Integrand size = 23, antiderivative size = 105 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=-\frac {\sqrt {e} \sqrt {d+e x^2}}{30 d x^5}+\frac {2 e^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {4 e^{5/2} \sqrt {d+e x^2}}{45 d^3 x}-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6} \]
-1/6*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^6+2/45*e^(3/2)*(e*x^2+d)^(1/2)/d ^2/x^3-4/45*e^(5/2)*(e*x^2+d)^(1/2)/d^3/x-1/30*e^(1/2)*(e*x^2+d)^(1/2)/d/x ^5
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=\frac {\sqrt {e} x \sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )-15 d^3 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{90 d^3 x^6} \]
(Sqrt[e]*x*Sqrt[d + e*x^2]*(-3*d^2 + 4*d*e*x^2 - 8*e^2*x^4) - 15*d^3*ArcTa nh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(90*d^3*x^6)
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6775, 245, 245, 242}
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.
\(\displaystyle \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx\) |
\(\Big \downarrow \) 6775 |
\(\displaystyle \frac {1}{6} \sqrt {e} \int \frac {1}{x^6 \sqrt {e x^2+d}}dx-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {1}{6} \sqrt {e} \left (-\frac {4 e \int \frac {1}{x^4 \sqrt {e x^2+d}}dx}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {1}{6} \sqrt {e} \left (-\frac {4 e \left (-\frac {2 e \int \frac {1}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle \frac {1}{6} \sqrt {e} \left (-\frac {4 e \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\) |
(Sqrt[e]*(-1/5*Sqrt[d + e*x^2]/(d*x^5) - (4*e*(-1/3*Sqrt[d + e*x^2]/(d*x^3 ) + (2*e*Sqrt[d + e*x^2])/(3*d^2*x)))/(5*d)))/6 - ArcTanh[(Sqrt[e]*x)/Sqrt [d + e*x^2]]/(6*x^6)
3.1.7.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
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]
Time = 0.01 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{6 x^{6}}-\frac {e^{\frac {3}{2}} \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{6 d}+\frac {\sqrt {e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{6 d}\) | \(110\) |
parts | \(-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{6 x^{6}}-\frac {e^{\frac {3}{2}} \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{6 d}+\frac {\sqrt {e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}\right )}{6 d}\) | \(110\) |
-1/6*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^6-1/6*e^(3/2)/d*(-1/3/d/x^3*(e*x ^2+d)^(1/2)+2/3*e/d^2/x*(e*x^2+d)^(1/2))+1/6*e^(1/2)/d*(-1/5/d/x^5*(e*x^2+ d)^(3/2)+2/15*e/d^2/x^3*(e*x^2+d)^(3/2))
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=-\frac {15 \, d^{3} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) + 2 \, {\left (8 \, e^{2} x^{5} - 4 \, d e x^{3} + 3 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {e}}{180 \, d^{3} x^{6}} \]
-1/180*(15*d^3*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) + 2*(8*e ^2*x^5 - 4*d*e*x^3 + 3*d^2*x)*sqrt(e*x^2 + d)*sqrt(e))/(d^3*x^6)
\[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=\int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{7}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=-\frac {{\left (2 \, e^{2} x^{4} + d e x^{2} - d^{2}\right )} e^{\frac {3}{2}}}{18 \, \sqrt {e x^{2} + d} d^{3} x^{3}} - \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{6 \, x^{6}} + \frac {{\left (2 \, e^{2} x^{4} - d e x^{2} - 3 \, d^{2}\right )} \sqrt {e x^{2} + d} \sqrt {e}}{90 \, d^{3} x^{5}} \]
-1/18*(2*e^2*x^4 + d*e*x^2 - d^2)*e^(3/2)/(sqrt(e*x^2 + d)*d^3*x^3) - 1/6* arctanh(sqrt(e)*x/sqrt(e*x^2 + d))/x^6 + 1/90*(2*e^2*x^4 - d*e*x^2 - 3*d^2 )*sqrt(e*x^2 + d)*sqrt(e)/(d^3*x^5)
\[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=\int { \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx=\int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^7} \,d x \]