Integrand size = 27, antiderivative size = 77 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\frac {2 a (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}-\frac {2 a \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \]
-2*a*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))/c^(1/2)+2*a*(a*x+1)/(-a^2*c*x^2 +c)^(1/2)-(-a^2*c*x^2+c)^(1/2)/c/x
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\frac {(1-3 a x) \sqrt {c-a^2 c x^2}}{c x (-1+a x)}+\frac {2 a \log (x)}{\sqrt {c}}-\frac {2 a \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )}{\sqrt {c}} \]
((1 - 3*a*x)*Sqrt[c - a^2*c*x^2])/(c*x*(-1 + a*x)) + (2*a*Log[x])/Sqrt[c] - (2*a*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/Sqrt[c]
Time = 0.37 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6701, 528, 534, 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.
\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle c \int \frac {(a x+1)^2}{x^2 \left (c-a^2 c x^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 528 |
\(\displaystyle c \left (\frac {\int \frac {2 a x+1}{x^2 \sqrt {c-a^2 c x^2}}dx}{c}+\frac {2 a (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 534 |
\(\displaystyle c \left (\frac {2 a \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{c x}}{c}+\frac {2 a (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c \left (\frac {a \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {\sqrt {c-a^2 c x^2}}{c x}}{c}+\frac {2 a (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c \left (\frac {-\frac {2 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a c}-\frac {\sqrt {c-a^2 c x^2}}{c x}}{c}+\frac {2 a (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c \left (\frac {-\frac {2 a \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {c-a^2 c x^2}}{c x}}{c}+\frac {2 a (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
c*((2*a*(1 + a*x))/(c*Sqrt[c - a^2*c*x^2]) + (-(Sqrt[c - a^2*c*x^2]/(c*x)) - (2*a*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/Sqrt[c])/c)
3.12.13.3.1 Defintions of rubi rules used
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[(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[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^(n/2) Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ [c, 0]) && IGtQ[n/2, 0]
Time = 0.34 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29
method | result | size |
default | \(-\frac {\sqrt {-a^{2} c \,x^{2}+c}}{c x}-\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {2 \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{c \left (x -\frac {1}{a}\right )}\) | \(99\) |
risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {2 \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{c \left (x -\frac {1}{a}\right )}\) | \(105\) |
-(-a^2*c*x^2+c)^(1/2)/c/x-2*a/c^(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/ 2))/x)-2/c/(x-1/a)*(-a^2*c*(x-1/a)^2-2*(x-1/a)*a*c)^(1/2)
Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.31 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\left [\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a x - 1\right )}}{a c x^{2} - c x}, -\frac {2 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a x - 1\right )}}{a c x^{2} - c x}\right ] \]
[((a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - sqrt(-a^2*c*x^2 + c)*(3*a*x - 1))/(a*c*x^2 - c*x), -(2*(a^2 *x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(3*a*x - 1))/(a*c*x^2 - c*x)]
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=- \int \frac {a x}{a x^{3} \sqrt {- a^{2} c x^{2} + c} - x^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a x^{3} \sqrt {- a^{2} c x^{2} + c} - x^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx \]
-Integral(a*x/(a*x**3*sqrt(-a**2*c*x**2 + c) - x**2*sqrt(-a**2*c*x**2 + c) ), x) - Integral(1/(a*x**3*sqrt(-a**2*c*x**2 + c) - x**2*sqrt(-a**2*c*x**2 + c)), x)
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=-\int \frac {{\left (a\,x+1\right )}^2}{x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )} \,d x \]