Integrand size = 23, antiderivative size = 72 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}+3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \] Output:
-c*(-a^2*x^2+1)^(1/2)/x/(-a*c*x+c)^(1/2)+3*a*c^(1/2)*arctanh(c^(1/2)*(-a^2 *x^2+1)^(1/2)/(-a*c*x+c)^(1/2))
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {1-a x} \left (-\frac {c \sqrt {1+a x}}{x}+3 a c \text {arctanh}\left (\sqrt {1+a x}\right )\right )}{\sqrt {c-a c x}} \] Input:
Integrate[Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^2),x]
Output:
(Sqrt[1 - a*x]*(-((c*Sqrt[1 + a*x])/x) + 3*a*c*ArcTanh[Sqrt[1 + a*x]]))/Sq rt[c - a*c*x]
Time = 0.57 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6678, 580, 573, 219}
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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle \frac {\int \frac {(c-a c x)^{3/2}}{x^2 \sqrt {1-a^2 x^2}}dx}{c}\) |
\(\Big \downarrow \) 580 |
\(\displaystyle \frac {-\frac {3}{2} a c \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}}dx-\frac {c^2 \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}}{c}\) |
\(\Big \downarrow \) 573 |
\(\displaystyle \frac {3 a c^2 \int \frac {1}{1-\frac {c \left (1-a^2 x^2\right )}{c-a c x}}d\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3 a c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )-\frac {c^2 \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}}{c}\) |
Input:
Int[Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^2),x]
Output:
(-((c^2*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a*c*x])) + 3*a*c^(3/2)*ArcTanh[(Sqr t[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]])/c
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2*c Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 2)*((a + b*x^2)^(p + 1)/(b*e*(m + 1))), x] + Simp[d*((2*m + p + 3)/(e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p - 1, 0] && LtQ[m, -1] && IntegerQ [p + 1/2]
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]
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (-3 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a c x +\sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x \sqrt {c}}\) | \(79\) |
risch | \(\frac {\left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{x \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {3 a \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(138\) |
Input:
int((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x,method=_RETURNVERBOS E)
Output:
(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(-3*arctanh((c*(a*x+1))^(1/2)/c^(1/2 ))*a*c*x+(c*(a*x+1))^(1/2)*c^(1/2))/(a*x-1)/(c*(a*x+1))^(1/2)/x/c^(1/2)
Time = 0.12 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.82 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{2 \, {\left (a x^{2} - x\right )}}, \frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x^{2} - x}\right ] \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm="fr icas")
Output:
[1/2*(3*(a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*s qrt(-a*c*x + c))/(a*x^2 - x), (3*(a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2 *x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + sqrt(-a^2*x^2 + 1)*sqrt (-a*c*x + c))/(a*x^2 - x)]
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2} \left (a x + 1\right )}\, dx \] Input:
integrate((-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)
Output:
Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(x**2*(a*x + 1)), x )
\[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{{\left (a x + 1\right )} x^{2}} \,d x } \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm="ma xima")
Output:
integrate(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/((a*x + 1)*x^2), x)
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-a {\left (\frac {3 \, \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {\sqrt {a c x + c}}{a c x}\right )} {\left | c \right |} + \frac {3 \, a \sqrt {c} {\left | c \right |} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + \sqrt {2} a \sqrt {-c} {\left | c \right |}}{\sqrt {-c} \sqrt {c}} \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x, algorithm="gi ac")
Output:
-a*(3*arctan(sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + sqrt(a*c*x + c)/(a*c*x)) *abs(c) + (3*a*sqrt(c)*abs(c)*arctan(sqrt(2)*sqrt(c)/sqrt(-c)) + sqrt(2)*a *sqrt(-c)*abs(c))/(sqrt(-c)*sqrt(c))
Timed out. \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^2\,\left (a\,x+1\right )} \,d x \] Input:
int(((1 - a^2*x^2)^(1/2)*(c - a*c*x)^(1/2))/(x^2*(a*x + 1)),x)
Output:
int(((1 - a^2*x^2)^(1/2)*(c - a*c*x)^(1/2))/(x^2*(a*x + 1)), x)
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-\text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {a x +1}-3 \,\mathrm {log}\left (\sqrt {a x +1}-1\right ) a x +3 \,\mathrm {log}\left (\sqrt {a x +1}+1\right ) a x \right )}{2 x} \] Input:
int((-a*c*x+c)^(1/2)/(a*x+1)*(-a^2*x^2+1)^(1/2)/x^2,x)
Output:
(sqrt(c)*( - 2*sqrt(a*x + 1) - 3*log(sqrt(a*x + 1) - 1)*a*x + 3*log(sqrt(a *x + 1) + 1)*a*x))/(2*x)