Integrand size = 25, antiderivative size = 78 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x}{a}-\frac {1}{2} \sqrt {c-\frac {c}{a^2 x^2}} x^2-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{2 a^2} \] Output:
2*(c-c/a^2/x^2)^(1/2)*x/a-1/2*(c-c/a^2/x^2)^(1/2)*x^2-3/2*c^(1/2)*arctanh( (c-c/a^2/x^2)^(1/2)/c^(1/2))/a^2
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.28 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {1+a x} \left (4-5 a x+a^2 x^2\right )-6 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \] Input:
Integrate[(Sqrt[c - c/(a^2*x^2)]*x)/E^(2*ArcTanh[a*x]),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[1 + a*x]*(4 - 5*a*x + a^2*x^2) - 6*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2] )
Time = 0.36 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6709, 466, 466, 223}
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.
\(\displaystyle \int x e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6709 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(a x+1)^2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 466 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {3}{2} \int \frac {\sqrt {1-a^2 x^2}}{a x+1}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (a x+1)}\right )}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 466 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{a}\right )+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (a x+1)}\right )}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {x \left (\frac {3}{2} \left (\frac {\sqrt {1-a^2 x^2}}{a}+\frac {\arcsin (a x)}{a}\right )+\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (a x+1)}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[(Sqrt[c - c/(a^2*x^2)]*x)/E^(2*ArcTanh[a*x]),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*((1 - a^2*x^2)^(3/2)/(2*a*(1 + a*x)) + (3*(Sqrt[1 - a^2*x^2]/a + ArcSin[a*x]/a))/2))/Sqrt[1 - a^2*x^2]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] - Simp[2*b*c*(p/(d^ 2*(n + 2*p + 1))) Int[(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; Fr eeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LeQ[-2, n, 0 ] || EqQ[n + p + 1, 0]) && NeQ[n + 2*p + 1, 0] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.46
method | result | size |
risch | \(-\frac {\left (a x -4\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 a}-\frac {3 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 \sqrt {a^{2} c}\, \left (a^{2} x^{2}-1\right )}\) | \(114\) |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (-x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}+\sqrt {c}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-4 \sqrt {c}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right )+4 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a \right )}{2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}}\) | \(147\) |
Input:
int((c-c/a^2/x^2)^(1/2)*x/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
Output:
-1/2*(a*x-4)/a*x*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)-3/2*ln(a^2*c*x/(a^2*c)^(1/2 )+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)*x*(c*(a^2*x^2-1))^(1/2)*(c*(a^2*x^2-1 )/a^2/x^2)^(1/2)/(a^2*x^2-1)
Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.41 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\left [-\frac {2 \, {\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{4 \, a^{2}}, -\frac {{\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{2 \, a^{2}}\right ] \] Input:
integrate((c-c/a^2/x^2)^(1/2)*x/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="frica s")
Output:
[-1/4*(2*(a^2*x^2 - 4*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(c)*log (2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a^2 , -1/2*((a^2*x^2 - 4*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 3*sqrt(-c)*arc tan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^2 ]
\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=- \int \left (- \frac {x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\right )\, dx - \int \frac {a x^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\, dx \] Input:
integrate((c-c/a**2/x**2)**(1/2)*x/(a*x+1)**2*(-a**2*x**2+1),x)
Output:
-Integral(-x*sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x**2*sqrt( c - c/(a**2*x**2))/(a*x + 1), x)
\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(1/2)*x/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxim a")
Output:
-integrate((a^2*x^2 - 1)*sqrt(c - c/(a^2*x^2))*x/(a*x + 1)^2, x)
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.36 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=-\frac {1}{4} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (\frac {x \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {4 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} - \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{2} {\left | a \right |}} + \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 8 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{3} {\left | a \right |}}\right )} {\left | a \right |} \] Input:
integrate((c-c/a^2/x^2)^(1/2)*x/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac" )
Output:
-1/4*(2*sqrt(a^2*c*x^2 - c)*(x*sgn(x)/a^2 - 4*sgn(x)/a^3) - 6*sqrt(c)*log( abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a^2*abs(a)) + (3*a*sqrt (c)*log(abs(c)) + 8*sqrt(-c)*abs(a))*sgn(x)/(a^3*abs(a)))*abs(a)
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=-\int \frac {x\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:
int(-(x*(c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)
Output:
-int((x*(c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)
Time = 0.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x \, dx=\frac {\sqrt {c}\, \left (-\sqrt {a^{2} x^{2}-1}\, a x +4 \sqrt {a^{2} x^{2}-1}-3 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )\right )}{2 a^{2}} \] Input:
int((c-c/a^2/x^2)^(1/2)*x/(a*x+1)^2*(-a^2*x^2+1),x)
Output:
(sqrt(c)*( - sqrt(a**2*x**2 - 1)*a*x + 4*sqrt(a**2*x**2 - 1) - 3*log(sqrt( a**2*x**2 - 1) + a*x)))/(2*a**2)