Integrand size = 20, antiderivative size = 104 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {6 \sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac {3 \arcsin (a x)}{a c^2} \]
1/3*(-a^2*x^2+1)^(3/2)/a/c^2/(-a*x+1)^3+(-a^2*x^2+1)^(3/2)/a/c^2/(-a*x+1)^ 2+3*arcsin(a*x)/a/c^2-6*(-a^2*x^2+1)^(1/2)/a/c^2/(-a*x+1)
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {\frac {\sqrt {1+a x} \left (-14+19 a x-3 a^2 x^2\right )}{(1-a x)^{3/2}}-18 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{3 a c^2} \]
((Sqrt[1 + a*x]*(-14 + 19*a*x - 3*a^2*x^2))/(1 - a*x)^(3/2) - 18*ArcSin[Sq rt[1 - a*x]/Sqrt[2]])/(3*a*c^2)
Time = 0.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6681, 6678, 581, 25, 671, 463, 25, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\) |
\(\Big \downarrow \) 6681 |
\(\displaystyle \frac {a^2 \int \frac {e^{\text {arctanh}(a x)} x^2}{(1-a x)^2}dx}{c^2}\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle \frac {a^2 \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^3}dx}{c^2}\) |
\(\Big \downarrow \) 581 |
\(\displaystyle \frac {a^2 \left (\frac {\int -\frac {(2-3 a x) \sqrt {1-a^2 x^2}}{(1-a x)^3}dx}{a^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^2}\right )}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {a^2 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^2}-\frac {\int \frac {(2-3 a x) \sqrt {1-a^2 x^2}}{(1-a x)^3}dx}{a^2}\right )}{c^2}\) |
\(\Big \downarrow \) 671 |
\(\displaystyle \frac {a^2 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^2}-\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^2}dx-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a (1-a x)^3}}{a^2}\right )}{c^2}\) |
\(\Big \downarrow \) 463 |
\(\displaystyle \frac {a^2 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^2}-\frac {3 \left (\int -\frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {2 \sqrt {1-a^2 x^2}}{a (1-a x)}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a (1-a x)^3}}{a^2}\right )}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {a^2 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^2}-\frac {3 \left (\frac {2 \sqrt {1-a^2 x^2}}{a (1-a x)}-\int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a (1-a x)^3}}{a^2}\right )}{c^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {a^2 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^2}-\frac {3 \left (\frac {2 \sqrt {1-a^2 x^2}}{a (1-a x)}-\frac {\arcsin (a x)}{a}\right )-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a (1-a x)^3}}{a^2}\right )}{c^2}\) |
(a^2*((1 - a^2*x^2)^(3/2)/(a^3*(1 - a*x)^2) - (-1/3*(1 - a^2*x^2)^(3/2)/(a *(1 - a*x)^3) + 3*((2*Sqrt[1 - a^2*x^2])/(a*(1 - a*x)) - ArcSin[a*x]/a))/a ^2))/c^2
3.5.53.3.1 Defintions of rubi rules used
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x ))), x] - Simp[d^(2*n + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ (2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b*x^ 2)^p*ExpandToSum[d^m*(m + n + 2*p + 1)*x^m - (m + n + 2*p + 1)*(c + d*x)^m + c*(c + d*x)^(m - 2)*(c*(m + n - 1) + c*(m + n + 2*p + 1) + 2*d*(m + n + p )*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] & & IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[2*p] || ILtQ[m + n, 0] )
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.44
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2}}{c^{2}}\) | \(150\) |
default | \(\frac {a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{3}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{a^{4}}+\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{4} \left (x -\frac {1}{a}\right )}\right )}{c^{2}}\) | \(182\) |
1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^2+(3/a^2/(a^2)^(1/2)*arctan((a^2)^(1/ 2)*x/(-a^2*x^2+1)^(1/2))+2/3/a^5/(x-1/a)^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1 /2)+13/3/a^4/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2))*a^2/c^2
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^ 3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {a^{2} \left (\int \frac {x^{2}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \]
a**2*(Integral(x**2/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x** 2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(a**2*x**2*sqrt(-a**2 *x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 3.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {2\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )} \]