Integrand size = 24, antiderivative size = 77 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {2 \sqrt {c-a^2 c x^2}}{a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}-\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a} \] 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 = 76, normalized size of antiderivative = 0.99 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left ((4+a x) \sqrt {1-a^2 x^2}+6 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]
Output:
(Sqrt[c - a^2*c*x^2]*((4 + a*x)*Sqrt[1 - a^2*x^2] + 6*ArcSin[Sqrt[1 - a*x] /Sqrt[2]]))/(2*a*Sqrt[1 - a^2*x^2])
Time = 0.61 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6717, 6691, 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.
\(\displaystyle \int \sqrt {c-a^2 c x^2} e^{2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}dx\) |
\(\Big \downarrow \) 6691 |
\(\displaystyle -c \int \frac {(a x+1)^2}{\sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle -c \left (\frac {3}{2} \int \frac {a x+1}{\sqrt {c-a^2 c x^2}}dx-\frac {(a x+1) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -c \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{a c}\right )-\frac {(a x+1) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -c \left (\frac {3}{2} \left (\int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{a c}\right )-\frac {(a x+1) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -c \left (\frac {3}{2} \left (\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}-\frac {\sqrt {c-a^2 c x^2}}{a c}\right )-\frac {(a x+1) \sqrt {c-a^2 c x^2}}{2 a c}\right )\) |
Input:
Int[E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]
Output:
-(c*(-1/2*((1 + a*x)*Sqrt[c - a^2*c*x^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[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]) && IGtQ[n/ 2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.18 (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 | \(\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}-\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a}\) | \(136\) |
Input:
int(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2),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)*ar ctan((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 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\left [\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a x + 4\right )} + 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{4 \, a}, \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 4\right )} + 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{2 \, a}\right ] \] Input:
integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2),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*sq rt(-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 ]
\[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \] Input:
integrate(1/(a*x-1)*(a*x+1)*(-a**2*c*x**2+c)**(1/2),x)
Output:
Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/(a*x - 1), x)
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} x - \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{2 \, a} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a} \] Input:
integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2),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 = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} {\left (x + \frac {4}{a}\right )} + \frac {3 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, \sqrt {-c} {\left | a \right |}} \] Input:
integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2),x, algorithm="giac")
Output:
1/2*sqrt(-a^2*c*x^2 + c)*(x + 4/a) + 3/2*c*log(abs(-sqrt(-a^2*c)*x + sqrt( -a^2*c*x^2 + c)))/(sqrt(-c)*abs(a))
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \] Input:
int(((c - a^2*c*x^2)^(1/2)*(a*x + 1))/(a*x - 1),x)
Output:
int(((c - a^2*c*x^2)^(1/2)*(a*x + 1))/(a*x - 1), x)
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.55 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c}\, \left (-3 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a x +4 \sqrt {-a^{2} x^{2}+1}-4\right )}{2 a} \] Input:
int(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(1/2),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)