Integrand size = 23, antiderivative size = 69 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {e^{n \text {arctanh}(a x)}}{a^2 c^2 \left (4-n^2\right )}+\frac {e^{n \text {arctanh}(a x)} (2-a n x)}{a^2 c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]
-exp(n*arctanh(a*x))/a^2/c^2/(-n^2+4)+exp(n*arctanh(a*x))*(-a*n*x+2)/a^2/c ^2/(-n^2+4)/(-a^2*x^2+1)
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (1-a n x+a^2 x^2\right )}{a^2 c^2 \left (-4+n^2\right )} \]
Time = 0.50 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.57, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6695, 27, 6686, 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.
\(\displaystyle \int \frac {x e^{n \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6695 |
\(\displaystyle \frac {e^{n \text {arctanh}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}-\frac {n \int \frac {e^{n \text {arctanh}(a x)}}{c^2 \left (1-a^2 x^2\right )^2}dx}{2 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^{n \text {arctanh}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}-\frac {n \int \frac {e^{n \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^2}dx}{2 a c^2}\) |
\(\Big \downarrow \) 6686 |
\(\displaystyle \frac {e^{n \text {arctanh}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}-\frac {n \left (\frac {2 \int \frac {e^{n \text {arctanh}(a x)}}{1-a^2 x^2}dx}{4-n^2}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{2 a c^2}\) |
\(\Big \downarrow \) 6687 |
\(\displaystyle \frac {e^{n \text {arctanh}(a x)}}{2 a^2 c^2 \left (1-a^2 x^2\right )}-\frac {n \left (\frac {2 e^{n \text {arctanh}(a x)}}{a n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \text {arctanh}(a x)}}{a \left (4-n^2\right ) \left (1-a^2 x^2\right )}\right )}{2 a c^2}\) |
E^(n*ArcTanh[a*x])/(2*a^2*c^2*(1 - a^2*x^2)) - (n*((2*E^(n*ArcTanh[a*x]))/ (a*n*(4 - n^2)) - (E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*(4 - n^2)*(1 - a^2*x ^2))))/(2*a*c^2)
3.14.21.3.1 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)^(p_), x_Symbol] :> S imp[(n + 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcTanh[a*x])/(a*c*(n^2 - 4*(p + 1)^2))), x] - Simp[2*(p + 1)*((2*p + 3)/(c*(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]
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_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x^2)^(p + 1)*(E^(n*ArcTanh[a*x])/(2*d*(p + 1))), x] - Simp[ a*c*(n/(2*d*(p + 1))) Int[(c + d*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; Fre eQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && LtQ[p, -1] && !IntegerQ[n] && IntegerQ[2*p]
Time = 2.74 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(\frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (a^{2} x^{2}-a n x +1\right )}{\left (a^{2} x^{2}-1\right ) c^{2} a^{2} \left (n^{2}-4\right )}\) | \(47\) |
risch | \(\frac {\left (a^{2} x^{2}-a n x +1\right ) \left (-a x +1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{\left (a^{2} x^{2}-1\right ) c^{2} a^{2} \left (n^{2}-4\right )}\) | \(59\) |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2}-x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n +{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{c^{2} \left (a^{2} x^{2}-1\right ) \left (n^{2}-4\right ) a^{2}}\) | \(60\) |
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {{\left (a^{2} x^{2} - a n x + 1\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - 4 \, a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - 4 \, a^{4} c^{2}\right )} x^{2}} \]
-(a^2*x^2 - a*n*x + 1)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c^2*n^2 - 4*a^2 *c^2 - (a^4*c^2*n^2 - 4*a^4*c^2)*x^2)
\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=\begin {cases} \frac {x^{2}}{2 c^{2}} & \text {for}\: a = 0 \\- \frac {a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {2 a x \operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} + \frac {a x}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} - \frac {\operatorname {atanh}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname {atanh}{\left (a x \right )}} - 4 a^{2} c^{2} e^{2 \operatorname {atanh}{\left (a x \right )}}} & \text {for}\: n = -2 \\\frac {\int \frac {x e^{2 \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text {for}\: n = 2 \\\frac {a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} - \frac {a n x e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} + \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} & \text {otherwise} \end {cases} \]
Piecewise((x**2/(2*c**2), Eq(a, 0)), (-a**2*x**2*atanh(a*x)/(4*a**4*c**2*x **2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) - 2*a*x*atanh(a*x)/ (4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) + a*x /(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) - at anh(a*x)/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x ))), Eq(n, -2)), (Integral(x*exp(2*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2, Eq(n, 2)), (a**2*x**2*exp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) - a*n*x*exp(n*atanh(a*x ))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) + exp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n **2 + 4*a**2*c**2), True))
\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
\[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {x \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]
Time = 3.92 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \frac {e^{n \text {arctanh}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (a^2\,x^2-n\,a\,x+1\right )}{a^2\,c^2\,\left (n^2-4\right )\,\left (a^2\,x^2-1\right )} \]