Integrand size = 24, antiderivative size = 65 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2 \sqrt {c-a^2 c x^2}}{a c (1+a x)}-\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}} \] Output:
-2*(-a^2*c*x^2+c)^(1/2)/a/c/(a*x+1)-arctan(a*c^(1/2)*x/(-a^2*c*x^2+c)^(1/2 ))/a/c^(1/2)
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {1-a^2 x^2} \left ((-1+a x) \sqrt {1+a x}+\sqrt {1-a x} (1+a x) \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-a x} (1+a x) \sqrt {c-a^2 c x^2}} \] Input:
Integrate[1/(E^(2*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2]),x]
Output:
(2*Sqrt[1 - a^2*x^2]*((-1 + a*x)*Sqrt[1 + a*x] + Sqrt[1 - a*x]*(1 + a*x)*A rcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*Sqrt[1 - a*x]*(1 + a*x)*Sqrt[c - a^2*c*x ^2])
Time = 0.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6692, 457, 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.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) |
\(\Big \downarrow \) 6692 |
\(\displaystyle c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 457 |
\(\displaystyle c \left (-\frac {\int \frac {1}{\sqrt {c-a^2 c x^2}}dx}{c}-\frac {2 (1-a x)}{a c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle c \left (-\frac {\int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}}{c}-\frac {2 (1-a x)}{a c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle c \left (-\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a c^{3/2}}-\frac {2 (1-a x)}{a c \sqrt {c-a^2 c x^2}}\right )\) |
Input:
Int[1/(E^(2*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2]),x]
Output:
c*((-2*(1 - a*x))/(a*c*Sqrt[c - a^2*c*x^2]) - ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/(a*c^(3/2)))
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_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[p, -1]
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 = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}{a^{2} c \left (x +\frac {1}{a}\right )}\) | \(74\) |
Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))-2/a^2/c/(x+1 /a)*(-(x+1/a)^2*a^2*c+2*(x+1/a)*a*c)^(1/2)
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.31 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\left [-\frac {{\left (a x + 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 4 \, \sqrt {-a^{2} c x^{2} + c}}{2 \, {\left (a^{2} c x + a c\right )}}, \frac {{\left (a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x + a c}\right ] \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2),x, algorithm="fric as")
Output:
[-1/2*((a*x + 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt( -c)*x - c) + 4*sqrt(-a^2*c*x^2 + c))/(a^2*c*x + a*c), ((a*x + 1)*sqrt(c)*a rctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - 2*sqrt(-a^2*c*x^ 2 + c))/(a^2*c*x + a*c)]
\[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=- \int \frac {a x}{a x \sqrt {- a^{2} c x^{2} + c} + \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{a x \sqrt {- a^{2} c x^{2} + c} + \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \] Input:
integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**(1/2),x)
Output:
-Integral(a*x/(a*x*sqrt(-a**2*c*x**2 + c) + sqrt(-a**2*c*x**2 + c)), x) - Integral(-1/(a*x*sqrt(-a**2*c*x**2 + c) + sqrt(-a**2*c*x**2 + c)), x)
Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.62 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x + a c} - \frac {\arcsin \left (a x\right )}{a \sqrt {c}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2),x, algorithm="maxi ma")
Output:
-2*sqrt(-a^2*c*x^2 + c)/(a^2*c*x + a*c) - arcsin(a*x)/(a*sqrt(c))
Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {2 \, {\left (\frac {{\left (c \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right ) - \sqrt {-c} \sqrt {c}\right )} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{c^{\frac {3}{2}}} - \frac {\frac {\arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{c}}{\mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}\right )}}{{\left | a \right |}} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2),x, algorithm="giac ")
Output:
-2*((c*arctan(sqrt(-c)/sqrt(c)) - sqrt(-c)*sqrt(c))*sgn(1/(a*x + 1))*sgn(a )/c^(3/2) - (arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))/sqrt(c) - sqrt(-c + 2*c/(a*x + 1))/c)/(sgn(1/(a*x + 1))*sgn(a)))/abs(a)
Timed out. \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\int \frac {a^2\,x^2-1}{\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2} \,d x \] Input:
int(-(a^2*x^2 - 1)/((c - a^2*c*x^2)^(1/2)*(a*x + 1)^2),x)
Output:
-int((a^2*x^2 - 1)/((c - a^2*c*x^2)^(1/2)*(a*x + 1)^2), x)
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {c}\, \left (-\mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-\mathit {asin} \left (a x \right )+4 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )}{a c \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+1\right )} \] Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*( - asin(a*x)*tan(asin(a*x)/2) - asin(a*x) + 4*tan(asin(a*x)/2))) /(a*c*(tan(asin(a*x)/2) + 1))