Integrand size = 24, antiderivative size = 146 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}{2 \left (1-a^2 x^2\right )^{3/2}}+\frac {3 a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{\left (1-a^2 x^2\right )^{3/2}}-\frac {a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^4}{\left (1-a^2 x^2\right )^{3/2}}+\frac {3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \log (x)}{\left (1-a^2 x^2\right )^{3/2}} \] Output:
-1/2*(c-c/a^2/x^2)^(3/2)*x/(-a^2*x^2+1)^(3/2)+3*a*(c-c/a^2/x^2)^(3/2)*x^2/ (-a^2*x^2+1)^(3/2)-a^3*(c-c/a^2/x^2)^(3/2)*x^4/(-a^2*x^2+1)^(3/2)+3*a^2*(c -c/a^2/x^2)^(3/2)*x^3*ln(x)/(-a^2*x^2+1)^(3/2)
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.44 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (1-6 a x+2 a^3 x^3-6 a^2 x^2 \log (x)\right )}{2 a^2 x \sqrt {1-a^2 x^2}} \] Input:
Integrate[(c - c/(a^2*x^2))^(3/2)/E^(3*ArcTanh[a*x]),x]
Output:
(c*Sqrt[c - c/(a^2*x^2)]*(1 - 6*a*x + 2*a^3*x^3 - 6*a^2*x^2*Log[x]))/(2*a^ 2*x*Sqrt[1 - a^2*x^2])
Time = 0.45 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.41, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6710, 6700, 49, 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.
\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \int \frac {e^{-3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{3/2}}{x^3}dx}{\left (1-a^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \int \frac {(1-a x)^3}{x^3}dx}{\left (1-a^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \int \left (-a^3+\frac {3 a^2}{x}-\frac {3 a}{x^2}+\frac {1}{x^3}\right )dx}{\left (1-a^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (a^3 (-x)+3 a^2 \log (x)+\frac {3 a}{x}-\frac {1}{2 x^2}\right )}{\left (1-a^2 x^2\right )^{3/2}}\) |
Input:
Int[(c - c/(a^2*x^2))^(3/2)/E^(3*ArcTanh[a*x]),x]
Output:
((c - c/(a^2*x^2))^(3/2)*x^3*(-1/2*1/x^2 + (3*a)/x - a^3*x + 3*a^2*Log[x]) )/(1 - a^2*x^2)^(3/2)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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 = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.48
method | result | size |
default | \(-\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} x \left (-2 a^{3} x^{3}+6 a^{2} \ln \left (x \right ) x^{2}+6 a x -1\right )}{2 \left (a^{2} x^{2}-1\right ) \sqrt {-a^{2} x^{2}+1}}\) | \(70\) |
Input:
int((c-c/a^2/x^2)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBO SE)
Output:
-1/2*(c*(a^2*x^2-1)/a^2/x^2)^(3/2)*x/(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)*(-2*a^ 3*x^3+6*a^2*ln(x)*x^2+6*a*x-1)
Time = 0.12 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.59 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\left [\frac {3 \, {\left (a^{3} c x^{3} - a c x\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - {\left (2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - 6 \, a + 1\right )} c x^{2} - 6 \, a c x + c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} x^{3} - a^{2} x\right )}}, \frac {6 \, {\left (a^{3} c x^{3} - a c x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} - a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} + {\left (a^{2} - 1\right )} c x^{2} - c}\right ) - {\left (2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - 6 \, a + 1\right )} c x^{2} - 6 \, a c x + c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{4} x^{3} - a^{2} x\right )}}\right ] \] Input:
integrate((c-c/a^2/x^2)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="f ricas")
Output:
[1/2*(3*(a^3*c*x^3 - a*c*x)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 - (a*x^5 - a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (2*a^3*c*x^3 - (2*a^3 - 6*a + 1)*c*x^2 - 6*a*c*x + c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x), 1/2*(6*(a^3*c*x^3 - a*c*x)*sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 - a*x )*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 + (a^2 - 1)*c*x^2 - c )) - (2*a^3*c*x^3 - (2*a^3 - 6*a + 1)*c*x^2 - 6*a*c*x + c)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^4*x^3 - a^2*x)]
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \] Input:
integrate((c-c/a**2/x**2)**(3/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))** (3/2)/(a*x + 1)**3, x)
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="m axima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^(3/2)/(a*x + 1)^3, x)
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.35 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {1}{2} \, {\left (\frac {2 \, c x \mathrm {sgn}\left (x\right )}{a} - \frac {6 \, c \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {6 \, a c x \mathrm {sgn}\left (x\right ) - c \mathrm {sgn}\left (x\right )}{a^{4} x^{2}}\right )} \sqrt {-c} {\left | a \right |} \] Input:
integrate((c-c/a^2/x^2)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="g iac")
Output:
1/2*(2*c*x*sgn(x)/a - 6*c*log(abs(x))*sgn(x)/a^2 - (6*a*c*x*sgn(x) - c*sgn (x))/(a^4*x^2))*sqrt(-c)*abs(a)
Timed out. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \] Input:
int(((c - c/(a^2*x^2))^(3/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
Output:
int(((c - c/(a^2*x^2))^(3/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3, x)
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.25 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {\sqrt {c}\, c i \left (6 \,\mathrm {log}\left (x \right ) a^{2} x^{2}-2 a^{3} x^{3}+6 a x -1\right )}{2 a^{3} x^{2}} \] Input:
int((c-c/a^2/x^2)^(3/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
Output:
(sqrt(c)*c*i*(6*log(x)*a**2*x**2 - 2*a**3*x**3 + 6*a*x - 1))/(2*a**3*x**2)