Integrand size = 19, antiderivative size = 89 \[ \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx=-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}-\frac {3 x}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}+\frac {15 \text {Chi}(2 \text {arctanh}(a x))}{16 a}+\frac {3 \text {Chi}(4 \text {arctanh}(a x))}{2 a}+\frac {9 \text {Chi}(6 \text {arctanh}(a x))}{16 a} \]
-1/2/a/(-a^2*x^2+1)^3/arctanh(a*x)^2-3*x/(-a^2*x^2+1)^3/arctanh(a*x)+15/16 *Chi(2*arctanh(a*x))/a+3/2*Chi(4*arctanh(a*x))/a+9/16*Chi(6*arctanh(a*x))/ a
Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx=\frac {1}{16} \left (\frac {8}{a \left (-1+a^2 x^2\right )^3 \text {arctanh}(a x)^2}+\frac {48 x}{\left (-1+a^2 x^2\right )^3 \text {arctanh}(a x)}+\frac {15 \text {Chi}(2 \text {arctanh}(a x))}{a}+\frac {24 \text {Chi}(4 \text {arctanh}(a x))}{a}+\frac {9 \text {Chi}(6 \text {arctanh}(a x))}{a}\right ) \]
(8/(a*(-1 + a^2*x^2)^3*ArcTanh[a*x]^2) + (48*x)/((-1 + a^2*x^2)^3*ArcTanh[ a*x]) + (15*CoshIntegral[2*ArcTanh[a*x]])/a + (24*CoshIntegral[4*ArcTanh[a *x]])/a + (9*CoshIntegral[6*ArcTanh[a*x]])/a)/16
Time = 1.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.67, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6528, 6594, 6530, 3042, 3793, 2009, 6596, 5971, 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.
\(\displaystyle \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx\) |
\(\Big \downarrow \) 6528 |
\(\displaystyle 3 a \int \frac {x}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^2}dx-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6594 |
\(\displaystyle 3 a \left (\frac {\int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)}dx}{a}+5 a \int \frac {x^2}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)}dx-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6530 |
\(\displaystyle 3 a \left (5 a \int \frac {x^2}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)}dx+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}+3 a \left (5 a \int \frac {x^2}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)}dx+\frac {\int \frac {\sin \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )^6}{\text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle 3 a \left (5 a \int \frac {x^2}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)}dx+\frac {\int \left (\frac {15 \cosh (2 \text {arctanh}(a x))}{32 \text {arctanh}(a x)}+\frac {3 \cosh (4 \text {arctanh}(a x))}{16 \text {arctanh}(a x)}+\frac {\cosh (6 \text {arctanh}(a x))}{32 \text {arctanh}(a x)}+\frac {5}{16 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 a \left (5 a \int \frac {x^2}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)}dx+\frac {\frac {15}{32} \text {Chi}(2 \text {arctanh}(a x))+\frac {3}{16} \text {Chi}(4 \text {arctanh}(a x))+\frac {1}{32} \text {Chi}(6 \text {arctanh}(a x))+\frac {5}{16} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 6596 |
\(\displaystyle 3 a \left (\frac {5 \int \frac {a^2 x^2}{\left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}d\text {arctanh}(a x)}{a^2}+\frac {\frac {15}{32} \text {Chi}(2 \text {arctanh}(a x))+\frac {3}{16} \text {Chi}(4 \text {arctanh}(a x))+\frac {1}{32} \text {Chi}(6 \text {arctanh}(a x))+\frac {5}{16} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle 3 a \left (\frac {5 \int \left (-\frac {\cosh (2 \text {arctanh}(a x))}{32 \text {arctanh}(a x)}+\frac {\cosh (4 \text {arctanh}(a x))}{16 \text {arctanh}(a x)}+\frac {\cosh (6 \text {arctanh}(a x))}{32 \text {arctanh}(a x)}-\frac {1}{16 \text {arctanh}(a x)}\right )d\text {arctanh}(a x)}{a^2}+\frac {\frac {15}{32} \text {Chi}(2 \text {arctanh}(a x))+\frac {3}{16} \text {Chi}(4 \text {arctanh}(a x))+\frac {1}{32} \text {Chi}(6 \text {arctanh}(a x))+\frac {5}{16} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 a \left (\frac {5 \left (-\frac {1}{32} \text {Chi}(2 \text {arctanh}(a x))+\frac {1}{16} \text {Chi}(4 \text {arctanh}(a x))+\frac {1}{32} \text {Chi}(6 \text {arctanh}(a x))-\frac {1}{16} \log (\text {arctanh}(a x))\right )}{a^2}+\frac {\frac {15}{32} \text {Chi}(2 \text {arctanh}(a x))+\frac {3}{16} \text {Chi}(4 \text {arctanh}(a x))+\frac {1}{32} \text {Chi}(6 \text {arctanh}(a x))+\frac {5}{16} \log (\text {arctanh}(a x))}{a^2}-\frac {x}{a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)}\right )-\frac {1}{2 a \left (1-a^2 x^2\right )^3 \text {arctanh}(a x)^2}\) |
-1/2*1/(a*(1 - a^2*x^2)^3*ArcTanh[a*x]^2) + 3*a*(-(x/(a*(1 - a^2*x^2)^3*Ar cTanh[a*x])) + (5*(-1/32*CoshIntegral[2*ArcTanh[a*x]] + CoshIntegral[4*Arc Tanh[a*x]]/16 + CoshIntegral[6*ArcTanh[a*x]]/32 - Log[ArcTanh[a*x]]/16))/a ^2 + ((15*CoshIntegral[2*ArcTanh[a*x]])/32 + (3*CoshIntegral[4*ArcTanh[a*x ]])/16 + CoshIntegral[6*ArcTanh[a*x]]/32 + (5*Log[ArcTanh[a*x]])/16)/a^2)
3.4.63.3.1 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[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
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 = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.47
method | result | size |
derivativedivides | \(\frac {-\frac {5}{32 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {15 \cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{64 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {15 \sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \,\operatorname {arctanh}\left (a x \right )}+\frac {15 \,\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{16}-\frac {3 \cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {3 \sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{2}-\frac {\cosh \left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{64 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {3 \sinh \left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \,\operatorname {arctanh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{16}}{a}\) | \(131\) |
default | \(\frac {-\frac {5}{32 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {15 \cosh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{64 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {15 \sinh \left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \,\operatorname {arctanh}\left (a x \right )}+\frac {15 \,\operatorname {Chi}\left (2 \,\operatorname {arctanh}\left (a x \right )\right )}{16}-\frac {3 \cosh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {3 \sinh \left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{8 \,\operatorname {arctanh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (4 \,\operatorname {arctanh}\left (a x \right )\right )}{2}-\frac {\cosh \left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{64 \operatorname {arctanh}\left (a x \right )^{2}}-\frac {3 \sinh \left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{32 \,\operatorname {arctanh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (6 \,\operatorname {arctanh}\left (a x \right )\right )}{16}}{a}\) | \(131\) |
1/a*(-5/32/arctanh(a*x)^2-15/64/arctanh(a*x)^2*cosh(2*arctanh(a*x))-15/32* sinh(2*arctanh(a*x))/arctanh(a*x)+15/16*Chi(2*arctanh(a*x))-3/32/arctanh(a *x)^2*cosh(4*arctanh(a*x))-3/8/arctanh(a*x)*sinh(4*arctanh(a*x))+3/2*Chi(4 *arctanh(a*x))-1/64/arctanh(a*x)^2*cosh(6*arctanh(a*x))-3/32/arctanh(a*x)* sinh(6*arctanh(a*x))+9/16*Chi(6*arctanh(a*x)))
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (79) = 158\).
Time = 0.24 (sec) , antiderivative size = 435, normalized size of antiderivative = 4.89 \[ \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx=\frac {192 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + 3 \, {\left (3 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 3 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 8 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 8 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 64}{32 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
1/32*(192*a*x*log(-(a*x + 1)/(a*x - 1)) + 3*(3*(a^6*x^6 - 3*a^4*x^4 + 3*a^ 2*x^2 - 1)*log_integral(-(a^3*x^3 + 3*a^2*x^2 + 3*a*x + 1)/(a^3*x^3 - 3*a^ 2*x^2 + 3*a*x - 1)) + 3*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log_integral (-(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)/(a^3*x^3 + 3*a^2*x^2 + 3*a*x + 1)) + 8 *(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log_integral((a^2*x^2 + 2*a*x + 1)/ (a^2*x^2 - 2*a*x + 1)) + 8*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log_integ ral((a^2*x^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x + 1)) + 5*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log_integral(-(a*x + 1)/(a*x - 1)) + 5*(a^6*x^6 - 3*a^4*x ^4 + 3*a^2*x^2 - 1)*log_integral(-(a*x - 1)/(a*x + 1)))*log(-(a*x + 1)/(a* x - 1))^2 + 64)/((a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log(-(a*x + 1)/(a*x - 1))^2)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx=\int \frac {1}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
2*(3*a*x*log(a*x + 1) - 3*a*x*log(-a*x + 1) + 1)/((a^7*x^6 - 3*a^5*x^4 + 3 *a^3*x^2 - a)*log(a*x + 1)^2 - 2*(a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log (a*x + 1)*log(-a*x + 1) + (a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*log(-a*x + 1)^2) - integrate(-6*(5*a^2*x^2 + 1)/((a^8*x^8 - 4*a^6*x^6 + 6*a^4*x^4 - 4*a^2*x^2 + 1)*log(a*x + 1) - (a^8*x^8 - 4*a^6*x^6 + 6*a^4*x^4 - 4*a^2*x^2 + 1)*log(-a*x + 1)), x)
\[ \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname {artanh}\left (a x\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (1-a^2 x^2\right )^4 \text {arctanh}(a x)^3} \, dx=\int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^4} \,d x \]