Integrand size = 24, antiderivative size = 77 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {2 \sqrt {c-a^2 c x^2}}{a}-\frac {1}{2} x \sqrt {c-a^2 c x^2}+\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a} \] Output:
2*(-a^2*c*x^2+c)^(1/2)/a-1/2*x*(-a^2*c*x^2+c)^(1/2)+3/2*c^(1/2)*arctan(a*c ^(1/2)*x/(-a^2*c*x^2+c)^(1/2))/a
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (4-5 a x+a^2 x^2\right )-6 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \] Input:
Integrate[Sqrt[c - a^2*c*x^2]/E^(2*ArcTanh[a*x]),x]
Output:
(Sqrt[c - a^2*c*x^2]*(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 = 0.47 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6692, 469, 455, 224, 216}
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.
\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6692 |
\(\displaystyle c \int \frac {(1-a x)^2}{\sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c \left (\frac {3}{2} \int \frac {1-a x}{\sqrt {c-a^2 c x^2}}dx+\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {\sqrt {c-a^2 c x^2}}{a c}\right )+\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle c \left (\frac {3}{2} \left (\int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {\sqrt {c-a^2 c x^2}}{a c}\right )+\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle c \left (\frac {3}{2} \left (\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}+\frac {\sqrt {c-a^2 c x^2}}{a c}\right )+\frac {(1-a x) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
Input:
Int[Sqrt[c - a^2*c*x^2]/E^(2*ArcTanh[a*x]),x]
Output:
c*(((1 - a*x)*Sqrt[c - a^2*c*x^2])/(2*a*c) + (3*(Sqrt[c - a^2*c*x^2]/(a*c) + ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/(a*Sqrt[c])))/2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* ((n + p)/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* p + 1, 0] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^(n/2) Int[(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[ n/2, 0]
Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {\left (a x -4\right ) \left (a^{2} x^{2}-1\right ) c}{2 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\) | \(69\) |
default | \(-\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}-\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}+\frac {2 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}+\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a}\) | \(127\) |
Input:
int((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/2*(a*x-4)*(a^2*x^2-1)/a/(-c*(a^2*x^2-1))^(1/2)*c+3/2*c/(a^2*c)^(1/2)*arc tan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.74 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=\left [-\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a x - 4\right )} - 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{4 \, a}, -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 4\right )} + 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{2 \, a}\right ] \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas ")
Output:
[-1/4*(2*sqrt(-a^2*c*x^2 + c)*(a*x - 4) - 3*sqrt(-c)*log(2*a^2*c*x^2 + 2*s qrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a, -1/2*(sqrt(-a^2*c*x^2 + c)*(a*x - 4) + 3*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c))) /a]
\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=- \int \left (- \frac {\sqrt {- a^{2} c x^{2} + c}}{a x + 1}\right )\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x + 1}\, dx \] Input:
integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)
Output:
-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x + 1), x) - Integral(a*x*sqrt(-a**2* c*x**2 + c)/(a*x + 1), x)
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} x + \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{2 \, a} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima ")
Output:
-1/2*sqrt(-a^2*c*x^2 + c)*x + 3/2*sqrt(c)*arcsin(a*x)/a + 2*sqrt(-a^2*c*x^ 2 + c)/a
Time = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.64 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {{\left (12 \, a^{3} \sqrt {c} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - \frac {{\left (3 \, a^{3} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 5 \, a^{3} c {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )\right )} {\left (a x + 1\right )}^{2}}{c^{2}}\right )} {\left | a \right |}}{4 \, a^{5}} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
Output:
-1/4*(12*a^3*sqrt(c)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn(1/(a*x + 1))*sgn(a) - (3*a^3*c^2*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 5*a^3*c*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))*sgn(a))*(a*x + 1)^2/ c^2)*abs(a)/a^5
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:
int(-((c - a^2*c*x^2)^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)
Output:
-int(((c - a^2*c*x^2)^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.56 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c}\, \left (3 \mathit {asin} \left (a x \right )-\sqrt {-a^{2} x^{2}+1}\, a x +4 \sqrt {-a^{2} x^{2}+1}-4\right )}{2 a} \] Input:
int((-a^2*c*x^2+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x)
Output:
(sqrt(c)*(3*asin(a*x) - sqrt( - a**2*x**2 + 1)*a*x + 4*sqrt( - a**2*x**2 + 1) - 4))/(2*a)