Integrand size = 27, antiderivative size = 226 \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=-\frac {4 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-a^2 x^2}}-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}-\frac {4 x^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {3 x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}-\frac {a x^5 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^4 \sqrt {1-a^2 x^2}} \]
-4*x*(-a^2*c*x^2+c)^(1/2)/a^3/(-a^2*x^2+1)^(1/2)-2*x^2*(-a^2*c*x^2+c)^(1/2 )/a^2/(-a^2*x^2+1)^(1/2)-4/3*x^3*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2) -3/4*x^4*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)-1/5*a*x^5*(-a^2*c*x^2+c)^ (1/2)/(-a^2*x^2+1)^(1/2)-4*ln(-a*x+1)*(-a^2*c*x^2+c)^(1/2)/a^4/(-a^2*x^2+1 )^(1/2)
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.35 \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=-\frac {\sqrt {c-a^2 c x^2} \left (a x \left (240+120 a x+80 a^2 x^2+45 a^3 x^3+12 a^4 x^4\right )+240 \log (1-a x)\right )}{60 a^4 \sqrt {1-a^2 x^2}} \]
-1/60*(Sqrt[c - a^2*c*x^2]*(a*x*(240 + 120*a*x + 80*a^2*x^2 + 45*a^3*x^3 + 12*a^4*x^4) + 240*Log[1 - a*x]))/(a^4*Sqrt[1 - a^2*x^2])
Time = 0.51 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.36, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6703, 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.
\(\displaystyle \int x^3 e^{3 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {1-a^2 x^2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \frac {x^3 (a x+1)^2}{1-a x}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \int \left (-a x^4-3 x^3-\frac {4 x^2}{a}-\frac {4 x}{a^2}-\frac {4}{a^3 (a x-1)}-\frac {4}{a^3}\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c-a^2 c x^2} \left (-\frac {4 \log (1-a x)}{a^4}-\frac {4 x}{a^3}-\frac {2 x^2}{a^2}-\frac {a x^5}{5}-\frac {4 x^3}{3 a}-\frac {3 x^4}{4}\right )}{\sqrt {1-a^2 x^2}}\) |
(Sqrt[c - a^2*c*x^2]*((-4*x)/a^3 - (2*x^2)/a^2 - (4*x^3)/(3*a) - (3*x^4)/4 - (a*x^5)/5 - (4*Log[1 - a*x])/a^4))/Sqrt[1 - a^2*x^2]
3.12.56.3.1 Defintions of rubi rules used
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_.))*(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 = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.39
method | result | size |
default | \(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (12 a^{5} x^{5}+45 a^{4} x^{4}+80 a^{3} x^{3}+120 a^{2} x^{2}+240 a x +240 \ln \left (a x -1\right )\right )}{60 \left (a^{2} x^{2}-1\right ) a^{4}}\) | \(88\) |
1/60*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(12*a^5*x^5+45*a^4*x^4+80*a ^3*x^3+120*a^2*x^2+240*a*x+240*ln(a*x-1))/(a^2*x^2-1)/a^4
Time = 0.29 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.77 \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\left [\frac {120 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (\frac {a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) + {\left (12 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} + 120 \, a^{2} x^{2} + 240 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{60 \, {\left (a^{6} x^{2} - a^{4}\right )}}, -\frac {240 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right ) - {\left (12 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} + 120 \, a^{2} x^{2} + 240 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{60 \, {\left (a^{6} x^{2} - a^{4}\right )}}\right ] \]
[1/60*(120*(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)) + (12*a^5*x^5 + 45*a^4*x^4 + 80*a^3*x^3 + 120*a^2*x^2 + 240*a* x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4), -1/60*(240*(a ^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 2*a*x + 2)*sqr t(-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)) - (12*a^5*x^5 + 45*a^4*x^4 + 80*a^3*x^3 + 120*a^2*x^2 + 240*a*x)*sqrt(-a^2 *c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4)]
\[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3} x^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int e^{3 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^3\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]