Integrand size = 18, antiderivative size = 83 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}} \] Output:
-2*(-a^2*x^2+1)^(1/2)/a/(-a*c*x+c)^(1/2)+2*2^(1/2)*arctanh(1/2*c^(1/2)*(-a ^2*x^2+1)^(1/2)*2^(1/2)/(-a*c*x+c)^(1/2))/a/c^(1/2)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \sqrt {c-a c x} \left (\sqrt {1+a x}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{a c \sqrt {1-a x}} \] Input:
Integrate[E^ArcTanh[a*x]/Sqrt[c - a*c*x],x]
Output:
(-2*Sqrt[c - a*c*x]*(Sqrt[1 + a*x] - Sqrt[2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2] ]))/(a*c*Sqrt[1 - a*x])
Time = 0.46 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6677, 466, 471, 221}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{3/2}}dx\) |
\(\Big \downarrow \) 466 |
\(\displaystyle c \left (\frac {2 \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}}dx}{c}-\frac {2 \sqrt {1-a^2 x^2}}{a c \sqrt {c-a c x}}\right )\) |
\(\Big \downarrow \) 471 |
\(\displaystyle c \left (-4 a \int \frac {1}{\frac {a^2 c^2 \left (1-a^2 x^2\right )}{c-a c x}-2 a^2 c}d\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {2 \sqrt {1-a^2 x^2}}{a c \sqrt {c-a c x}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c \left (\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a c^{3/2}}-\frac {2 \sqrt {1-a^2 x^2}}{a c \sqrt {c-a c x}}\right )\) |
Input:
Int[E^ArcTanh[a*x]/Sqrt[c - a*c*x],x]
Output:
c*((-2*Sqrt[1 - a^2*x^2])/(a*c*Sqrt[c - a*c*x]) + (2*Sqrt[2]*ArcTanh[(Sqrt [c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(a*c^(3/2)))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/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[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[2*d Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] ], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\sqrt {c \left (a x +1\right )}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, c a}\) | \(84\) |
risch | \(\frac {2 \left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a \sqrt {c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(147\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(c^(1/2)*2^(1/2)*arctanh(1/2*(c*( a*x+1))^(1/2)*2^(1/2)/c^(1/2))-(c*(a*x+1))^(1/2))/(a*x-1)/(c*(a*x+1))^(1/2 )/c/a
Time = 0.08 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.55 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\left [\frac {\frac {\sqrt {2} {\left (a c x - c\right )} \log \left (-\frac {a^{2} x^{2} + 2 \, a x - \frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{\sqrt {c}} + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a^{2} c x - a c}, \frac {2 \, {\left (\sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}}{2 \, {\left (a x - 1\right )}}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{a^{2} c x - a c}\right ] \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2),x, algorithm="fricas ")
Output:
[(sqrt(2)*(a*c*x - c)*log(-(a^2*x^2 + 2*a*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1) *sqrt(-a*c*x + c)/sqrt(c) - 3)/(a^2*x^2 - 2*a*x + 1))/sqrt(c) + 2*sqrt(-a^ 2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*c*x - a*c), 2*(sqrt(2)*(a*c*x - c)*sqrt( -1/c)*arctan(1/2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-1/c)/(a *x - 1)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*c*x - a*c)]
\[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {a x + 1}{\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(1/2),x)
Output:
Integral((a*x + 1)/(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2),x, algorithm="maxima ")
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)), x)
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {2 \, c {\left (\frac {\frac {\sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + \sqrt {a c x + c}}{c} - \frac {\sqrt {2} {\left (c \arctan \left (\frac {\sqrt {c}}{\sqrt {-c}}\right ) + \sqrt {-c} \sqrt {c}\right )}}{\sqrt {-c} c}\right )}}{a {\left | c \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2),x, algorithm="giac")
Output:
-2*c*((sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + s qrt(a*c*x + c))/c - sqrt(2)*(c*arctan(sqrt(c)/sqrt(-c)) + sqrt(-c)*sqrt(c) )/(sqrt(-c)*c))/(a*abs(c))
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {a\,x+1}{\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}} \,d x \] Input:
int((a*x + 1)/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^(1/2)),x)
Output:
int((a*x + 1)/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^(1/2)), x)
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \sqrt {c}\, \left (-\sqrt {a x +1}-\sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )+\sqrt {2}\right )}{a c} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^(1/2),x)
Output:
(2*sqrt(c)*( - sqrt(a*x + 1) - sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt( 2))/2)) + sqrt(2)))/(a*c)