Integrand size = 23, antiderivative size = 79 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=-\frac {\sqrt {e} \sqrt {d+e x^2}}{12 d x^3}+\frac {e^{3/2} \sqrt {d+e x^2}}{6 d^2 x}-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 x^4} \] Output:
-1/12*e^(1/2)*(e*x^2+d)^(1/2)/d/x^3+1/6*e^(3/2)*(e*x^2+d)^(1/2)/d^2/x-1/4* arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/x^4
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=\frac {\sqrt {e} x \sqrt {d+e x^2} \left (-d+2 e x^2\right )-3 d^2 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{12 d^2 x^4} \] Input:
Integrate[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^5,x]
Output:
(Sqrt[e]*x*Sqrt[d + e*x^2]*(-d + 2*e*x^2) - 3*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt [d + e*x^2]])/(12*d^2*x^4)
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6775, 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.
\(\displaystyle \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 6775 |
\(\displaystyle \frac {1}{4} \sqrt {e} \int \frac {1}{x^4 \sqrt {e x^2+d}}dx-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 x^4}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle \frac {1}{4} \sqrt {e} \left (-\frac {2 e \int \frac {1}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 x^4}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle \frac {1}{4} \sqrt {e} \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 x^4}\) |
Input:
Int[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^5,x]
Output:
(Sqrt[e]*(-1/3*Sqrt[d + e*x^2]/(d*x^3) + (2*e*Sqrt[d + e*x^2])/(3*d^2*x))) /4 - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(4*x^4)
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 = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{4 x^{4}}+\frac {e^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{4 d^{2} x}-\frac {\sqrt {e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{12 d^{2} x^{3}}\) | \(62\) |
parts | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{4 x^{4}}+\frac {e^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{4 d^{2} x}-\frac {\sqrt {e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{12 d^{2} x^{3}}\) | \(62\) |
Input:
int(arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/x^5,x,method=_RETURNVERBOSE)
Output:
-1/4*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/x^4+1/4*e^(3/2)*(e*x^2+d)^(1/2)/d^ 2/x-1/12*e^(1/2)/d^2/x^3*(e*x^2+d)^(3/2)
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=-\frac {3 \, d^{2} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) - 2 \, {\left (2 \, e x^{3} - d x\right )} \sqrt {e x^{2} + d} \sqrt {e}}{24 \, d^{2} x^{4}} \] Input:
integrate(arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/x^5,x, algorithm="fricas")
Output:
-1/24*(3*d^2*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) - 2*(2*e*x ^3 - d*x)*sqrt(e*x^2 + d)*sqrt(e))/(d^2*x^4)
\[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=\int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x^{5}}\, dx \] Input:
integrate(atanh(e**(1/2)*x/(e*x**2+d)**(1/2))/x**5,x)
Output:
Integral(atanh(sqrt(e)*x/sqrt(d + e*x**2))/x**5, x)
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.77 \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=\frac {\sqrt {e x^{2} + d} e^{\frac {3}{2}}}{4 \, d^{2} x} - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} \sqrt {e}}{12 \, d^{2} x^{3}} - \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{4 \, x^{4}} \] Input:
integrate(arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/x^5,x, algorithm="maxima")
Output:
1/4*sqrt(e*x^2 + d)*e^(3/2)/(d^2*x) - 1/12*(e*x^2 + d)^(3/2)*sqrt(e)/(d^2* x^3) - 1/4*arctanh(sqrt(e)*x/sqrt(e*x^2 + d))/x^4
Timed out. \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=\text {Timed out} \] Input:
integrate(arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/x^5,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=\int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^5} \,d x \] Input:
int(atanh((e^(1/2)*x)/(d + e*x^2)^(1/2))/x^5,x)
Output:
int(atanh((e^(1/2)*x)/(d + e*x^2)^(1/2))/x^5, x)
\[ \int \frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^5} \, dx=\int \frac {\mathit {atanh} \left (\frac {\sqrt {e}\, x}{\sqrt {e \,x^{2}+d}}\right )}{x^{5}}d x \] Input:
int(atanh(e^(1/2)*x/(e*x^2+d)^(1/2))/x^5,x)
Output:
int(atanh((sqrt(e)*x)/sqrt(d + e*x**2))/x**5,x)