Integrand size = 20, antiderivative size = 108 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {2 c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{a}+c \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x+\frac {c \csc ^{-1}(a x)}{a}-\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]
c*arccsc(a*x)/a-c*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a+c*(1-1/a/x)^( 3/2)*x*(1+1/a/x)^(1/2)+2*c*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)/a
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.51 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (\sqrt {1-\frac {1}{a^2 x^2}} (1+a x)+\arcsin \left (\frac {1}{a x}\right )-\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{a} \]
(c*(Sqrt[1 - 1/(a^2*x^2)]*(1 + a*x) + ArcSin[1/(a*x)] - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6748, 108, 25, 27, 171, 140, 39, 103, 221, 223}
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.
| \(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right ) e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -c \int \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -c \left (\int -\frac {\left (a+\frac {2}{x}\right ) \sqrt {1-\frac {1}{a x}} x}{a^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-x \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\int \frac {\left (a+\frac {2}{x}\right ) \sqrt {1-\frac {1}{a x}} x}{a^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\frac {\int \frac {\left (a+\frac {2}{x}\right ) \sqrt {1-\frac {1}{a x}} x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a^2}\right )\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\frac {a \int \frac {\sqrt {1+\frac {1}{a x}} x}{\sqrt {1-\frac {1}{a x}}}d\frac {1}{x}+2 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{a^2}\right )\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\frac {a \left (\frac {\int \frac {1}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}+\int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )+2 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{a^2}\right )\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\frac {a \left (\frac {\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}+\int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )+2 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{a^2}\right )\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\frac {a \left (\frac {\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}-\frac {\int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\right )+2 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{a^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\frac {a \left (\frac {\int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{a}-\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+2 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{a^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -c \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (1-\frac {1}{a x}\right )^{3/2}-\frac {a \left (\arcsin \left (\frac {1}{a x}\right )-\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+2 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{a^2}\right )\) |
-(c*(-((1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x) - (2*a*Sqrt[1 - 1/(a*x)]*S qrt[1 + 1/(a*x)] + a*(ArcSin[1/(a*x)] - ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]))/a^2))
3.9.9.3.1 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_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
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[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^2), x], x , 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[ n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2]
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{x \,a^{2}}+\frac {\left (-\frac {a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\sqrt {\left (a x -1\right ) \left (a x +1\right )}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \left (a x -1\right )}\) | \(127\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\) | \(166\) |
(a*x+1)/x*c/a^2*((a*x-1)/(a*x+1))^(1/2)+1/a*(-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)+arctan(1/(a^2*x^2-1)^(1/ 2)))*c*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x-1)
Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {2 \, a c x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \]
-(2*a*c*x*arctan(sqrt((a*x - 1)/(a*x + 1))) + a*c*x*log(sqrt((a*x - 1)/(a* x + 1)) + 1) - a*c*x*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (a^2*c*x^2 + 2*a *c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*x)
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (\int a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\right )\, dx\right )}{a^{2}} \]
c*(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))/a**2
Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-a {\left (\frac {4 \, c \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac {2 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
-a*(4*c*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)^2*a^2/(a*x + 1)^2 - a^2) + 2* c*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + c*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - c*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {2 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {2 \, c \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |}} \]
-2*c*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + c*log(abs(-x*a bs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1)*c*sgn( a*x + 1)/a + 2*c*sgn(a*x + 1)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)*abs( a))
Time = 3.87 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.78 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {4\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}}-\frac {2\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]