Integrand size = 23, antiderivative size = 101 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {1+a x}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {5+4 a x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {15+8 a x}{15 c^3 \sqrt {1-a^2 x^2}}-\frac {\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \]
1/5*(a*x+1)/c^3/(-a^2*x^2+1)^(5/2)+1/15*(4*a*x+5)/c^3/(-a^2*x^2+1)^(3/2)-a rctanh((-a^2*x^2+1)^(1/2))/c^3+1/15*(8*a*x+15)/c^3/(-a^2*x^2+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {23-8 a x-27 a^2 x^2+7 a^3 x^3+8 a^4 x^4-15 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{15 c^3 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \]
(23 - 8*a*x - 27*a^2*x^2 + 7*a^3*x^3 + 8*a^4*x^4 - 15*(-1 + a*x)^2*(1 + a* x)*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(15*c^3*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6698, 532, 25, 532, 25, 532, 27, 243, 73, 221}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {a x+1}{x \left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int -\frac {4 a x+5}{x \left (1-a^2 x^2\right )^{5/2}}dx}{c^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{5} \int \frac {4 a x+5}{x \left (1-a^2 x^2\right )^{5/2}}dx+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {8 a x+15}{x \left (1-a^2 x^2\right )^{3/2}}dx\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {8 a x+15}{x \left (1-a^2 x^2\right )^{3/2}}dx+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {8 a x+15}{\sqrt {1-a^2 x^2}}-\int -\frac {15}{x \sqrt {1-a^2 x^2}}dx\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (15 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {8 a x+15}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {15}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\frac {8 a x+15}{\sqrt {1-a^2 x^2}}\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {8 a x+15}{\sqrt {1-a^2 x^2}}-\frac {15 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {8 a x+15}{\sqrt {1-a^2 x^2}}-15 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )+\frac {4 a x+5}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
((1 + a*x)/(5*(1 - a^2*x^2)^(5/2)) + ((5 + 4*a*x)/(3*(1 - a^2*x^2)^(3/2)) + ((15 + 8*a*x)/Sqrt[1 - a^2*x^2] - 15*ArcTanh[Sqrt[1 - a^2*x^2]])/3)/5)/c ^3
3.10.15.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
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]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(383\) vs. \(2(87)=174\).
Time = 0.25 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.80
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{2}}+\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a}-\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{2 a}-\frac {5 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a \left (x +\frac {1}{a}\right )}+\frac {11 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{16 a \left (x -\frac {1}{a}\right )}}{c^{3}}\) | \(384\) |
-1/c^3*(arctanh(1/(-a^2*x^2+1)^(1/2))+1/4/a^2*(1/5/a/(x-1/a)^3*(-(x-1/a)^2 *a^2-2*(x-1/a)*a)^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a )^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)))+1/8/a*(-1/3/a/(x+ 1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-1/3/(x+1/a)*(-a^2*(x+1/a)^2+2*a* (x+1/a))^(1/2))-1/2/a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)- 1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2))-5/16/a/(x+1/a)*(-a^2*(x+1/ a)^2+2*a*(x+1/a))^(1/2)+11/16/a/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2) )
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (87) = 174\).
Time = 0.25 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.96 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {23 \, a^{5} x^{5} - 23 \, a^{4} x^{4} - 46 \, a^{3} x^{3} + 46 \, a^{2} x^{2} + 23 \, a x + 15 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (8 \, a^{4} x^{4} + 7 \, a^{3} x^{3} - 27 \, a^{2} x^{2} - 8 \, a x + 23\right )} \sqrt {-a^{2} x^{2} + 1} - 23}{15 \, {\left (a^{5} c^{3} x^{5} - a^{4} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x - c^{3}\right )}} \]
1/15*(23*a^5*x^5 - 23*a^4*x^4 - 46*a^3*x^3 + 46*a^2*x^2 + 23*a*x + 15*(a^5 *x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (8*a^4*x^4 + 7*a^3*x^3 - 27*a^2*x^2 - 8*a*x + 23)*sqrt(-a^2*x^2 + 1) - 23)/(a^5*c^3*x^5 - a^4*c^3*x^4 - 2*a^3*c^3*x^3 + 2*a^2*c^3*x^2 + a* c^3*x - c^3)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {a}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
(Integral(a/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Int egral(1/(-a**6*x**7*sqrt(-a**2*x**2 + 1) + 3*a**4*x**5*sqrt(-a**2*x**2 + 1 ) - 3*a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c**3
\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
Time = 0.09 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^3} \, dx=\frac {a^2\,\sqrt {1-a^2\,x^2}}{5\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {17\,a\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {71\,a\,\sqrt {1-a^2\,x^2}}{80\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^3} \]
(atan((1 - a^2*x^2)^(1/2)*1i)*1i)/c^3 + (a^2*(1 - a^2*x^2)^(1/2))/(5*(a^2* c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) + (a^2*(1 - a^2*x^2)^(1/2))/(24*(a^2*c^3 + 2*a^3*c^3*x + a^4*c^3*x^2)) - (17*a*(1 - a^2*x^2)^(1/2))/(48*(-a^2)^(1/ 2)*(c^3*x*(-a^2)^(1/2) + (c^3*(-a^2)^(1/2))/a)) + (71*a*(1 - a^2*x^2)^(1/2 ))/(80*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a)) + (a*(1 - a^2*x^2)^(1/2))/(20*(-a^2)^(1/2)*(3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2 ))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c^3*x^2*(-a^2)^(1/2)))