Integrand size = 23, antiderivative size = 111 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx=-\frac {\sqrt {e} \sqrt {d+e x^2}}{20 d x^4}+\frac {3 e^{3/2} \sqrt {d+e x^2}}{40 d^2 x^2}-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}-\frac {3 e^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{40 d^{5/2}} \]
-1/5*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^5-3/40*e^(5/2)*arctanh((e*x^2+d) ^(1/2)/d^(1/2))/d^(5/2)+3/40*e^(3/2)*(e*x^2+d)^(1/2)/d^2/x^2-1/20*e^(1/2)* (e*x^2+d)^(1/2)/d/x^4
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx=\frac {-8 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {\sqrt {e} x \left (\sqrt {d} \sqrt {d+e x^2} \left (-2 d+3 e x^2\right )+3 e^2 x^4 \log (x)-3 e^2 x^4 \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )\right )}{d^{5/2}}}{40 x^5} \]
(-8*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + (Sqrt[e]*x*(Sqrt[d]*Sqrt[d + e* x^2]*(-2*d + 3*e*x^2) + 3*e^2*x^4*Log[x] - 3*e^2*x^4*Log[d + Sqrt[d]*Sqrt[ d + e*x^2]]))/d^(5/2))/(40*x^5)
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6775, 243, 52, 52, 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.
| \(\displaystyle \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 6775 |
| \(\displaystyle \frac {1}{5} \sqrt {e} \int \frac {1}{x^5 \sqrt {e x^2+d}}dx-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{10} \sqrt {e} \int \frac {1}{x^6 \sqrt {e x^2+d}}dx^2-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{10} \sqrt {e} \left (-\frac {3 e \int \frac {1}{x^4 \sqrt {e x^2+d}}dx^2}{4 d}-\frac {\sqrt {d+e x^2}}{2 d x^4}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{10} \sqrt {e} \left (-\frac {3 e \left (-\frac {e \int \frac {1}{x^2 \sqrt {e x^2+d}}dx^2}{2 d}-\frac {\sqrt {d+e x^2}}{d x^2}\right )}{4 d}-\frac {\sqrt {d+e x^2}}{2 d x^4}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{10} \sqrt {e} \left (-\frac {3 e \left (-\frac {\int \frac {1}{\frac {x^4}{e}-\frac {d}{e}}d\sqrt {e x^2+d}}{d}-\frac {\sqrt {d+e x^2}}{d x^2}\right )}{4 d}-\frac {\sqrt {d+e x^2}}{2 d x^4}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{10} \sqrt {e} \left (-\frac {3 e \left (\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {\sqrt {d+e x^2}}{d x^2}\right )}{4 d}-\frac {\sqrt {d+e x^2}}{2 d x^4}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 x^5}\) |
-1/5*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^5 + (Sqrt[e]*(-1/2*Sqrt[d + e* x^2]/(d*x^4) - (3*e*(-(Sqrt[d + e*x^2]/(d*x^2)) + (e*ArcTanh[Sqrt[d + e*x^ 2]/Sqrt[d]])/d^(3/2)))/(4*d)))/10
3.1.15.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[ArcTanh[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_ Symbol] :> Simp[(d*x)^(m + 1)*(ArcTanh[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1))), x] - Simp[c/(d*(m + 1)) Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; Fre eQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(83)=166\).
Time = 0.02 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.54
| method | result | size |
| default | \(-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{5 x^{5}}-\frac {e^{\frac {3}{2}} \left (-\frac {\sqrt {e \,x^{2}+d}}{2 x^{2} d}+\frac {e \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{2 d^{\frac {3}{2}}}\right )}{5 d}+\frac {\sqrt {e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {e \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {e \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{2 d}\right )}{4 d}\right )}{5 d}\) | \(171\) |
| parts | \(-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{5 x^{5}}-\frac {e^{\frac {3}{2}} \left (-\frac {\sqrt {e \,x^{2}+d}}{2 x^{2} d}+\frac {e \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{2 d^{\frac {3}{2}}}\right )}{5 d}+\frac {\sqrt {e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {e \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {e \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{2 d}\right )}{4 d}\right )}{5 d}\) | \(171\) |
-1/5*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^5-1/5*e^(3/2)/d*(-1/2*(e*x^2+d)^ (1/2)/x^2/d+1/2*e/d^(3/2)*ln((2*d+2*d^(1/2)*(e*x^2+d)^(1/2))/x))+1/5*e^(1/ 2)/d*(-1/4/d/x^4*(e*x^2+d)^(3/2)-1/4*e/d*(-1/2/d/x^2*(e*x^2+d)^(3/2)+1/2*e /d*((e*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(e*x^2+d)^(1/2))/x))))
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (83) = 166\).
Time = 0.30 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.45 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx=\left [\frac {3 \, e^{2} x^{5} \sqrt {\frac {e}{d}} \log \left (-\frac {e^{2} x^{2} - 2 \, \sqrt {e x^{2} + d} d \sqrt {e} \sqrt {\frac {e}{d}} + 2 \, d e}{x^{2}}\right ) - 8 \, d^{2} x^{5} \log \left (\frac {e x + \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) + 8 \, d^{2} x^{5} \log \left (\frac {e x - \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) + 2 \, {\left (3 \, e x^{3} - 2 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + 8 \, {\left (d^{2} x^{5} - d^{2}\right )} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{80 \, d^{2} x^{5}}, \frac {3 \, e^{2} x^{5} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {e x^{2} + d} d \sqrt {e} \sqrt {-\frac {e}{d}}}{e^{2} x^{2} + d e}\right ) - 4 \, d^{2} x^{5} \log \left (\frac {e x + \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) + 4 \, d^{2} x^{5} \log \left (\frac {e x - \sqrt {e x^{2} + d} \sqrt {e}}{x}\right ) + {\left (3 \, e x^{3} - 2 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + 4 \, {\left (d^{2} x^{5} - d^{2}\right )} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{40 \, d^{2} x^{5}}\right ] \]
[1/80*(3*e^2*x^5*sqrt(e/d)*log(-(e^2*x^2 - 2*sqrt(e*x^2 + d)*d*sqrt(e)*sqr t(e/d) + 2*d*e)/x^2) - 8*d^2*x^5*log((e*x + sqrt(e*x^2 + d)*sqrt(e))/x) + 8*d^2*x^5*log((e*x - sqrt(e*x^2 + d)*sqrt(e))/x) + 2*(3*e*x^3 - 2*d*x)*sqr t(e*x^2 + d)*sqrt(e) + 8*(d^2*x^5 - d^2)*log((2*e*x^2 + 2*sqrt(e*x^2 + d)* sqrt(e)*x + d)/d))/(d^2*x^5), 1/40*(3*e^2*x^5*sqrt(-e/d)*arctan(sqrt(e*x^2 + d)*d*sqrt(e)*sqrt(-e/d)/(e^2*x^2 + d*e)) - 4*d^2*x^5*log((e*x + sqrt(e* x^2 + d)*sqrt(e))/x) + 4*d^2*x^5*log((e*x - sqrt(e*x^2 + d)*sqrt(e))/x) + (3*e*x^3 - 2*d*x)*sqrt(e*x^2 + d)*sqrt(e) + 4*(d^2*x^5 - d^2)*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d))/(d^2*x^5)]
\[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx=\int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{6}}\, dx \]
\[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx=\int { \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{6}} \,d x } \]
d*sqrt(e)*integrate(-1/5*sqrt(e*x^2 + d)/(e^2*x^9 + d*e*x^7 - (e*x^7 + d*x ^5)*(e*x^2 + d)), x) - 1/10*(log(sqrt(e)*x + sqrt(e*x^2 + d)) - log(-sqrt( e)*x + sqrt(e*x^2 + d)))/x^5
\[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx=\int { \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^6} \, dx=\int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^6} \,d x \]