Integrand size = 22, antiderivative size = 64 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Chi}(2 \text {arctanh}(a x))}{a^3} \]
-1/2*x^2/a/(-a^2*x^2+1)/arctanh(a*x)^2-x/a^2/(-a^2*x^2+1)/arctanh(a*x)+Chi (2*arctanh(a*x))/a^3
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\frac {\frac {a x (a x+2 \text {arctanh}(a x))}{\left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2}+2 \text {Chi}(2 \text {arctanh}(a x))}{2 a^3} \]
((a*x*(a*x + 2*ArcTanh[a*x]))/((-1 + a^2*x^2)*ArcTanh[a*x]^2) + 2*CoshInte gral[2*ArcTanh[a*x]])/(2*a^3)
Time = 1.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.69, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6568, 6594, 6530, 3042, 3793, 2009, 6596, 3042, 25, 3793, 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.
\(\displaystyle \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx\) |
\(\Big \downarrow \) 6568 |
\(\displaystyle \frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6594 |
\(\displaystyle \frac {\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx}{a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6530 |
\(\displaystyle \frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+\frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6596 |
\(\displaystyle \frac {\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+\frac {\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+\frac {-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}}{a}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
-1/2*x^2/(a*(1 - a^2*x^2)*ArcTanh[a*x]^2) + (-(x/(a*(1 - a^2*x^2)*ArcTanh[ a*x])) + (CoshIntegral[2*ArcTanh[a*x]]/2 - Log[ArcTanh[a*x]]/2)/a^2 + (Cos hIntegral[2*ArcTanh[a*x]]/2 + Log[ArcTanh[a*x]]/2)/a^2)/a
3.3.93.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x _Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && I LtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Arc Tanh[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[f*(m/(b*c*(p + 1))) Int[(f *x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 2, 0] && Lt Q[p, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^( p + 1)/(b*c*d*(p + 1))), x] + (Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^( m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p + 1), x], x]) / ; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && LtQ[q, - 1] && LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sinh[x]^ m/Cosh[x]^(m + 2*(q + 1))), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (In tegerQ[q] || GtQ[d, 0])
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{3}}\) | \(51\) |
default | \(\frac {\frac {1}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a^{3}}\) | \(51\) |
1/a^3*(1/4/arctanh(a*x)^2-1/4/arctanh(a*x)^2*cosh(2*arctanh(a*x))-1/2*sinh (2*arctanh(a*x))/arctanh(a*x)+Chi(2*arctanh(a*x)))
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (59) = 118\).
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.05 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\frac {4 \, a^{2} x^{2} + 4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{2 \, {\left (a^{5} x^{2} - a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
1/2*(4*a^2*x^2 + 4*a*x*log(-(a*x + 1)/(a*x - 1)) + ((a^2*x^2 - 1)*log_inte gral(-(a*x + 1)/(a*x - 1)) + (a^2*x^2 - 1)*log_integral(-(a*x - 1)/(a*x + 1)))*log(-(a*x + 1)/(a*x - 1))^2)/((a^5*x^2 - a^3)*log(-(a*x + 1)/(a*x - 1 ))^2)
\[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int \frac {x^{2}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
2*(a*x^2 + x*log(a*x + 1) - x*log(-a*x + 1))/((a^4*x^2 - a^2)*log(a*x + 1) ^2 - 2*(a^4*x^2 - a^2)*log(a*x + 1)*log(-a*x + 1) + (a^4*x^2 - a^2)*log(-a *x + 1)^2) - integrate(-2*(a^2*x^2 + 1)/((a^6*x^4 - 2*a^4*x^2 + a^2)*log(a *x + 1) - (a^6*x^4 - 2*a^4*x^2 + a^2)*log(-a*x + 1)), x)
\[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int \frac {x^2}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \]