Integrand size = 19, antiderivative size = 58 \[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}+\frac {\text {Chi}(2 \text {arctanh}(a x))}{a} \]
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\frac {1+2 a x \text {arctanh}(a x)+2 \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2 \text {Chi}(2 \text {arctanh}(a x))}{2 a \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2} \]
(1 + 2*a*x*ArcTanh[a*x] + 2*(-1 + a^2*x^2)*ArcTanh[a*x]^2*CoshIntegral[2*A rcTanh[a*x]])/(2*a*(-1 + a^2*x^2)*ArcTanh[a*x]^2)
Time = 1.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.78, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6528, 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.
| \(\displaystyle \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx\) |
| \(\Big \downarrow \) 6528 |
| \(\displaystyle a \int \frac {x}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^2}dx-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6594 |
| \(\displaystyle a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx}{a}+a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6530 |
| \(\displaystyle a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}+\frac {1}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \left (a \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)}dx+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 6596 |
| \(\displaystyle a \left (\frac {\int \frac {a^2 x^2}{\left (1-a^2 x^2\right ) \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+a \left (\frac {\int -\frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}+a \left (-\frac {\int \frac {\sin (i \text {arctanh}(a x))^2}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle a \left (-\frac {\int \left (\frac {1}{2 \text {arctanh}(a x)}-\frac {\cosh (2 \text {arctanh}(a x))}{2 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \left (\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))-\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}+\frac {\frac {1}{2} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{2} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right ) \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}\) |
-1/2*1/(a*(1 - a^2*x^2)*ArcTanh[a*x]^2) + a*(-(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)
3.3.95.3.1 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 + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + Simp[2*c*((q + 1)/(b*(p + 1))) Int[x*(d + e*x^2)^q*(a + b*A rcTanh[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && LtQ[p, -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_.)*(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 = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a}\) | \(51\) |
| default | \(\frac {-\frac {1}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{4 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {\sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{2 \,\operatorname {arctanh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{a}\) | \(51\) |
1/a*(-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. 122 vs. \(2 (53) = 106\).
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.10 \[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\frac {4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4}{2 \, {\left (a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
1/2*(4*a*x*log(-(a*x + 1)/(a*x - 1)) + ((a^2*x^2 - 1)*log_integral(-(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 + 4)/((a^3*x^2 - a)*log(-(a*x + 1)/(a*x - 1))^2)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int \frac {1}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
2*(a*x*log(a*x + 1) - a*x*log(-a*x + 1) + 1)/((a^3*x^2 - a)*log(a*x + 1)^2 - 2*(a^3*x^2 - a)*log(a*x + 1)*log(-a*x + 1) + (a^3*x^2 - a)*log(-a*x + 1 )^2) - integrate(-2*(a^2*x^2 + 1)/((a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) - (a^4*x^4 - 2*a^2*x^2 + 1)*log(-a*x + 1)), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (1-a^2 x^2\right )^2 \text {arctanh}(a x)^3} \, dx=\int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \]