Integrand size = 20, antiderivative size = 104 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {2 \sqrt {1+\frac {1}{a x}}}{a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \sqrt {1-\frac {1}{a x}}}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c} \]
arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c-2*(1+1/a/x)^(1/2)/a/c/(1-1/a/ x)^(1/2)+x*(1+1/a/x)^(1/2)/c/(1-1/a/x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {\frac {\sqrt {1-\frac {1}{a^2 x^2}} x (-2+a x)}{-1+a x}+\frac {\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{c} \]
Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6748, 114, 25, 27, 35, 105, 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.
\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6748 |
\(\displaystyle -\frac {\int \frac {x^2}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{c}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {-\int -\frac {\left (a+\frac {1}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {x \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right ) x}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {x \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\int \frac {\left (a+\frac {1}{x}\right ) x}{\left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a^2}-\frac {x \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{c}\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -\frac {\frac {\int \frac {\sqrt {1+\frac {1}{a x}} x}{\left (1-\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{c}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\frac {\int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {2 \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{c}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle -\frac {\frac {\frac {2 \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}-\frac {\int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\frac {\frac {2 \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}-\text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a}-\frac {x \sqrt {\frac {1}{a x}+1}}{\sqrt {1-\frac {1}{a x}}}}{c}\) |
-((-((Sqrt[1 + 1/(a*x)]*x)/Sqrt[1 - 1/(a*x)]) + ((2*Sqrt[1 + 1/(a*x)])/Sqr t[1 - 1/(a*x)] - ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a)/c)
3.8.76.3.1 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[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[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 = 0.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.38
method | result | size |
risch | \(\frac {a x -1}{a c \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(144\) |
default | \(-\frac {-3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{2 a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\) | \(250\) |
1/a*(a*x-1)/c/((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)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^ 2/c/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {{\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]
((a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - (a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (a^2*x^2 - a*x - 2)*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c *x - a*c)
\[ \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {a^{2} \int \frac {x^{2}}{a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c} \]
a**2*Integral(x**2/(a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - sqrt(a*x /(a*x + 1) - 1/(a*x + 1))), x)/c
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-a {\left (\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
-a*((3*(a*x - 1)/(a*x + 1) - 1)/(a^2*c*((a*x - 1)/(a*x + 1))^(3/2) - a^2*c *sqrt((a*x - 1)/(a*x + 1))) - log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) + log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c))
\[ \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {2\,a\,x+4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}-4}{2\,a\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]