Integrand size = 25, antiderivative size = 150 \[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=-\frac {4 x \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {a x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^2 \sqrt {1-a^2 x^2}} \] Output:
-4*x*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3/2*x^2*(-a^2*c*x^2+c)^(1/2 )/(-a^2*x^2+1)^(1/2)-1/3*a*x^3*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)-4*( -a^2*c*x^2+c)^(1/2)*ln(-a*x+1)/a^2/(-a^2*x^2+1)^(1/2)
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.43 \[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (-\frac {4 x}{a}-\frac {3 x^2}{2}-\frac {a x^3}{3}-\frac {4 \log (1-a x)}{a^2}\right )}{\sqrt {1-a^2 x^2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])*x*Sqrt[c - a^2*c*x^2],x]
Output:
(Sqrt[c - a^2*c*x^2]*((-4*x)/a - (3*x^2)/2 - (a*x^3)/3 - (4*Log[1 - a*x])/ a^2))/Sqrt[1 - a^2*x^2]
Time = 0.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6703, 6700, 86, 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.
\(\displaystyle \int x e^{3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int e^{3 \text {arctanh}(a x)} x \sqrt {1-a^2 x^2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {x (a x+1)^2}{1-a x}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \left (-a x^2-3 x-\frac {4}{a}-\frac {4}{a (a x-1)}\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \left (-\frac {4 \log (1-a x)}{a^2}-\frac {a x^3}{3}-\frac {4 x}{a}-\frac {3 x^2}{2}\right )}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[E^(3*ArcTanh[a*x])*x*Sqrt[c - a^2*c*x^2],x]
Output:
(Sqrt[c - a^2*c*x^2]*((-4*x)/a - (3*x^2)/2 - (a*x^3)/3 - (4*Log[1 - a*x])/ a^2))/Sqrt[1 - a^2*x^2]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.41
method | result | size |
default | \(-\frac {\left (2 a^{3} x^{3}+9 a^{2} x^{2}+24 a x +24 \ln \left (a x -1\right )\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{6 \sqrt {-a^{2} x^{2}+1}\, a^{2}}\) | \(61\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^(1/2),x,method=_RETURNVE RBOSE)
Output:
-1/6*(2*a^3*x^3+9*a^2*x^2+24*a*x+24*ln(a*x-1))/(-a^2*x^2+1)^(1/2)*(-c*(a^2 *x^2-1))^(1/2)/a^2
Time = 0.11 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.45 \[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\left [\frac {12 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (\frac {a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) + {\left (2 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 24 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{4} x^{2} - a^{2}\right )}}, -\frac {24 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right ) - {\left (2 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 24 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{4} x^{2} - a^{2}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^(1/2),x, algorithm ="fricas")
Output:
[1/6*(12*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 - 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 + 4*a*c*x + (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x)*sqrt(- a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 - 2*a^3*x^3 + 2* a*x - 1)) + (2*a^3*x^3 + 9*a^2*x^2 + 24*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^ 2*x^2 + 1))/(a^4*x^2 - a^2), -1/6*(24*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(- a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^ 4 - 2*a^3*c*x^3 - a^2*c*x^2 + 2*a*c*x)) - (2*a^3*x^3 + 9*a^2*x^2 + 24*a*x) *sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^4*x^2 - a^2)]
\[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\int \frac {x \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x*(-a**2*c*x**2+c)**(1/2),x)
Output:
Integral(x*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1) )**(3/2), x)
Exception generated. \[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^(1/2),x, algorithm ="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.14 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.30 \[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=-\frac {1}{6} \, \sqrt {c} {\left (\frac {24 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{2}} + \frac {2 \, a^{4} x^{3} + 9 \, a^{3} x^{2} + 24 \, a^{2} x}{a^{3}}\right )} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^(1/2),x, algorithm ="giac")
Output:
-1/6*sqrt(c)*(24*log(abs(a*x - 1))/a^2 + (2*a^4*x^3 + 9*a^3*x^2 + 24*a^2*x )/a^3)
Timed out. \[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\int \frac {x\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((x*(c - a^2*c*x^2)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
int((x*(c - a^2*c*x^2)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2), x)
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24 \[ \int e^{3 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c}\, \left (-24 \,\mathrm {log}\left (a x -1\right )-2 a^{3} x^{3}-9 a^{2} x^{2}-24 a x \right )}{6 a^{2}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x*(-a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*( - 24*log(a*x - 1) - 2*a**3*x**3 - 9*a**2*x**2 - 24*a*x))/(6*a** 2)