Integrand size = 26, antiderivative size = 41 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 e^{\frac {1}{2} \text {arctanh}(a x)} (1-2 a x)}{3 a c \sqrt {c-a^2 c x^2}} \] Output:
-2/3*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)*(-2*a*x+1)/a/c/(-a^2*c*x^2+c)^(1/2 )
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {2 (-1+2 a x) \sqrt {1-a^2 x^2}}{3 a c (1-a x)^{3/4} \sqrt [4]{1+a x} \sqrt {c-a^2 c x^2}} \] Input:
Integrate[E^(ArcTanh[a*x]/2)/(c - a^2*c*x^2)^(3/2),x]
Output:
(2*(-1 + 2*a*x)*Sqrt[1 - a^2*x^2])/(3*a*c*(1 - a*x)^(3/4)*(1 + a*x)^(1/4)* Sqrt[c - a^2*c*x^2])
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6685}
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.
\(\displaystyle \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6685 |
\(\displaystyle -\frac {2 (1-2 a x) e^{\frac {1}{2} \text {arctanh}(a x)}}{3 a c \sqrt {c-a^2 c x^2}}\) |
Input:
Int[E^(ArcTanh[a*x]/2)/(c - a^2*c*x^2)^(3/2),x]
Output:
(-2*E^(ArcTanh[a*x]/2)*(1 - 2*a*x))/(3*a*c*Sqrt[c - a^2*c*x^2])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(n - a*x)*(E^(n*ArcTanh[a*x])/(a*c*(n^2 - 1)*Sqrt[c + d*x^2])), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n]
Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.34
method | result | size |
gosper | \(-\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2 a x -1\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{3 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(55\) |
orering | \(-\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2 a x -1\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{3 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(55\) |
Input:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(3/2),x,method=_RETU RNVERBOSE)
Output:
-2/3*(a*x-1)*(a*x+1)*(2*a*x-1)*((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/a/(-a^2* c*x^2+c)^(3/2)
Timed out. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(3/2),x, algor ithm="fricas")
Output:
Timed out
\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(3/2),x )
Output:
Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/(-c*(a*x - 1)*(a*x + 1))**(3 /2), x)
Exception generated. \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(3/2),x, algor ithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(3/2),x, algor ithm="giac")
Output:
integrate(sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1))/(-a^2*c*x^2 + c)^(3/2), x)
Time = 26.82 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {4\,x}{3\,c}-\frac {2}{3\,a\,c}\right )}{\sqrt {c-a^2\,c\,x^2}} \] Input:
int(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)/(c - a^2*c*x^2)^(3/2),x)
Output:
(((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2)*((4*x)/(3*c) - 2/(3*a*c)))/(c - a^2 *c*x^2)^(1/2)
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {1}{2} \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {c}\, \left (-a x +1\right )^{\frac {1}{4}} \left (-2 a x +1\right )}{3 \left (a x +1\right )^{\frac {1}{4}} a \,c^{2} \left (a x -1\right )} \] Input:
int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2)/(-a^2*c*x^2+c)^(3/2),x)
Output:
(2*sqrt(c)*( - a*x + 1)**(1/4)*( - 2*a*x + 1))/(3*(a*x + 1)**(1/4)*a*c**2* (a*x - 1))