Integrand size = 22, antiderivative size = 71 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \] Output:
(a-1/x)/a^2/c^2/(1-1/a^2/x^2)^(1/2)+(1-1/a^2/x^2)^(1/2)*x/c^2-arctanh((1-1 /a^2/x^2)^(1/2))/a/c^2
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {-2+a x+a^2 x^2-a \sqrt {1-\frac {1}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \] Input:
Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^2),x]
Output:
(-2 + a*x + a^2*x^2 - a*Sqrt[1 - 1/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - 1/(a^2*x^ 2)]])/(a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.53 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6731, 27, 528, 27, 534, 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.
\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -\frac {\int \frac {c \left (a-\frac {1}{x}\right ) x^2}{a \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\left (a-\frac {1}{x}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a c^2}\) |
\(\Big \downarrow \) 528 |
\(\displaystyle -\frac {a^2 \int \frac {\left (a-\frac {1}{x}\right ) x^2}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a-\frac {1}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}}{a c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\left (a-\frac {1}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a-\frac {1}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}}{a c^2}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -\frac {-\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a-\frac {1}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-a x \sqrt {1-\frac {1}{a^2 x^2}}}{a c^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {-\frac {1}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-\frac {a-\frac {1}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-a x \sqrt {1-\frac {1}{a^2 x^2}}}{a c^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a-\frac {1}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}}{a c^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {a-\frac {1}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-a x \sqrt {1-\frac {1}{a^2 x^2}}}{a c^2}\) |
Input:
Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^2),x]
Output:
-((-((a - x^(-1))/(a*Sqrt[1 - 1/(a^2*x^2)])) - a*Sqrt[1 - 1/(a^2*x^2)]*x + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^2))
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)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(65)=130\).
Time = 0.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.94
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2}}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{4} \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{2} \left (a x -1\right )}\) | \(138\) |
default | \(-\frac {\left (-3 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +4 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -3 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+2 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a \sqrt {a^{2}}\, c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(250\) |
Input:
int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x+1)/c^2*((a*x-1)/(a*x+1))^(1/2)+(-1/a^2*ln(a^2*x/(a^2)^(1/2)+(a^2* x^2-1)^(1/2))/(a^2)^(1/2)+1/a^4/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)) *a^2/c^2*((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {{\left (a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}} - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c^{2}} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x, algorithm="fricas")
Output:
((a*x + 2)*sqrt((a*x - 1)/(a*x + 1)) - log(sqrt((a*x - 1)/(a*x + 1)) + 1) + log(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a*c^2)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a^{2} \left (\int \left (- \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\right )\, dx + \int \frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx\right )}{c^{2}} \] Input:
integrate(((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**2,x)
Output:
a**2*(Integral(-x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**3*x**3 - a**2*x **2 - a*x + 1), x) + Integral(a*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a* *3*x**3 - a**2*x**2 - a*x + 1), x))/c**2
Time = 0.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-a {\left (\frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c^{2}}{a x + 1} - a^{2} c^{2}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}} - \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x, algorithm="maxima")
Output:
-a*(2*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2*c^2/(a*x + 1) - a^2*c^2) + log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^2) - log(sqrt((a*x - 1)/(a*x + 1 )) - 1)/(a^2*c^2) - sqrt((a*x - 1)/(a*x + 1))/(a^2*c^2))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a x}\right )}^{2}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x, algorithm="giac")
Output:
undef
Time = 13.64 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.27 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2-\frac {a\,c^2\,\left (a\,x-1\right )}{a\,x+1}}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2} \] Input:
int(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a*x))^2,x)
Output:
(2*((a*x - 1)/(a*x + 1))^(1/2))/(a*c^2 - (a*c^2*(a*x - 1))/(a*x + 1)) + (( a*x - 1)/(a*x + 1))^(1/2)/(a*c^2) - (2*atanh(((a*x - 1)/(a*x + 1))^(1/2))) /(a*c^2)
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.34 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +4 \sqrt {a x +1}\, \sqrt {a x -1}-4 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -4 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+3 a x +3}{2 a \,c^{2} \left (a x +1\right )} \] Input:
int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^2,x)
Output:
(2*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x + 4*sqrt(a*x + 1)*sqrt(a*x - 1) - 4*log ((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - 4*log((sqrt(a*x - 1) + sqr t(a*x + 1))/sqrt(2)) + 3*a*x + 3)/(2*a*c**2*(a*x + 1))