Integrand size = 27, antiderivative size = 216 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=a d \log (x)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(1+c x)-\frac {1}{2} b d \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b d \operatorname {PolyLog}(2,c x)-\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}(2,c x)-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )-b e \log (1-c x) \operatorname {PolyLog}(2,1-c x)+b e \log (1+c x) \operatorname {PolyLog}(2,1+c x)+b e \operatorname {PolyLog}(3,1-c x)-b e \operatorname {PolyLog}(3,1+c x) \]
a*d*ln(x)-1/2*b*e*ln(c*x)*ln(-c*x+1)^2+1/2*b*e*ln(-c*x)*ln(c*x+1)^2-1/2*b* d*polylog(2,-c*x)+1/2*b*e*(ln(-c*x+1)+ln(c*x+1)-ln(-c^2*x^2+1))*polylog(2, -c*x)+1/2*b*d*polylog(2,c*x)-1/2*b*e*(ln(-c*x+1)+ln(c*x+1)-ln(-c^2*x^2+1)) *polylog(2,c*x)-1/2*a*e*polylog(2,c^2*x^2)-b*e*ln(-c*x+1)*polylog(2,-c*x+1 )+b*e*ln(c*x+1)*polylog(2,c*x+1)+b*e*polylog(3,-c*x+1)-b*e*polylog(3,c*x+1 )
\[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx \]
Time = 1.14 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6641, 6446, 6639, 2838, 6637, 2843, 2881, 2821, 6446, 7143}
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.
\(\displaystyle \int \frac {(a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )}{x} \, dx\) |
\(\Big \downarrow \) 6641 |
\(\displaystyle e \int \frac {(a+b \text {arctanh}(c x)) \log \left (1-c^2 x^2\right )}{x}dx+d \int \frac {a+b \text {arctanh}(c x)}{x}dx\) |
\(\Big \downarrow \) 6446 |
\(\displaystyle e \int \frac {(a+b \text {arctanh}(c x)) \log \left (1-c^2 x^2\right )}{x}dx+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 6639 |
\(\displaystyle e \left (a \int \frac {\log \left (1-c^2 x^2\right )}{x}dx+b \int \frac {\text {arctanh}(c x) \log \left (1-c^2 x^2\right )}{x}dx\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle e \left (b \int \frac {\text {arctanh}(c x) \log \left (1-c^2 x^2\right )}{x}dx-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 6637 |
\(\displaystyle e \left (b \left (-\left (\left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right ) \int \frac {\text {arctanh}(c x)}{x}dx\right )-\frac {1}{2} \int \frac {\log ^2(1-c x)}{x}dx+\frac {1}{2} \int \frac {\log ^2(c x+1)}{x}dx\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 2843 |
\(\displaystyle e \left (b \left (-\left (\left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right ) \int \frac {\text {arctanh}(c x)}{x}dx\right )+\frac {1}{2} \left (-2 c \int \frac {\log (c x) \log (1-c x)}{1-c x}dx-\log (c x) \log ^2(1-c x)\right )+\frac {1}{2} \left (\log (-c x) \log ^2(c x+1)-2 c \int \frac {\log (-c x) \log (c x+1)}{c x+1}dx\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 2881 |
\(\displaystyle e \left (b \left (-\left (\left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right ) \int \frac {\text {arctanh}(c x)}{x}dx\right )+\frac {1}{2} \left (2 \int \frac {\log (c x) \log (1-c x)}{1-c x}d(1-c x)-\log (c x) \log ^2(1-c x)\right )+\frac {1}{2} \left (\log (-c x) \log ^2(c x+1)-2 \int \frac {\log (-c x) \log (c x+1)}{c x+1}d(c x+1)\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle e \left (b \left (-\left (\left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right ) \int \frac {\text {arctanh}(c x)}{x}dx\right )+\frac {1}{2} \left (2 \left (\int \frac {\operatorname {PolyLog}(2,1-c x)}{1-c x}d(1-c x)-\operatorname {PolyLog}(2,1-c x) \log (1-c x)\right )-\log (c x) \log ^2(1-c x)\right )+\frac {1}{2} \left (\log (-c x) \log ^2(c x+1)-2 \left (\int \frac {\operatorname {PolyLog}(2,c x+1)}{c x+1}d(c x+1)-\operatorname {PolyLog}(2,c x+1) \log (c x+1)\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 6446 |
\(\displaystyle e \left (b \left (\frac {1}{2} \left (2 \left (\int \frac {\operatorname {PolyLog}(2,1-c x)}{1-c x}d(1-c x)-\operatorname {PolyLog}(2,1-c x) \log (1-c x)\right )-\log (c x) \log ^2(1-c x)\right )+\frac {1}{2} \left (\log (-c x) \log ^2(c x+1)-2 \left (\int \frac {\operatorname {PolyLog}(2,c x+1)}{c x+1}d(c x+1)-\operatorname {PolyLog}(2,c x+1) \log (c x+1)\right )\right )-\left (\left (\frac {\operatorname {PolyLog}(2,c x)}{2}-\frac {\operatorname {PolyLog}(2,-c x)}{2}\right ) \left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle e \left (b \left (-\left (\left (\frac {\operatorname {PolyLog}(2,c x)}{2}-\frac {\operatorname {PolyLog}(2,-c x)}{2}\right ) \left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right )\right )+\frac {1}{2} \left (2 (\operatorname {PolyLog}(3,1-c x)-\operatorname {PolyLog}(2,1-c x) \log (1-c x))-\log (c x) \log ^2(1-c x)\right )+\frac {1}{2} \left (\log (-c x) \log ^2(c x+1)-2 (\operatorname {PolyLog}(3,c x+1)-\operatorname {PolyLog}(2,c x+1) \log (c x+1))\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)-\frac {1}{2} b \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b \operatorname {PolyLog}(2,c x)\right )\) |
d*(a*Log[x] - (b*PolyLog[2, -(c*x)])/2 + (b*PolyLog[2, c*x])/2) + e*(-1/2* (a*PolyLog[2, c^2*x^2]) + b*(-((Log[1 - c*x] + Log[1 + c*x] - Log[1 - c^2* x^2])*(-1/2*PolyLog[2, -(c*x)] + PolyLog[2, c*x]/2)) + (-(Log[c*x]*Log[1 - c*x]^2) + 2*(-(Log[1 - c*x]*PolyLog[2, 1 - c*x]) + PolyLog[3, 1 - c*x]))/ 2 + (Log[-(c*x)]*Log[1 + c*x]^2 - 2*(-(Log[1 + c*x]*PolyLog[2, 1 + c*x]) + PolyLog[3, 1 + c*x]))/2))
3.6.27.3.1 Defintions of rubi rules used
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Simp[b*e*n*(p/g) Int[Log[(e*(f + g*x))/(e*f - d*g)] *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym bol] :> Simp[1/e Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* ((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x ] + (-Simp[(b/2)*PolyLog[2, (-c)*x], x] + Simp[(b/2)*PolyLog[2, c*x], x]) / ; FreeQ[{a, b, c}, x]
Int[(ArcTanh[(c_.)*(x_)]*Log[(f_.) + (g_.)*(x_)^2])/(x_), x_Symbol] :> Simp [(Log[f + g*x^2] - Log[1 - c*x] - Log[1 + c*x]) Int[ArcTanh[c*x]/x, x], x ] + (-Simp[1/2 Int[Log[1 - c*x]^2/x, x], x] + Simp[1/2 Int[Log[1 + c*x] ^2/x, x], x]) /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
Int[(Log[(f_.) + (g_.)*(x_)^2]*(ArcTanh[(c_.)*(x_)]*(b_.) + (a_)))/(x_), x_ Symbol] :> Simp[a Int[Log[f + g*x^2]/x, x], x] + Simp[b Int[Log[f + g*x ^2]*(ArcTanh[c*x]/x), x], x] /; FreeQ[{a, b, c, f, g}, x]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(Log[(f_.) + (g_.)*(x_)^2]*(e_.) + (d_)))/(x_), x_Symbol] :> Simp[d Int[(a + b*ArcTanh[c*x])/x, x], x] + Si mp[e Int[Log[f + g*x^2]*((a + b*ArcTanh[c*x])/x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.09 (sec) , antiderivative size = 1227, normalized size of antiderivative = 5.68
-1/8*(-2*Pi^2*b*e*csgn(I*(c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2+8*I*Pi*a*e*csg n(I*(c*x-1)*(c*x+1))^2-4*I*Pi*b*d*csgn(I*(c*x-1))^2+4*I*Pi*b*d*csgn(I*(c*x -1))^3-2*Pi^2*b*e*csgn(I*(c*x-1))^4*csgn(I*(c*x-1)*(c*x+1))^2-2*Pi^2*b*e*c sgn(I*(c*x-1))^3*csgn(I*(c*x-1)*(c*x+1))^3-2*Pi^2*b*e*csgn(I*(c*x+1))*csgn (I*(c*x-1)*(c*x+1))^2-8*I*Pi*a*e+4*I*Pi*b*d-2*Pi^2*b*e*csgn(I*(c*x-1))^3*c sgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))^2-2*Pi^2*b*e*csgn(I*(c*x-1)*(c*x+1) )^3-4*Pi^2*b*e*csgn(I*(c*x-1))^3-4*I*Pi*a*e*csgn(I*(c*x-1)*(c*x+1))^3+2*Pi ^2*b*e*csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+2*Pi^2*b*e* csgn(I*(c*x-1))^2*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))^2+2*Pi^2*b*e*csg n(I*(c*x-1))^4*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))-4*I*Pi*a*e*csgn(I*( c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2-4*I*Pi*a*e*csgn(I*(c*x+1))*csgn(I*(c*x-1 )*(c*x+1))^2+6*Pi^2*b*e*csgn(I*(c*x-1))^3*csgn(I*(c*x-1)*(c*x+1))^2+2*Pi^2 *b*e*csgn(I*(c*x-1))^2*csgn(I*(c*x-1)*(c*x+1))^3+4*Pi^2*b*e*csgn(I*(c*x-1) )^2+4*Pi^2*b*e*csgn(I*(c*x-1)*(c*x+1))^2-2*Pi^2*b*e*csgn(I*(c*x-1))^3*csgn (I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))-8*a*d-4*Pi^2*b*e-4*Pi^2*b*e*csgn(I*(c* x-1))^2*csgn(I*(c*x-1)*(c*x+1))^2+4*I*Pi*a*e*csgn(I*(c*x-1))*csgn(I*(c*x+1 ))*csgn(I*(c*x-1)*(c*x+1)))*ln(c*x)-1/8*(-4*I*Pi*b*e*csgn(I*(c*x-1))^2-4*I *Pi*b*e*csgn(I*(c*x-1)*(c*x+1))^2+4*I*Pi*b*e*csgn(I*(c*x-1))^3-2*I*Pi*b*e* csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+2*I*Pi*b*e*csgn(I* (c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2+2*I*Pi*b*e*csgn(I*(c*x+1))*csgn(I*(c...
\[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \]
\[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x}\, dx \]
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=-\frac {1}{2} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 2 \, {\rm Li}_2\left (-c x + 1\right ) \log \left (-c x + 1\right ) - 2 \, {\rm Li}_{3}(-c x + 1)\right )} b e + \frac {1}{2} \, {\left (\log \left (c x + 1\right )^{2} \log \left (-c x\right ) + 2 \, {\rm Li}_2\left (c x + 1\right ) \log \left (c x + 1\right ) - 2 \, {\rm Li}_{3}(c x + 1)\right )} b e + a d \log \left (x\right ) - \frac {1}{2} \, {\left (b d - 2 \, a e\right )} {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} + \frac {1}{2} \, {\left (b d + 2 \, a e\right )} {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} \]
-1/2*(log(c*x)*log(-c*x + 1)^2 + 2*dilog(-c*x + 1)*log(-c*x + 1) - 2*polyl og(3, -c*x + 1))*b*e + 1/2*(log(c*x + 1)^2*log(-c*x) + 2*dilog(c*x + 1)*lo g(c*x + 1) - 2*polylog(3, c*x + 1))*b*e + a*d*log(x) - 1/2*(b*d - 2*a*e)*( log(c*x)*log(-c*x + 1) + dilog(-c*x + 1)) + 1/2*(b*d + 2*a*e)*(log(c*x + 1 )*log(-c*x) + dilog(c*x + 1))
\[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x} \,d x \]