Integrand size = 25, antiderivative size = 165 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-a^2 x^2}}{2 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log (x)}{c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2} \log (1-a x)}{4 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{4 c \sqrt {c-a^2 c x^2}} \] Output:
1/2*(-a^2*x^2+1)^(1/2)/c/(-a*x+1)/(-a^2*c*x^2+c)^(1/2)+(-a^2*x^2+1)^(1/2)* ln(x)/c/(-a^2*c*x^2+c)^(1/2)-3/4*(-a^2*x^2+1)^(1/2)*ln(-a*x+1)/c/(-a^2*c*x ^2+c)^(1/2)-1/4*(-a^2*x^2+1)^(1/2)*ln(a*x+1)/c/(-a^2*c*x^2+c)^(1/2)
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2-2 a x}+\log (x)-\frac {3}{4} \log (1-a x)-\frac {1}{4} \log (1+a x)\right )}{c \sqrt {c-a^2 c x^2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x*(c - a^2*c*x^2)^(3/2)),x]
Output:
(Sqrt[1 - a^2*x^2]*((2 - 2*a*x)^(-1) + Log[x] - (3*Log[1 - a*x])/4 - Log[1 + a*x]/4))/(c*Sqrt[c - a^2*c*x^2])
Time = 0.77 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.42, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6703, 6700, 93, 2009}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{\text {arctanh}(a x)}}{x \left (1-a^2 x^2\right )^{3/2}}dx}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {1}{x (1-a x)^2 (a x+1)}dx}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \left (-\frac {3 a}{4 (a x-1)}-\frac {a}{4 (a x+1)}+\frac {a}{2 (a x-1)^2}+\frac {1}{x}\right )dx}{c \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2 (1-a x)}-\frac {3}{4} \log (1-a x)-\frac {1}{4} \log (a x+1)+\log (x)\right )}{c \sqrt {c-a^2 c x^2}}\) |
Input:
Int[E^ArcTanh[a*x]/(x*(c - a^2*c*x^2)^(3/2)),x]
Output:
(Sqrt[1 - a^2*x^2]*(1/(2*(1 - a*x)) + Log[x] - (3*Log[1 - a*x])/4 - Log[1 + a*x]/4))/(c*Sqrt[c - a^2*c*x^2])
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.52
method | result | size |
default | \(-\frac {\left (\ln \left (a x +1\right ) x a +3 a \ln \left (a x -1\right ) x -4 a \ln \left (x \right ) x -\ln \left (a x +1\right )-3 \ln \left (a x -1\right )+4 \ln \left (x \right )+2\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{4 \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}\, c^{2}}\) | \(86\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(3/2),x,method=_RETURNVERB OSE)
Output:
-1/4*(ln(a*x+1)*x*a+3*a*ln(a*x-1)*x-4*a*ln(x)*x-ln(a*x+1)-3*ln(a*x-1)+4*ln (x)+2)/(a*x-1)/(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^2
\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm=" fricas")
Output:
integral(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^5*c^2*x^6 - a^4*c^2*x ^5 - 2*a^3*c^2*x^4 + 2*a^2*c^2*x^3 + a*c^2*x^2 - c^2*x), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {a x + 1}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a**2*c*x**2+c)**(3/2),x)
Output:
Integral((a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))* *(3/2)), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm=" maxima")
Output:
integrate((a*x + 1)/((-a^2*c*x^2 + c)^(3/2)*sqrt(-a^2*x^2 + 1)*x), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {-a^{2} x^{2} + 1} x} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm=" giac")
Output:
integrate((a*x + 1)/((-a^2*c*x^2 + c)^(3/2)*sqrt(-a^2*x^2 + 1)*x), x)
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {a\,x+1}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((a*x + 1)/(x*(c - a^2*c*x^2)^(3/2)*(1 - a^2*x^2)^(1/2)),x)
Output:
int((a*x + 1)/(x*(c - a^2*c*x^2)^(3/2)*(1 - a^2*x^2)^(1/2)), x)
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.38 \[ \int \frac {e^{\text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-3 \,\mathrm {log}\left (a x -1\right ) a x +3 \,\mathrm {log}\left (a x -1\right )-\mathrm {log}\left (a x +1\right ) a x +\mathrm {log}\left (a x +1\right )+4 \,\mathrm {log}\left (x \right ) a x -4 \,\mathrm {log}\left (x \right )-2 a x \right )}{4 c^{2} \left (a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(3/2),x)
Output:
(sqrt(c)*( - 3*log(a*x - 1)*a*x + 3*log(a*x - 1) - log(a*x + 1)*a*x + log( a*x + 1) + 4*log(x)*a*x - 4*log(x) - 2*a*x))/(4*c**2*(a*x - 1))