*(-a^2)^(1/2))) + (1 - a^2*x^2)^(1/2)/(8 0*(-a^2)^(1/2)*(3*c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a + a^2*c^4*x^3* (-a^2)^(1/2) - 3*a*c^4*x^2*(-a^2)^(1/2)))
Time = 0.16 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.23 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{5} x^{5}+105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{4} x^{4}-210 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{3} x^{3}-210 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x +105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+279 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+279 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-558 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-558 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+279 \sqrt {-a^{2} x^{2}+1}\, a x +279 \sqrt {-a^{2} x^{2}+1}-105 a^{7} x^{7}-281 a^{6} x^{6}+559 a^{5} x^{5}+965 a^{4} x^{4}-715 a^{3} x^{3}-1065 a^{2} x^{2}+279 a x +384}{105 \sqrt {-a^{2} x^{2}+1}\, a \,c^{4} \left (a^{5} x^{5}+a^{4} x^{4}-2 a^{3} x^{3}-2 a^{2} x^{2}+a x +1\right )} \] Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a^2/x^2)^4,x)
Output:
(105*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**5*x**5 + 105*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**4*x**4 - 210*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**3*x**3 - 210*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 + 105*sqrt( - a**2*x**2 + 1 )*asin(a*x)*a*x + 105*sqrt( - a**2*x**2 + 1)*asin(a*x) + 279*sqrt( - a**2* x**2 + 1)*a**5*x**5 + 279*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 558*sqrt( - a **2*x**2 + 1)*a**3*x**3 - 558*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 279*sqrt( - a**2*x**2 + 1)*a*x + 279*sqrt( - a**2*x**2 + 1) - 105*a**7*x**7 - 281*a **6*x**6 + 559*a**5*x**5 + 965*a**4*x**4 - 715*a**3*x**3 - 1065*a**2*x**2 + 279*a*x + 384)/(105*sqrt( - a**2*x**2 + 1)*a*c**4*(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1))