Integrand size = 25, antiderivative size = 79 \[ \int \frac {e^{n \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {e^{n \text {arctanh}(a x)} \left (2-n^2\right )}{a^3 c^2 n \left (4-n^2\right )}-\frac {e^{n \text {arctanh}(a x)} (n-2 a x)}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]
-exp(n*arctanh(a*x))*(-n^2+2)/a^3/c^2/n/(-n^2+4)-exp(n*arctanh(a*x))*(-2*a *x+n)/a^3/c^2/(-n^2+4)/(-a^2*x^2+1)
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82 \[ \int \frac {e^{n \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (-2+2 a n x-a^2 \left (-2+n^2\right ) x^2\right )}{a^3 c^2 n \left (-4+n^2\right )} \]
-(((1 - a*x)^(-1 - n/2)*(1 + a*x)^(-1 + n/2)*(-2 + 2*a*n*x - a^2*(-2 + n^2 )*x^2))/(a^3*c^2*n*(-4 + n^2)))
Time = 0.41 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6697, 27, 6687}
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.
| \(\displaystyle \int \frac {x^2 e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6697 |
| \(\displaystyle -\frac {\left (2-n^2\right ) \int \frac {e^{n \text {arctanh}(a x)}}{c \left (1-a^2 x^2\right )}dx}{a^2 c \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (2-n^2\right ) \int \frac {e^{n \text {arctanh}(a x)}}{1-a^2 x^2}dx}{a^2 c^2 \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\) |
| \(\Big \downarrow \) 6687 |
| \(\displaystyle -\frac {\left (2-n^2\right ) e^{n \text {arctanh}(a x)}}{a^3 c^2 n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a^3 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\) |
-((E^(n*ArcTanh[a*x])*(2 - n^2))/(a^3*c^2*n*(4 - n^2))) - (E^(n*ArcTanh[a* x])*(n - 2*a*x))/(a^3*c^2*(4 - n^2)*(1 - a^2*x^2))
3.14.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[ E^(n*ArcTanh[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^2*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(-(n + 2*(p + 1)*a*x))*(c + d*x^2)^(p + 1)*(E^(n*ArcTanh[a*x])/( a*d*(n^2 - 4*(p + 1)^2))), x] + Simp[(n^2 + 2*(p + 1))/(d*(n^2 - 4*(p + 1)^ 2)) Int[(c + d*x^2)^(p + 1)*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && !IntegerQ[n] && NeQ[n^2 - 4* (p + 1)^2, 0] && IntegerQ[2*p]
Time = 2.71 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78
| method | result | size |
| gosper | \(-\frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (a^{2} n^{2} x^{2}-2 a^{2} x^{2}-2 a n x +2\right )}{\left (a^{2} x^{2}-1\right ) c^{2} a^{3} n \left (n^{2}-4\right )}\) | \(62\) |
| risch | \(-\frac {\left (a^{2} n^{2} x^{2}-2 a^{2} x^{2}-2 a n x +2\right ) \left (-a x +1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{\left (a^{2} x^{2}-1\right ) c^{2} a^{3} n \left (n^{2}-4\right )}\) | \(74\) |
| parallelrisch | \(\frac {-x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2} n^{2}+2 x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2}+2 x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n -2 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{c^{2} \left (a^{2} x^{2}-1\right ) a^{3} n \left (n^{2}-4\right )}\) | \(84\) |
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16 \[ \int \frac {e^{n \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {{\left (2 \, a n x - {\left (a^{2} n^{2} - 2 \, a^{2}\right )} x^{2} - 2\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n - {\left (a^{5} c^{2} n^{3} - 4 \, a^{5} c^{2} n\right )} x^{2}} \]
-(2*a*n*x - (a^2*n^2 - 2*a^2)*x^2 - 2)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^3 *c^2*n^3 - 4*a^3*c^2*n - (a^5*c^2*n^3 - 4*a^5*c^2*n)*x^2)
\[ \int \frac {e^{n \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx=\begin {cases} \frac {x^{3}}{3 c^{2}} & \text {for}\: a = 0 \\- \frac {a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{5} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{3} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {2 a x \operatorname {atanh}{\left (a x \right )}}{4 a^{5} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{3} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {3 a x}{4 a^{5} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{3} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{5} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{3} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {2}{4 a^{5} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{3} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} & \text {for}\: n = -2 \\\frac {a^{2} x^{2} \log {\left (x - \frac {1}{a} \right )}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} - \frac {a^{2} x^{2} \log {\left (x + \frac {1}{a} \right )}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} - \frac {2 a x}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} - \frac {\log {\left (x - \frac {1}{a} \right )}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} + \frac {\log {\left (x + \frac {1}{a} \right )}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} & \text {for}\: n = 0 \\\frac {\int \frac {x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text {for}\: n = 2 \\- \frac {a^{2} n^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} + \frac {2 a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} + \frac {2 a n x e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} - \frac {2 e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} & \text {otherwise} \end {cases} \]
Piecewise((x**3/(3*c**2), Eq(a, 0)), (-a**2*x**2*atanh(a*x)/(4*a**5*c**2*x **2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))) - 2*a*x*atanh(a*x)/ (4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))) - 3*a *x/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))) - atanh(a*x)/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a *x))) - 2/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a* x))), Eq(n, -2)), (a**2*x**2*log(x - 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2) - a**2*x**2*log(x + 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2) - 2*a*x/(4*a**5 *c**2*x**2 - 4*a**3*c**2) - log(x - 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2) + log(x + 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2), Eq(n, 0)), (Integral(x**2 *exp(2*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2, Eq(n, 2)), (-a* *2*n**2*x**2*exp(n*atanh(a*x))/(a**5*c**2*n**3*x**2 - 4*a**5*c**2*n*x**2 - a**3*c**2*n**3 + 4*a**3*c**2*n) + 2*a**2*x**2*exp(n*atanh(a*x))/(a**5*c** 2*n**3*x**2 - 4*a**5*c**2*n*x**2 - a**3*c**2*n**3 + 4*a**3*c**2*n) + 2*a*n *x*exp(n*atanh(a*x))/(a**5*c**2*n**3*x**2 - 4*a**5*c**2*n*x**2 - a**3*c**2 *n**3 + 4*a**3*c**2*n) - 2*exp(n*atanh(a*x))/(a**5*c**2*n**3*x**2 - 4*a**5 *c**2*n*x**2 - a**3*c**2*n**3 + 4*a**3*c**2*n), True))
\[ \int \frac {e^{n \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {x^{2} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {x^{2} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
Time = 3.70 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.18 \[ \int \frac {e^{n \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {2}{a^5\,c^2\,n\,\left (n^2-4\right )}-\frac {2\,x}{a^4\,c^2\,\left (n^2-4\right )}+\frac {x^2\,\left (n^2-2\right )}{a^3\,c^2\,n\,\left (n^2-4\right )}\right )}{\left (\frac {1}{a^2}-x^2\right )\,{\left (1-a\,x\right )}^{n/2}} \]