Integrand size = 27, antiderivative size = 187 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=-\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2 \sqrt {1-a^2 x^2}}-\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}} x \log (1-a x)}{\sqrt {1-a^2 x^2}} \] Output:
-4*a^2*(c-c/a^2/x^2)^(1/2)/(-a^2*x^2+1)^(1/2)-1/3*(c-c/a^2/x^2)^(1/2)/x^2/ (-a^2*x^2+1)^(1/2)-3/2*a*(c-c/a^2/x^2)^(1/2)/x/(-a^2*x^2+1)^(1/2)+4*a^3*(c -c/a^2/x^2)^(1/2)*x*ln(x)/(-a^2*x^2+1)^(1/2)-4*a^3*(c-c/a^2/x^2)^(1/2)*x*l n(-a*x+1)/(-a^2*x^2+1)^(1/2)
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.40 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (-\frac {1}{3 x^3}-\frac {3 a}{2 x^2}-\frac {4 a^2}{x}+4 a^3 \log (x)-4 a^3 \log (1-a x)\right )}{\sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3,x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-1/3*1/x^3 - (3*a)/(2*x^2) - (4*a^2)/x + 4*a^3*L og[x] - 4*a^3*Log[1 - a*x]))/Sqrt[1 - a^2*x^2]
Time = 0.83 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.40, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6710, 6700, 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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {1-a^2 x^2}}{x^4}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(a x+1)^2}{x^4 (1-a x)}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \left (-\frac {4 a^4}{a x-1}+\frac {4 a^3}{x}+\frac {4 a^2}{x^2}+\frac {3 a}{x^3}+\frac {1}{x^4}\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (4 a^3 \log (x)-4 a^3 \log (1-a x)-\frac {4 a^2}{x}-\frac {3 a}{2 x^2}-\frac {1}{3 x^3}\right )}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3,x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-1/3*1/x^3 - (3*a)/(2*x^2) - (4*a^2)/x + 4*a^3*L og[x] - 4*a^3*Log[1 - a*x]))/Sqrt[1 - a^2*x^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^(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_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[(u/x^(2*p))*(1 - a ^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && !IntegerQ[n/2]
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.40
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (24 a^{3} \ln \left (a x -1\right ) x^{3}-24 \ln \left (x \right ) x^{3} a^{3}+24 a^{2} x^{2}+9 a x +2\right )}{6 x^{2} \sqrt {-a^{2} x^{2}+1}}\) | \(75\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^3,x,method=_RETURNV ERBOSE)
Output:
-1/6*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x^2*(24*a^3*ln(a*x-1)*x^3-24*ln(x)*x^3* a^3+24*a^2*x^2+9*a*x+2)/(-a^2*x^2+1)^(1/2)
Time = 0.19 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.83 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\left [\frac {12 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {-c} \log \left (-\frac {4 \, a^{5} c x^{5} - {\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} - {\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x - {\left (4 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - {\left (4 \, a^{4} - 6 \, a^{3} + 4 \, a^{2} - a\right )} x^{5} + 4 \, a^{2} x^{2} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) + {\left (24 \, a^{2} x^{2} - {\left (24 \, a^{2} + 9 \, a + 2\right )} x^{3} + 9 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left (a^{2} x^{4} - x^{2}\right )}}, \frac {24 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {c} \arctan \left (\frac {{\left (2 \, a^{2} x^{2} - {\left (2 \, a^{3} - 2 \, a^{2} + a\right )} x^{3} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - a^{2}\right )} c x^{4} - {\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) + {\left (24 \, a^{2} x^{2} - {\left (24 \, a^{2} + 9 \, a + 2\right )} x^{3} + 9 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, {\left (a^{2} x^{4} - x^{2}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^3,x, algorith m="fricas")
Output:
[1/6*(12*(a^4*x^4 - a^2*x^2)*sqrt(-c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6*a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5* a^2*c*x^2 - 4*a*c*x - (4*a^4*x^4 - 6*a^3*x^3 - (4*a^4 - 6*a^3 + 4*a^2 - a) *x^5 + 4*a^2*x^2 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/( a^2*x^2)) + c)/(a^4*x^6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + (24*a^2*x^2 - (24* a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a ^2*x^2)))/(a^2*x^4 - x^2), 1/6*(24*(a^4*x^4 - a^2*x^2)*sqrt(c)*arctan((2*a ^2*x^2 - (2*a^3 - 2*a^2 + a)*x^3 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(c)*sqrt((a ^2*c*x^2 - c)/(a^2*x^2))/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) + (24*a^2*x^2 - (24*a^2 + 9*a + 2)*x^3 + 9*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x^4 - x^2)]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2)/x**3,x)
Output:
Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.98 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=-\frac {1}{2} \, a^{3} {\left (-\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a} + \frac {2 i \, \sqrt {c} \log \left (x\right )}{a}\right )} - \frac {1}{2} i \, a^{2} \sqrt {c} \log \left (a x + 1\right ) + \frac {1}{2} i \, a^{2} \sqrt {c} \log \left (a x - 1\right ) - \frac {3}{2} \, {\left (i \, \sqrt {c} \log \left (a x + 1\right ) - i \, \sqrt {c} \log \left (a x - 1\right ) - \frac {2 i \, \sqrt {c}}{a x}\right )} a^{2} - \frac {3}{2} \, {\left (-i \, a \sqrt {c} \log \left (a x + 1\right ) - i \, a \sqrt {c} \log \left (a x - 1\right ) + 2 i \, a \sqrt {c} \log \left (x\right ) - \frac {i \, \sqrt {c}}{a x^{2}}\right )} a + \frac {3 i \, a^{2} \sqrt {c} x^{2} + i \, \sqrt {c}}{3 \, a x^{3}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^3,x, algorith m="maxima")
Output:
-1/2*a^3*(-I*sqrt(c)*log(a*x + 1)/a - I*sqrt(c)*log(a*x - 1)/a + 2*I*sqrt( c)*log(x)/a) - 1/2*I*a^2*sqrt(c)*log(a*x + 1) + 1/2*I*a^2*sqrt(c)*log(a*x - 1) - 3/2*(I*sqrt(c)*log(a*x + 1) - I*sqrt(c)*log(a*x - 1) - 2*I*sqrt(c)/ (a*x))*a^2 - 3/2*(-I*a*sqrt(c)*log(a*x + 1) - I*a*sqrt(c)*log(a*x - 1) + 2 *I*a*sqrt(c)*log(x) - I*sqrt(c)/(a*x^2))*a + 1/3*(3*I*a^2*sqrt(c)*x^2 + I* sqrt(c))/(a*x^3)
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.31 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=-\frac {1}{6} \, {\left (24 \, a \log \left ({\left | a x - 1 \right |}\right ) \mathrm {sgn}\left (x\right ) - 24 \, a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) + \frac {24 \, a^{2} x^{2} \mathrm {sgn}\left (x\right ) + 9 \, a x \mathrm {sgn}\left (x\right ) + 2 \, \mathrm {sgn}\left (x\right )}{a^{2} x^{3}}\right )} \sqrt {-c} {\left | a \right |} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^3,x, algorith m="giac")
Output:
-1/6*(24*a*log(abs(a*x - 1))*sgn(x) - 24*a*log(abs(x))*sgn(x) + (24*a^2*x^ 2*sgn(x) + 9*a*x*sgn(x) + 2*sgn(x))/(a^2*x^3))*sqrt(-c)*abs(a)
Timed out. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^3}{x^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^3)/(x^3*(1 - a^2*x^2)^(3/2)),x)
Output:
int(((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^3)/(x^3*(1 - a^2*x^2)^(3/2)), x)
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.26 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx=\frac {\sqrt {c}\, i \left (-24 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+24 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-24 a^{2} x^{2}-9 a x -2\right )}{6 a \,x^{3}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^(1/2)/x^3,x)
Output:
(sqrt(c)*i*( - 24*log(a*x - 1)*a**3*x**3 + 24*log(x)*a**3*x**3 - 24*a**2*x **2 - 9*a*x - 2))/(6*a*x**3)