Integrand size = 23, antiderivative size = 102 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {10 a \sqrt {c-a c x}}{\sqrt {1-a^2 x^2}}-\frac {(c-a c x)^{3/2}}{c x \sqrt {1-a^2 x^2}}+7 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \] Output:
-10*a*(-a*c*x+c)^(1/2)/(-a^2*x^2+1)^(1/2)-(-a*c*x+c)^(3/2)/c/x/(-a^2*x^2+1 )^(1/2)+7*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 = 64, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {c \sqrt {1-a x} \left (-1-9 a x+7 a x \sqrt {1+a x} \text {arctanh}\left (\sqrt {1+a x}\right )\right )}{x \sqrt {1+a x} \sqrt {c-a c x}} \] Input:
Integrate[Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
(c*Sqrt[1 - a*x]*(-1 - 9*a*x + 7*a*x*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 + a*x]]) )/(x*Sqrt[1 + a*x]*Sqrt[c - a*c*x])
Time = 0.51 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6680, 37, 100, 27, 87, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(1-a x)^{3/2} \sqrt {c-a c x}}{x^2 (a x+1)^{3/2}}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \frac {(1-a x)^2}{x^2 (a x+1)^{3/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\int -\frac {a (7-2 a x)}{2 x (a x+1)^{3/2}}dx-\frac {1}{x \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{2} a \int \frac {7-2 a x}{x (a x+1)^{3/2}}dx-\frac {1}{x \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{2} a \left (7 \int \frac {1}{x \sqrt {a x+1}}dx+\frac {18}{\sqrt {a x+1}}\right )-\frac {1}{x \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{2} a \left (\frac {14 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}+\frac {18}{\sqrt {a x+1}}\right )-\frac {1}{x \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\left (-\frac {1}{2} a \left (\frac {18}{\sqrt {a x+1}}-14 \text {arctanh}\left (\sqrt {a x+1}\right )\right )-\frac {1}{x \sqrt {a x+1}}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
Input:
Int[Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
(Sqrt[c - a*c*x]*(-(1/(x*Sqrt[1 + a*x])) - (a*(18/Sqrt[1 + a*x] - 14*ArcTa nh[Sqrt[1 + a*x]]))/2))/Sqrt[1 - a*x]
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] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\left (-7 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a x \sqrt {c \left (a x +1\right )}+9 \sqrt {c}\, a x +\sqrt {c}\right ) \sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}}{\left (a x -1\right ) \sqrt {c}\, \left (a x +1\right ) x}\) | \(82\) |
risch | \(\frac {\left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{x \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}+\frac {a \left (\frac {16}{\sqrt {a c x +c}}-\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(152\) |
Input:
int((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x,method=_RETURNVERB OSE)
Output:
(-7*arctanh((c*(a*x+1))^(1/2)/c^(1/2))*a*x*(c*(a*x+1))^(1/2)+9*c^(1/2)*a*x +c^(1/2))*(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)/(a*x-1)/c^(1/2)/(a*x+1)/x
Time = 0.08 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.15 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {7 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (9 \, a x + 1\right )}}{2 \, {\left (a^{2} x^{3} - x\right )}}, \frac {7 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (9 \, a x + 1\right )}}{a^{2} x^{3} - x}\right ] \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=" fricas")
Output:
[1/2*(7*(a^3*x^3 - 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)*(9*a*x + 1))/(a^2*x^3 - x), (7*(a^3*x^3 - a*x)*sqrt(-c)*ar ctan(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)*(9*a*x + 1))/(a^2*x^3 - x)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)
Output:
Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**2*(a*x + 1)* *3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {-a c x + c}}{{\left (a x + 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=" maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*sqrt(-a*c*x + c)/((a*x + 1)^3*x^2), x)
Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm=" giac")
Output:
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
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}}{x^2\,{\left (a\,x+1\right )}^3} \,d x \] Input:
int(((1 - a^2*x^2)^(3/2)*(c - a*c*x)^(1/2))/(x^2*(a*x + 1)^3),x)
Output:
int(((1 - a^2*x^2)^(3/2)*(c - a*c*x)^(1/2))/(x^2*(a*x + 1)^3), x)
Time = 0.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {c}\, \left (-7 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}-1\right ) a x +7 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}+1\right ) a x -18 a x -2\right )}{2 \sqrt {a x +1}\, x} \] Input:
int((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x)
Output:
(sqrt(c)*( - 7*sqrt(a*x + 1)*log(sqrt(a*x + 1) - 1)*a*x + 7*sqrt(a*x + 1)* log(sqrt(a*x + 1) + 1)*a*x - 18*a*x - 2))/(2*sqrt(a*x + 1)*x)