Integrand size = 22, antiderivative size = 108 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx=-\frac {1}{2 c x^2}-\frac {\sqrt {1-c x}}{2 c x^2 \sqrt {\frac {1}{1+c x}}}-\frac {1}{2} c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c x} \sqrt {1+c x}\right )+c \log (x)-\frac {1}{2} c \log \left (1-c^2 x^2\right ) \]
-1/2/c/x^2+c*ln(x)-1/2*c*ln(-c^2*x^2+1)-1/2*(-c*x+1)^(1/2)/c/x^2/(1/(c*x+1 ))^(1/2)-1/2*c*arctanh((-c*x+1)^(1/2)*(c*x+1)^(1/2))*(1/(c*x+1))^(1/2)*(c* x+1)^(1/2)
Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx=\frac {1}{2} \left (-\frac {1}{c x^2}-\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x^2}+3 c \log (x)-c \log \left (1-c^2 x^2\right )-c \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )\right ) \]
(-(1/(c*x^2)) - (Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(c*x^2) + 3*c*Log[x] - c*Log[1 - c^2*x^2] - c*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/2
Time = 0.44 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6895, 243, 54, 2009, 2044, 114, 25, 27, 103, 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.
| \(\displaystyle \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6895 |
| \(\displaystyle \frac {\int \frac {1}{x^3 \left (1-c^2 x^2\right )}dx}{c}+\frac {\int \frac {\sqrt {\frac {1}{c x+1}}}{x^3 \sqrt {1-c x}}dx}{c}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\int \frac {1}{x^4 \left (1-c^2 x^2\right )}dx^2}{2 c}+\frac {\int \frac {\sqrt {\frac {1}{c x+1}}}{x^3 \sqrt {1-c x}}dx}{c}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\int \left (-\frac {c^4}{c^2 x^2-1}+\frac {c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{2 c}+\frac {\int \frac {\sqrt {\frac {1}{c x+1}}}{x^3 \sqrt {1-c x}}dx}{c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {\sqrt {\frac {1}{c x+1}}}{x^3 \sqrt {1-c x}}dx}{c}+\frac {c^2 \log \left (x^2\right )-c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}}{2 c}\) |
| \(\Big \downarrow \) 2044 |
| \(\displaystyle \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x^3 \sqrt {1-c x} \sqrt {c x+1}}dx}{c}+\frac {c^2 \log \left (x^2\right )-c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}}{2 c}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {1}{2} \int -\frac {c^2}{x \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )}{c}+\frac {c^2 \log \left (x^2\right )-c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}}{2 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{2} \int \frac {c^2}{x \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )}{c}+\frac {c^2 \log \left (x^2\right )-c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{2} c^2 \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1}}dx-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )}{c}+\frac {c^2 \log \left (x^2\right )-c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}}{2 c}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {1}{2} c^3 \int \frac {1}{c-c (1-c x) (c x+1)}d\left (\sqrt {1-c x} \sqrt {c x+1}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )}{c}+\frac {c^2 \log \left (x^2\right )-c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}}{2 c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {1}{2} c^2 \text {arctanh}\left (\sqrt {1-c x} \sqrt {c x+1}\right )-\frac {\sqrt {1-c x} \sqrt {c x+1}}{2 x^2}\right )}{c}+\frac {c^2 \log \left (x^2\right )-c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}}{2 c}\) |
(Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(-1/2*(Sqrt[1 - c*x]*Sqrt[1 + c*x])/x^ 2 - (c^2*ArcTanh[Sqrt[1 - c*x]*Sqrt[1 + c*x]])/2))/c + (-x^(-2) + c^2*Log[ x^2] - c^2*Log[1 - c^2*x^2])/(2*c)
3.1.95.3.1 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] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
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[(u_.)*((c_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[S imp[(c*(a + b*x^n)^q)^p/(a + b*x^n)^(p*q)] Int[u*(a + b*x^n)^(p*q), x], x ] /; FreeQ[{a, b, c, n, p, q}, x] && GeQ[a, 0]
Int[(E^ArcSech[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Sym bol] :> Simp[d/(a*c) Int[(d*x)^(m - 1)*(Sqrt[1/(1 + c*x)]/Sqrt[1 - c*x]), x], x] + Simp[d/c Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b + a*c^2, 0]
Time = 0.73 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10
| method | result | size |
| default | \(-\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{2 x \sqrt {-c^{2} x^{2}+1}}+\frac {-\frac {c^{2} \ln \left (c x +1\right )}{2}-\frac {1}{2 x^{2}}+c^{2} \ln \left (x \right )-\frac {c^{2} \ln \left (c x -1\right )}{2}}{c}\) | \(119\) |
-1/2*(-(c*x-1)/c/x)^(1/2)/x*((c*x+1)/c/x)^(1/2)*(arctanh(1/(-c^2*x^2+1)^(1 /2))*c^2*x^2+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)+1/c*(-1/2*c^2*ln(c*x+1 )-1/2/x^2+c^2*ln(x)-1/2*c^2*ln(c*x-1))
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (70) = 140\).
Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx=-\frac {2 \, c^{2} x^{2} \log \left (c^{2} x^{2} - 1\right ) + c^{2} x^{2} \log \left (c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + 1\right ) - c^{2} x^{2} \log \left (c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} - 1\right ) - 4 \, c^{2} x^{2} \log \left (x\right ) + 2 \, c x \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} + 2}{4 \, c x^{2}} \]
-1/4*(2*c^2*x^2*log(c^2*x^2 - 1) + c^2*x^2*log(c*x*sqrt((c*x + 1)/(c*x))*s qrt(-(c*x - 1)/(c*x)) + 1) - c^2*x^2*log(c*x*sqrt((c*x + 1)/(c*x))*sqrt(-( c*x - 1)/(c*x)) - 1) - 4*c^2*x^2*log(x) + 2*c*x*sqrt((c*x + 1)/(c*x))*sqrt (-(c*x - 1)/(c*x)) + 2)/(c*x^2)
\[ \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx=- \frac {\int \frac {c x \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {1}{c^{2} x^{5} - x^{3}}\, dx}{c} \]
-(Integral(c*x*sqrt(-1 + 1/(c*x))*sqrt(1 + 1/(c*x))/(c**2*x**5 - x**3), x) + Integral(1/(c**2*x**5 - x**3), x))/c
\[ \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx=\int { -\frac {\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}}{{\left (c^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
c*integrate(1/x, x) - 1/2*c*log(c*x + 1) - 1/2*c*log(c*x - 1) + integrate( x^(-3), x)/c - integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^3*x^5 - c*x^3), x )
\[ \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx=\int { -\frac {\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}}{{\left (c^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
Time = 17.89 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.06 \[ \int \frac {e^{\text {sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx=c\,\ln \left (x\right )+\frac {\frac {2\,c\,\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{c\,x}+1}-1}+\frac {14\,c\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^3}+\frac {14\,c\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^5}+\frac {2\,c\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^7}}{1+\frac {6\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^6}+\frac {{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{c\,x}+1}-1\right )}^2}}-\frac {c\,\ln \left (c^2\,x^2-1\right )}{2}-2\,c\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{c\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{c\,x}+1}-1}\right )-\frac {1}{2\,c\,x^2} \]
((2*c*((1/(c*x) - 1)^(1/2) - 1i))/((1/(c*x) + 1)^(1/2) - 1) + (14*c*((1/(c *x) - 1)^(1/2) - 1i)^3)/((1/(c*x) + 1)^(1/2) - 1)^3 + (14*c*((1/(c*x) - 1) ^(1/2) - 1i)^5)/((1/(c*x) + 1)^(1/2) - 1)^5 + (2*c*((1/(c*x) - 1)^(1/2) - 1i)^7)/((1/(c*x) + 1)^(1/2) - 1)^7)/((6*((1/(c*x) - 1)^(1/2) - 1i)^4)/((1/ (c*x) + 1)^(1/2) - 1)^4 - (4*((1/(c*x) - 1)^(1/2) - 1i)^2)/((1/(c*x) + 1)^ (1/2) - 1)^2 - (4*((1/(c*x) - 1)^(1/2) - 1i)^6)/((1/(c*x) + 1)^(1/2) - 1)^ 6 + ((1/(c*x) - 1)^(1/2) - 1i)^8/((1/(c*x) + 1)^(1/2) - 1)^8 + 1) - 2*c*at anh(((1/(c*x) - 1)^(1/2) - 1i)/((1/(c*x) + 1)^(1/2) - 1)) - (c*log(c^2*x^2 - 1))/2 + c*log(x) - 1/(2*c*x^2)