Integrand size = 25, antiderivative size = 214 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (x)}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {5 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1-a x)}{4 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \log (1+a x)}{4 \left (c-a^2 c x^2\right )^{3/2}} \] Output:
-a^3*(1-1/a^2/x^2)^(3/2)*x^2/(-a^2*c*x^2+c)^(3/2)+1/2*a^4*(1-1/a^2/x^2)^(3 /2)*x^3/(-a*x+1)/(-a^2*c*x^2+c)^(3/2)+a^4*(1-1/a^2/x^2)^(3/2)*x^3*ln(x)/(- a^2*c*x^2+c)^(3/2)-5/4*a^4*(1-1/a^2/x^2)^(3/2)*x^3*ln(-a*x+1)/(-a^2*c*x^2+ c)^(3/2)+1/4*a^4*(1-1/a^2/x^2)^(3/2)*x^3*ln(a*x+1)/(-a^2*c*x^2+c)^(3/2)
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.37 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \left (-\frac {4}{x}+\frac {2 a}{1-a x}+4 a \log (x)-5 a \log (1-a x)+a \log (1+a x)\right )}{4 \left (c-a^2 c x^2\right )^{3/2}} \] Input:
Integrate[E^ArcCoth[a*x]/(x^2*(c - a^2*c*x^2)^(3/2)),x]
Output:
(a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*(-4/x + (2*a)/(1 - a*x) + 4*a*Log[x] - 5* a*Log[1 - a*x] + a*Log[1 + a*x]))/(4*(c - a^2*c*x^2)^(3/2))
Time = 0.86 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.38, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6746, 6747, 99, 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.
\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^5}dx}{\left (c-a^2 c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \int \frac {1}{x^2 (1-a x)^2 (a x+1)}dx}{\left (c-a^2 c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \int \left (-\frac {5 a^2}{4 (a x-1)}+\frac {a^2}{4 (a x+1)}+\frac {a^2}{2 (a x-1)^2}+\frac {a}{x}+\frac {1}{x^2}\right )dx}{\left (c-a^2 c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {a}{2 (1-a x)}+a \log (x)-\frac {5}{4} a \log (1-a x)+\frac {1}{4} a \log (a x+1)-\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\) |
Input:
Int[E^ArcCoth[a*x]/(x^2*(c - a^2*c*x^2)^(3/2)),x]
Output:
(a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*(-x^(-1) + a/(2*(1 - a*x)) + a*Log[x] - ( 5*a*Log[1 - a*x])/4 + (a*Log[1 + a*x])/4))/(c - a^2*c*x^2)^(3/2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[c^p/a^(2*p) Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.55
method | result | size |
default | \(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x +1\right ) x^{2} a^{2}-5 a^{2} \ln \left (a x -1\right ) x^{2}+4 a^{2} \ln \left (x \right ) x^{2}-\ln \left (a x +1\right ) x a +5 a \ln \left (a x -1\right ) x -4 a \ln \left (x \right ) x -6 a x +4\right )}{4 \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-1\right ) c^{2} x}\) | \(118\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/x^2/(-a^2*c*x^2+c)^(3/2),x,method=_RETURNVER BOSE)
Output:
1/4/((a*x-1)/(a*x+1))^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(ln(a*x+1)*x^2*a^2-5*a^ 2*ln(a*x-1)*x^2+4*a^2*ln(x)*x^2-ln(a*x+1)*x*a+5*a*ln(a*x-1)*x-4*a*ln(x)*x- 6*a*x+4)/(a^2*x^2-1)/c^2/x
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.43 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-a^{2} c} {\left (6 \, a x - {\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right ) + 5 \, {\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 4 \, {\left (a^{2} x^{2} - a x\right )} \log \left (x\right ) - 4\right )}}{4 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm= "fricas")
Output:
-1/4*sqrt(-a^2*c)*(6*a*x - (a^2*x^2 - a*x)*log(a*x + 1) + 5*(a^2*x^2 - a*x )*log(a*x - 1) - 4*(a^2*x^2 - a*x)*log(x) - 4)/(a^2*c^2*x^2 - a*c^2*x)
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)/x**2/(-a**2*c*x**2+c)**(3/2),x)
Output:
Timed out
\[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm= "maxima")
Output:
integrate(1/((-a^2*c*x^2 + c)^(3/2)*x^2*sqrt((a*x - 1)/(a*x + 1))), x)
Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/x^2/(-a^2*c*x^2+c)^(3/2),x, algorithm= "giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int(1/(x^2*(c - a^2*c*x^2)^(3/2)*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
int(1/(x^2*(c - a^2*c*x^2)^(3/2)*((a*x - 1)/(a*x + 1))^(1/2)), x)
Time = 0.15 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, i \left (-\mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a^{2} x^{2}+\mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a x -4 \,\mathrm {log}\left (\sqrt {-a x +1}-1\right ) a^{2} x^{2}+4 \,\mathrm {log}\left (\sqrt {-a x +1}-1\right ) a x -\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a^{2} x^{2}+\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a x -4 \,\mathrm {log}\left (\sqrt {-a x +1}+1\right ) a^{2} x^{2}+4 \,\mathrm {log}\left (\sqrt {-a x +1}+1\right ) a x +10 \,\mathrm {log}\left (\sqrt {-a x +1}\right ) a^{2} x^{2}-10 \,\mathrm {log}\left (\sqrt {-a x +1}\right ) a x -6 a^{2} x^{2}+12 a x -4\right )}{4 c^{2} x \left (a x -1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/x^2/(-a^2*c*x^2+c)^(3/2),x)
Output:
(sqrt(c)*i*( - log(sqrt( - a*x + 1) - sqrt(2))*a**2*x**2 + log(sqrt( - a*x + 1) - sqrt(2))*a*x - 4*log(sqrt( - a*x + 1) - 1)*a**2*x**2 + 4*log(sqrt( - a*x + 1) - 1)*a*x - log(sqrt( - a*x + 1) + sqrt(2))*a**2*x**2 + log(sqr t( - a*x + 1) + sqrt(2))*a*x - 4*log(sqrt( - a*x + 1) + 1)*a**2*x**2 + 4*l og(sqrt( - a*x + 1) + 1)*a*x + 10*log(sqrt( - a*x + 1))*a**2*x**2 - 10*log (sqrt( - a*x + 1))*a*x - 6*a**2*x**2 + 12*a*x - 4))/(4*c**2*x*(a*x - 1))