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 = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {\sqrt {c-a^2 c x^2} \left ((4+a x) \sqrt {1-a^2 x^2}+6 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^(2*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2],x]
Output:
-1/2*(Sqrt[c - a^2*c*x^2]*((4 + a*x)*Sqrt[1 - a^2*x^2] + 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*Sqrt[1 - a^2*x^2])
Time = 0.31 (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 = {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.
\(\displaystyle \int e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 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*ArcTanh[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]
Time = 0.27 (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 | \(-\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}-\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}-\frac {2 \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}-\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )}{a}\) | \(136\) |
Input:
int((a*x+1)^2/(-a^2*x^2+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)*arc tan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))
Time = 0.09 (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*x+1)^2/(-a^2*x^2+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*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]
\[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=- \int \frac {\sqrt {- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x - 1}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2),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 = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {1}{2} \, a {\left (\frac {\sqrt {-a^{2} c x^{2} + c} x}{a} - \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{a^{2}} + \frac {4 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2}}\right )} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(1/2),x, algorithm="maxima ")
Output:
-1/2*a*(sqrt(-a^2*c*x^2 + c)*x/a - 3*sqrt(c)*arcsin(a*x)/a^2 + 4*sqrt(-a^2 *c*x^2 + c)/a^2)
Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} {\left (x + \frac {4}{a}\right )} - \frac {3 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, \sqrt {-c} {\left | a \right |}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+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. \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx=\int -\frac {\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \] Input:
int(-((c - a^2*c*x^2)^(1/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
int(-((c - a^2*c*x^2)^(1/2)*(a*x + 1)^2)/(a^2*x^2 - 1), 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*x+1)^2/(-a^2*x^2+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)