Integrand size = 23, antiderivative size = 138 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}+\frac {7 a \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arcsinh}\left (\sqrt {\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \] Output:
-8*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)/x-(1+1/a/x)^(1/2)*(-a* c*x+c)^(1/2)/(1-1/a/x)^(1/2)/x+7*a*(1/a/x)^(1/2)*(-a*c*x+c)^(1/2)*arcsinh( (1/a/x)^(1/2))/(1-1/a/x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.57 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {c-a c x} \left (-1-9 a x+\frac {7 a^{3/2} \sqrt {1+\frac {1}{a x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{3/2}}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}} x^2} \] Input:
Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^2),x]
Output:
(Sqrt[c - a*c*x]*(-1 - 9*a*x + (7*a^(3/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x ^(-1)]/Sqrt[a]])/(x^(-1))^(3/2)))/(a*Sqrt[1 - 1/(a^2*x^2)]*x^2)
Time = 0.55 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6730, 27, 100, 27, 90, 63, 222}
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.
\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2}{a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2}{\left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}-2 a^2 \int \frac {3 a-\frac {1}{x}}{2 a \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}-a \int \frac {3 a-\frac {1}{x}}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {7}{2} a \int \frac {1}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}-a \left (7 a \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}-a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {8 a^2 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}-a \left (7 a^{3/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )-a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )\right ) \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
Input:
Int[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^2),x]
Output:
-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((8*a^2*Sqrt[x^(-1)])/Sqrt[1 + 1/(a*x)] - a*(-(a*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) + 7*a^(3/2)*ArcSinh[Sqrt[x^(-1)]/Sq rt[a]])))/(a^2*Sqrt[1 - 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[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.62
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \left (7 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a x \sqrt {-c \left (a x +1\right )}+9 \sqrt {c}\, a x +\sqrt {c}\right ) \sqrt {-c \left (a x -1\right )}}{\left (a x -1\right )^{2} \sqrt {c}\, x}\) | \(86\) |
risch | \(\frac {\left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{x \sqrt {-c \left (a x -1\right )}}-\frac {\left (-\frac {8 a}{\sqrt {-a c x -c}}-\frac {7 a \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\) | \(112\) |
Input:
int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-((a*x-1)/(a*x+1))^(3/2)*(a*x+1)*(7*arctan((-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*x-1)^2/c^ (1/2)/x
Time = 0.09 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.72 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {7 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a c x + c} {\left (9 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{2} - x\right )}}, \frac {7 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (9 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x}\right ] \] Input:
integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="frica s")
Output:
[1/2*(7*(a^2*x^2 - a*x)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x)) - 2*s qrt(-a*c*x + c)*(9*a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^2 - x), (7*(a^ 2*x^2 - a*x)*sqrt(c)*arctan(sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - sqrt(-a*c*x + c)*(9*a*x + 1)*sqrt((a*x - 1)/ (a*x + 1)))/(a*x^2 - x)]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\text {Timed out} \] Input:
integrate((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**2,x)
Output:
Timed out
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int { \frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^2,x, algorithm="maxim a")
Output:
integrate(sqrt(-a*c*x + c)*((a*x - 1)/(a*x + 1))^(3/2)/x^2, x)
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+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. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^2} \,d x \] Input:
int(((c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^2,x)
Output:
int(((c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^2, x)
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {c}\, i \left (7 \sqrt {a x +1}\, \mathrm {log}\left (\frac {2 \sqrt {a x +1}-2}{\sqrt {2}}\right ) a x -7 \sqrt {a x +1}\, \mathrm {log}\left (\frac {2 \sqrt {a x +1}+2}{\sqrt {2}}\right ) a x +18 a x +2\right )}{2 \sqrt {a x +1}\, x} \] Input:
int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^2,x)
Output:
(sqrt(c)*i*(7*sqrt(a*x + 1)*log((2*sqrt(a*x + 1) - 2)/sqrt(2))*a*x - 7*sqr t(a*x + 1)*log((2*sqrt(a*x + 1) + 2)/sqrt(2))*a*x + 18*a*x + 2))/(2*sqrt(a *x + 1)*x)