Integrand size = 22, antiderivative size = 144 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {5 \sqrt {1-\frac {1}{a x}}}{3 a c \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{3 a c \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {1-\frac {1}{a x}} x}{c \left (1+\frac {1}{a x}\right )^{3/2}}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c} \]
-3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c+5/3*(1-1/a/x)^(1/2)/a/c/(1 +1/a/x)^(3/2)+x*(1-1/a/x)^(1/2)/c/(1+1/a/x)^(3/2)+14/3*(1-1/a/x)^(1/2)/a/c /(1+1/a/x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.48 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (14+19 a x+3 a^2 x^2\right )}{(1+a x)^2}-\frac {9 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{3 c} \]
((Sqrt[1 - 1/(a^2*x^2)]*x*(14 + 19*a*x + 3*a^2*x^2))/(1 + a*x)^2 - (9*Log[ (1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/(3*c)
Time = 0.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6748, 110, 25, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 6748 |
| \(\displaystyle -\frac {\int \frac {\sqrt {1-\frac {1}{a x}} x^2}{\left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle -\frac {\int -\frac {\left (3 a-\frac {2}{x}\right ) x}{a^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\int \frac {\left (3 a-\frac {2}{x}\right ) x}{a^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\left (3 a-\frac {2}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}d\frac {1}{x}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} a \int \frac {\left (9 a-\frac {5}{x}\right ) x}{a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \int \frac {\left (9 a-\frac {5}{x}\right ) x}{\sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (a \int \frac {9 x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {14 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (9 a \int \frac {x}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {14 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (\frac {14 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-9 \int \frac {1}{\frac {1}{a}-\frac {1}{a x^2}}d\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {1}{3} \left (\frac {14 a \sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-9 a \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )\right )+\frac {5 a \sqrt {1-\frac {1}{a x}}}{3 \left (\frac {1}{a x}+1\right )^{3/2}}}{a^2}-\frac {x \sqrt {1-\frac {1}{a x}}}{\left (\frac {1}{a x}+1\right )^{3/2}}}{c}\) |
-((-((Sqrt[1 - 1/(a*x)]*x)/(1 + 1/(a*x))^(3/2)) - ((5*a*Sqrt[1 - 1/(a*x)]) /(3*(1 + 1/(a*x))^(3/2)) + ((14*a*Sqrt[1 - 1/(a*x)])/Sqrt[1 + 1/(a*x)] - 9 *a*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/3)/a^2)/c)
3.9.26.3.1 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[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[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 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_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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[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 = 0.15 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}{a c}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {2 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{3 a^{5} \left (x +\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{3 a^{4} \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \left (a x -1\right )}\) | \(173\) |
| default | \(-\frac {\left (-9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+9 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+6 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -27 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+5 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-27 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x +27 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+9 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 a \left (a x +1\right ) \sqrt {a^{2}}\, c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )}\) | \(346\) |
1/a*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)/c+(-3/a^2*ln(a^2*x/(a^2)^(1/2)+(a^2*x^ 2-1)^(1/2))/(a^2)^(1/2)-2/3/a^5/(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2 )+13/3/a^4/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/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 = 96, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {9 \, {\left (a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (3 \, a^{2} x^{2} + 19 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} c x + a c\right )}} \]
-1/3*(9*(a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 9*(a*x + 1)*log(sqr t((a*x - 1)/(a*x + 1)) - 1) - (3*a^2*x^2 + 19*a*x + 14)*sqrt((a*x - 1)/(a* x + 1)))/(a^2*c*x + a*c)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {a^{2} \left (\int \left (- \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\right )\, dx + \int \frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx\right )}{c} \]
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
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {1}{3} \, a {\left (\frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} - \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 12 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
-1/3*a*(6*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*a^2*c/(a*x + 1) - a^2*c) - (((a*x - 1)/(a*x + 1))^(3/2) + 12*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c) + 9*l og(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 9*log(sqrt((a*x - 1)/(a*x + 1) ) - 1)/(a^2*c))
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {3 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{c {\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a c} \]
3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/(c*abs(a)) + sqrt(a ^2*x^2 - 1)*sgn(a*x + 1)/(a*c)
Time = 3.92 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c-\frac {a\,c\,\left (a\,x-1\right )}{a\,x+1}}+\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3\,a\,c}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c} \]