Integrand size = 22, antiderivative size = 77 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+c^2 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3 c^2 \csc ^{-1}(a x)}{a}-\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
-3*c^2*arccsc(a*x)/a-3*c^2*arctanh((1-1/a^2/x^2)^(1/2))/a-c^2*(1-1/a^2/x^2 )^(1/2)/a+c^2*x*(1-1/a^2/x^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^2 \left (\sqrt {1-\frac {1}{a^2 x^2}} (-1+a x)-3 \arcsin \left (\frac {1}{a x}\right )-3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a} \]
(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(-1 + a*x) - 3*ArcSin[1/(a*x)] - 3*ArcTanh[Sqr t[1 - 1/(a^2*x^2)]]))/a
Time = 0.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6731, 27, 540, 2340, 27, 538, 223, 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.
| \(\displaystyle \int \left (c-\frac {c}{a x}\right )^2 e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {c^3 \left (a-\frac {1}{x}\right )^3 x^2}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \int \frac {\left (a-\frac {1}{x}\right )^3 x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a^3}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {c^2 \left (a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )-\int \frac {\left (3 a^2-\frac {3 a}{x}+\frac {1}{x^2}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )}{a^3}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle -\frac {c^2 \left (a^2 \int -\frac {3 \left (a-\frac {1}{x}\right ) x}{a \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+a^2 \sqrt {1-\frac {1}{a^2 x^2}}+a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \left (-3 a \int \frac {\left (a-\frac {1}{x}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+a^2 \sqrt {1-\frac {1}{a^2 x^2}}+a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {c^2 \left (-3 a \left (a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}}+a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {c^2 \left (-3 a \left (a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a \arcsin \left (\frac {1}{a x}\right )\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}}+a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {c^2 \left (-3 a \left (\frac {1}{2} a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a \arcsin \left (\frac {1}{a x}\right )\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}}+a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^2 \left (-3 a \left (a^3 \left (-\int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )-a \arcsin \left (\frac {1}{a x}\right )\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}}+a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^2 \left (-3 a \left (-a \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a \arcsin \left (\frac {1}{a x}\right )\right )+a^2 \sqrt {1-\frac {1}{a^2 x^2}}+a^3 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^3}\) |
-((c^2*(a^2*Sqrt[1 - 1/(a^2*x^2)] - a^3*Sqrt[1 - 1/(a^2*x^2)]*x - 3*a*(-(a *ArcSin[1/(a*x)]) - a*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])))/a^3)
3.5.15.3.1 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] :> 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
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]
Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.74
| method | result | size |
| risch | \(-\frac {\left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{x \,a^{2}}+\frac {\left (-\frac {3 a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}-3 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \left (a x -1\right )}\) | \(134\) |
| default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{2} \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -3 a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+4 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\) | \(227\) |
-(a*x+1)/x*c^2/a^2*((a*x-1)/(a*x+1))^(1/2)+1/a*(-3*a*ln(a^2*x/(a^2)^(1/2)+ (a^2*x^2-1)^(1/2))/(a^2)^(1/2)+((a*x-1)*(a*x+1))^(1/2)-3*arctan(1/(a^2*x^2 -1)^(1/2)))*c^2/(a*x-1)*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {6 \, a c^{2} x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 3 \, a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, a c^{2} x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c^{2} x^{2} - c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]
(6*a*c^2*x*arctan(sqrt((a*x - 1)/(a*x + 1))) - 3*a*c^2*x*log(sqrt((a*x - 1 )/(a*x + 1)) + 1) + 3*a*c^2*x*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (a^2*c^ 2*x^2 - c^2)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*x)
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {c^{2} \left (\int a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\, dx + \int \left (- \frac {2 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x}\right )\, dx\right )}{a^{2}} \]
c**2*(Integral(a**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(sqrt( a*x/(a*x + 1) - 1/(a*x + 1))/x**2, x) + Integral(-2*a*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/x, x))/a**2
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.64 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=-{\left (\frac {4 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac {6 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \]
-(4*c^2*((a*x - 1)/(a*x + 1))^(3/2)/((a*x - 1)^2*a^2/(a*x + 1)^2 - a^2) - 6*c^2*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 3*c^2*log(sqrt((a*x - 1)/(a* x + 1)) + 1)/a^2 - 3*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)*a
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.69 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {6 \, c^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {3 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{2} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {2 \, c^{2} \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |}} \]
6*c^2*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + 3*c^2*log(abs (-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1)*c ^2*sgn(a*x + 1)/a - 2*c^2*sgn(a*x + 1)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)*abs(a))
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.17 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx=\frac {4\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}+\frac {6\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {6\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]