Integrand size = 27, antiderivative size = 123 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {46 a^2 \sqrt {c-\frac {c}{a x}} x}{3 \sqrt {1-a x} \sqrt {1+a x}} \] Output:
20/3*a*(c-c/a/x)^(1/2)/(-a*x+1)^(1/2)/(a*x+1)^(1/2)-2/3*(c-c/a/x)^(1/2)/x/ (-a*x+1)^(1/2)/(a*x+1)^(1/2)+46/3*a^2*(c-c/a/x)^(1/2)*x/(-a*x+1)^(1/2)/(a* x+1)^(1/2)
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (-1+10 a x+23 a^2 x^2\right )}{3 x \sqrt {1-a^2 x^2}} \] Input:
Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
(2*Sqrt[c - c/(a*x)]*(-1 + 10*a*x + 23*a^2*x^2))/(3*x*Sqrt[1 - a^2*x^2])
Time = 0.47 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6684, 6679, 100, 27, 87, 48}
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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\) |
\(\Big \downarrow \) 6684 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {1-a x}}{x^{5/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(1-a x)^2}{x^{5/2} (a x+1)^{3/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {2}{3} \int -\frac {a (10-3 a x)}{2 x^{3/2} (a x+1)^{3/2}}dx-\frac {2}{3 x^{3/2} \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {1}{3} a \int \frac {10-3 a x}{x^{3/2} (a x+1)^{3/2}}dx-\frac {2}{3 x^{3/2} \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {1}{3} a \left (-23 a \int \frac {1}{\sqrt {x} (a x+1)^{3/2}}dx-\frac {20}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {2}{3 x^{3/2} \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\sqrt {x} \left (-\frac {2}{3 x^{3/2} \sqrt {a x+1}}-\frac {1}{3} a \left (-\frac {46 a \sqrt {x}}{\sqrt {a x+1}}-\frac {20}{\sqrt {x} \sqrt {a x+1}}\right )\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\) |
Input:
Int[Sqrt[c - c/(a*x)]/(E^(3*ArcTanh[a*x])*x^2),x]
Output:
(Sqrt[c - c/(a*x)]*Sqrt[x]*(-2/(3*x^(3/2)*Sqrt[1 + a*x]) - (a*(-20/(Sqrt[x ]*Sqrt[1 + a*x]) - (46*a*Sqrt[x])/Sqrt[1 + a*x]))/3))/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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.48
method | result | size |
orering | \(\frac {2 \left (23 a^{2} x^{2}+10 a x -1\right ) \sqrt {c -\frac {c}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 \left (a x +1\right )^{2} x \left (a x -1\right )^{2}}\) | \(59\) |
gosper | \(\frac {2 \left (23 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x \left (a x +1\right )^{2} \left (a x -1\right )^{2}}\) | \(61\) |
default | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (23 a^{2} x^{2}+10 a x -1\right )}{3 x \left (a x +1\right ) \left (a x -1\right )}\) | \(61\) |
risch | \(\frac {2 \left (11 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{3 x \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {8 a^{2} x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}\) | \(151\) |
Input:
int((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x,method=_RETURNVERBO SE)
Output:
2/3*(23*a^2*x^2+10*a*x-1)/(a*x+1)^2/x/(a*x-1)^2*(c-c/a/x)^(1/2)*(-a^2*x^2+ 1)^(3/2)
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {2 \, {\left (23 \, a^{2} x^{2} + 10 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{2} x^{3} - x\right )}} \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm="f ricas")
Output:
-2/3*(23*a^2*x^2 + 10*a*x - 1)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x))/ (a^2*x^3 - x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)
Output:
Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**2*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm="m axima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*sqrt(c - c/(a*x))/((a*x + 1)^3*x^2), x)
Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x, algorithm="g iac")
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
Time = 14.52 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.65 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {46\,x^2\,\sqrt {1-a^2\,x^2}}{3}-\frac {2\,\sqrt {1-a^2\,x^2}}{3\,a^2}+\frac {20\,x\,\sqrt {1-a^2\,x^2}}{3\,a}\right )}{\frac {x}{a^2}-x^3} \] Input:
int(((c - c/(a*x))^(1/2)*(1 - a^2*x^2)^(3/2))/(x^2*(a*x + 1)^3),x)
Output:
((c - c/(a*x))^(1/2)*((46*x^2*(1 - a^2*x^2)^(1/2))/3 - (2*(1 - a^2*x^2)^(1 /2))/(3*a^2) + (20*x*(1 - a^2*x^2)^(1/2))/(3*a)))/(x/a^2 - x^3)
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.48 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c}\, i \left (26 \sqrt {a x +1}\, a^{2} x^{2}-23 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}-10 \sqrt {x}\, \sqrt {a}\, a x +\sqrt {x}\, \sqrt {a}\right )}{3 \sqrt {a x +1}\, a \,x^{2}} \] Input:
int((c-c/a/x)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^2,x)
Output:
(2*sqrt(c)*i*(26*sqrt(a*x + 1)*a**2*x**2 - 23*sqrt(x)*sqrt(a)*a**2*x**2 - 10*sqrt(x)*sqrt(a)*a*x + sqrt(x)*sqrt(a)))/(3*sqrt(a*x + 1)*a*x**2)