Integrand size = 21, antiderivative size = 599 \[ \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=-2 a e x+\frac {2 a e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-2 b e x \text {arctanh}(c x)+\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \log (1+c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac {b e \sqrt {-f} \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \sqrt {-f} \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {g}}-\frac {b e \sqrt {-f} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1+c x)}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {g}}+\frac {b e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c} \]
-2*a*e*x-2*b*e*x*arctanh(c*x)-b*e*ln(-c^2*x^2+1)/c+x*(a+b*arctanh(c*x))*(d +e*ln(g*x^2+f))+1/2*b*ln(g*(-c^2*x^2+1)/(c^2*f+g))*(d+e*ln(g*x^2+f))/c+1/2 *b*e*polylog(2,c^2*(g*x^2+f)/(c^2*f+g))/c+1/2*b*e*ln(-c*x+1)*ln(c*((-f)^(1 /2)-x*g^(1/2))/(c*(-f)^(1/2)-g^(1/2)))*(-f)^(1/2)/g^(1/2)-1/2*b*e*ln(c*x+1 )*ln(c*((-f)^(1/2)-x*g^(1/2))/(c*(-f)^(1/2)+g^(1/2)))*(-f)^(1/2)/g^(1/2)+1 /2*b*e*ln(c*x+1)*ln(c*((-f)^(1/2)+x*g^(1/2))/(c*(-f)^(1/2)-g^(1/2)))*(-f)^ (1/2)/g^(1/2)-1/2*b*e*ln(-c*x+1)*ln(c*((-f)^(1/2)+x*g^(1/2))/(c*(-f)^(1/2) +g^(1/2)))*(-f)^(1/2)/g^(1/2)+1/2*b*e*polylog(2,-(-c*x+1)*g^(1/2)/(c*(-f)^ (1/2)-g^(1/2)))*(-f)^(1/2)/g^(1/2)+1/2*b*e*polylog(2,-(c*x+1)*g^(1/2)/(c*( -f)^(1/2)-g^(1/2)))*(-f)^(1/2)/g^(1/2)-1/2*b*e*polylog(2,(-c*x+1)*g^(1/2)/ (c*(-f)^(1/2)+g^(1/2)))*(-f)^(1/2)/g^(1/2)-1/2*b*e*polylog(2,(c*x+1)*g^(1/ 2)/(c*(-f)^(1/2)+g^(1/2)))*(-f)^(1/2)/g^(1/2)+2*a*e*arctan(x*g^(1/2)/f^(1/ 2))*f^(1/2)/g^(1/2)
Result contains complex when optimal does not.
Time = 2.59 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.09 \[ \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx =\text {Too large to display} \]
a*d*x - 2*a*e*x + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + b* d*x*ArcTanh[c*x] + (b*d*Log[1 - c^2*x^2])/(2*c) + a*e*x*Log[f + g*x^2] + b *e*(x*ArcTanh[c*x] + Log[1 - c^2*x^2]/(2*c))*Log[f + g*x^2] - (b*e*g*(((-L og[-c^(-1) + x] - Log[c^(-1) + x] + Log[1 - c^2*x^2])*Log[f + g*x^2])/(2*g ) + (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/((-I)*Sqrt[f] - Sqrt [g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1) + x))/((-I)*Sqrt[f] - Sqrt[g]/c)])/ (2*g) + (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sqr t[g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sqrt[g]/c)])/(2 *g) + (Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt [g]/c)] + PolyLog[2, (Sqrt[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt[g]/c)])/( 2*g) + (Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(c^(-1) + x))/(I*Sqrt[f] + Sqrt[g ]/c)] + PolyLog[2, (Sqrt[g]*(c^(-1) + x))/(I*Sqrt[f] + Sqrt[g]/c)])/(2*g)) )/c - (b*e*(4*c*x*ArcTanh[c*x] - 4*Log[1/Sqrt[1 - c^2*x^2]] + (Sqrt[c^2*f* g]*((-2*I)*ArcCos[(-(c^2*f) + g)/(c^2*f + g)]*ArcTan[(c*g*x)/Sqrt[c^2*f*g] ] + 4*ArcTan[Sqrt[c^2*f*g]/(c*g*x)]*ArcTanh[c*x] - (ArcCos[(-(c^2*f) + g)/ (c^2*f + g)] - 2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[(2*c^2*f*(g + I*Sqrt[c ^2*f*g])*(1 + c*x))/((c^2*f + g)*(c^2*f + I*c*Sqrt[c^2*f*g]*x))] - (ArcCos [(-(c^2*f) + g)/(c^2*f + g)] + 2*ArcTan[(c*g*x)/Sqrt[c^2*f*g]])*Log[(2*c^2 *f*(I*g + Sqrt[c^2*f*g])*(-1 + c*x))/((c^2*f + g)*((-I)*c^2*f + c*Sqrt[c^2 *f*g]*x))] + (ArcCos[(-(c^2*f) + g)/(c^2*f + g)] + 2*(ArcTan[Sqrt[c^2*f...
Time = 1.78 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6635, 2925, 2841, 2840, 2838, 6542, 2009, 6536, 218, 6534, 2856, 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.
\(\displaystyle \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 6635 |
\(\displaystyle -2 e g \int \frac {x^2 (a+b \text {arctanh}(c x))}{g x^2+f}dx-b c \int \frac {x \left (d+e \log \left (g x^2+f\right )\right )}{1-c^2 x^2}dx+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )\) |
\(\Big \downarrow \) 2925 |
\(\displaystyle -2 e g \int \frac {x^2 (a+b \text {arctanh}(c x))}{g x^2+f}dx-\frac {1}{2} b c \int \frac {d+e \log \left (g x^2+f\right )}{1-c^2 x^2}dx^2+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle -2 e g \int \frac {x^2 (a+b \text {arctanh}(c x))}{g x^2+f}dx-\frac {1}{2} b c \left (\frac {e g \int \frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{f c^2+g}\right )}{g x^2+f}dx^2}{c^2}-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )\) |
\(\Big \downarrow \) 2840 |
\(\displaystyle -2 e g \int \frac {x^2 (a+b \text {arctanh}(c x))}{g x^2+f}dx-\frac {1}{2} b c \left (\frac {e \int \frac {\log \left (1-\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{x^2}d\left (g x^2+f\right )}{c^2}-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -2 e g \int \frac {x^2 (a+b \text {arctanh}(c x))}{g x^2+f}dx+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle -2 e g \left (\frac {\int (a+b \text {arctanh}(c x))dx}{g}-\frac {f \int \frac {a+b \text {arctanh}(c x)}{g x^2+f}dx}{g}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 e g \left (\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{g}-\frac {f \int \frac {a+b \text {arctanh}(c x)}{g x^2+f}dx}{g}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 6536 |
\(\displaystyle -2 e g \left (\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{g}-\frac {f \left (a \int \frac {1}{g x^2+f}dx+b \int \frac {\text {arctanh}(c x)}{g x^2+f}dx\right )}{g}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -2 e g \left (\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{g}-\frac {f \left (b \int \frac {\text {arctanh}(c x)}{g x^2+f}dx+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 6534 |
\(\displaystyle -2 e g \left (\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{g}-\frac {f \left (b \left (\frac {1}{2} \int \frac {\log (c x+1)}{g x^2+f}dx-\frac {1}{2} \int \frac {\log (1-c x)}{g x^2+f}dx\right )+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 2856 |
\(\displaystyle -2 e g \left (\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{g}-\frac {f \left (b \left (\frac {1}{2} \int \left (\frac {\sqrt {-f} \log (c x+1)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (c x+1)}{2 f \left (\sqrt {g} x+\sqrt {-f}\right )}\right )dx-\frac {1}{2} \int \left (\frac {\sqrt {-f} \log (1-c x)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (1-c x)}{2 f \left (\sqrt {g} x+\sqrt {-f}\right )}\right )dx\right )+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 e g \left (\frac {a x+b x \text {arctanh}(c x)+\frac {b \log \left (1-c^2 x^2\right )}{2 c}}{g}-\frac {f \left (\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+b \left (\frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (1-c x)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-c x)}{\sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log (1-c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log (1-c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}\right )+\frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c x+1)}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c x+1)}{\sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log (c x+1) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log (c x+1) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}\right )\right )\right )}{g}\right )+x (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
x*(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]) - 2*e*g*((a*x + b*x*ArcTanh[ c*x] + (b*Log[1 - c^2*x^2])/(2*c))/g - (f*((a*ArcTan[(Sqrt[g]*x)/Sqrt[f]]) /(Sqrt[f]*Sqrt[g]) + b*((-1/2*(Log[1 - c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x)) /(c*Sqrt[-f] - Sqrt[g])])/(Sqrt[-f]*Sqrt[g]) + (Log[1 - c*x]*Log[(c*(Sqrt[ -f] + Sqrt[g]*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - PolyLog[ 2, -((Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] - Sqrt[g]))]/(2*Sqrt[-f]*Sqrt[g]) + P olyLog[2, (Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] + Sqrt[g])]/(2*Sqrt[-f]*Sqrt[g]) )/2 + ((Log[1 + c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + Sqrt[g]) ])/(2*Sqrt[-f]*Sqrt[g]) - (Log[1 + c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c* Sqrt[-f] - Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - PolyLog[2, -((Sqrt[g]*(1 + c* x))/(c*Sqrt[-f] - Sqrt[g]))]/(2*Sqrt[-f]*Sqrt[g]) + PolyLog[2, (Sqrt[g]*(1 + c*x))/(c*Sqrt[-f] + Sqrt[g])]/(2*Sqrt[-f]*Sqrt[g]))/2)))/g) - (b*c*(-(( Log[(g*(1 - c^2*x^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]))/c^2) - (e*PolyL og[2, (c^2*(f + g*x^2))/(c^2*f + g)])/c^2))/2
3.6.34.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ Symbol] :> Simp[1/g Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c *(e*f - d*g), 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ )), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x )^n])/g), x] - Simp[b*e*(n/g) Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Int[ArcTanh[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/2 Int [Log[1 + c*x]/(d + e*x^2), x], x] - Simp[1/2 Int[Log[1 - c*x]/(d + e*x^2) , x], x] /; FreeQ[{c, d, e}, x]
Int[(ArcTanh[(c_.)*(x_)]*(b_.) + (a_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[a Int[1/(d + e*x^2), x], x] + Simp[b Int[ArcTanh[c*x]/(d + e*x^2) , x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]* (e_.)), x_Symbol] :> Simp[x*(d + e*Log[f + g*x^2])*(a + b*ArcTanh[c*x]), x] + (-Simp[b*c Int[x*((d + e*Log[f + g*x^2])/(1 - c^2*x^2)), x], x] - Simp [2*e*g Int[x^2*((a + b*ArcTanh[c*x])/(f + g*x^2)), x], x]) /; FreeQ[{a, b , c, d, e, f, g}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.70 (sec) , antiderivative size = 3550, normalized size of antiderivative = 5.93
1/2*b*d*ln(-c*x+1)/c+1/4*I*e*b*ln(c*x+1)*Pi*csgn(I*(c^2*f+((c*x+1)^2-2*c*x -1)*g))*csgn(I/c^2*(c^2*f+((c*x+1)^2-2*c*x-1)*g))^2*x+1/4*I*e*b*Pi*csgn(I/ c^2)*csgn(I*(c^2*f+((c*x+1)^2-2*c*x-1)*g))*csgn(I/c^2*(c^2*f+((c*x+1)^2-2* c*x-1)*g))*x+1/4*I*e*b*ln(c*x+1)*Pi*csgn(I/c^2)*csgn(I/c^2*(c^2*f+((c*x+1) ^2-2*c*x-1)*g))^2*x+1/4*I*e*b*ln(c*x+1)*Pi*csgn(I*c)^2*csgn(I*c^2)*x+1/4*I *e*b/c*Pi*csgn(I/c^2)*csgn(I*(c^2*f+((c*x+1)^2-2*c*x-1)*g))*csgn(I/c^2*(c^ 2*f+((c*x+1)^2-2*c*x-1)*g))+1/4*I*e*b/c*ln(c*x+1)*Pi*csgn(I/c^2)*csgn(I/c^ 2*(c^2*f+((c*x+1)^2-2*c*x-1)*g))^2+1/4*I*e*b/c*ln(c*x+1)*Pi*csgn(I*c)^2*cs gn(I*c^2)-1/2*I*e*b/c*ln(c*x+1)*Pi*csgn(I*c)*csgn(I*c^2)^2+1/4*I*e*b/c*ln( c*x+1)*Pi*csgn(I*(c^2*f+((c*x+1)^2-2*c*x-1)*g))*csgn(I/c^2*(c^2*f+((c*x+1) ^2-2*c*x-1)*g))^2+4*e*b/c-1/4*I*e*b*Pi*csgn(I/c^2)*csgn(I*(c^2*f+((c*x-1)^ 2+2*c*x-1)*g))*csgn(I/c^2*(c^2*f+((c*x-1)^2+2*c*x-1)*g))*x-1/4*I*e*b*ln(-c *x+1)*Pi*csgn(I/c^2)*csgn(I/c^2*(c^2*f+((c*x-1)^2+2*c*x-1)*g))^2*x-1/4*I*e *b*ln(-c*x+1)*Pi*csgn(I*c)^2*csgn(I*c^2)*x+1/2*I*e*b*ln(-c*x+1)*Pi*csgn(I* c)*csgn(I*c^2)^2*x-1/4*I*e*b*ln(-c*x+1)*Pi*csgn(I*(c^2*f+((c*x-1)^2+2*c*x- 1)*g))*csgn(I/c^2*(c^2*f+((c*x-1)^2+2*c*x-1)*g))^2*x+1/4*I*e*b/c*Pi*csgn(I /c^2)*csgn(I*(c^2*f+((c*x-1)^2+2*c*x-1)*g))*csgn(I/c^2*(c^2*f+((c*x-1)^2+2 *c*x-1)*g))+1/4*I*e*b/c*ln(-c*x+1)*Pi*csgn(I/c^2)*csgn(I/c^2*(c^2*f+((c*x- 1)^2+2*c*x-1)*g))^2+1/4*I*e*b/c*ln(-c*x+1)*Pi*csgn(I*c)^2*csgn(I*c^2)-1/2* I*e*b/c*ln(-c*x+1)*Pi*csgn(I*c)*csgn(I*c^2)^2+1/4*I*e*b/c*ln(-c*x+1)*Pi...
\[ \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} \,d x } \]
Timed out. \[ \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\text {Timed out} \]
\[ \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} \,d x } \]
(2*g*(f*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*g) - x/g) + x*log(g*x^2 + f))*a*e + a*d*x + 1/2*b*e*(((c*x + 1)*log(c*x + 1) - (c*x - 1)*log(-c*x + 1))*log (g*x^2 + f)/c + integrate(-2*((c*g*x^2 + g*x)*log(c*x + 1) - (c*g*x^2 - g* x)*log(-c*x + 1))/(c*g*x^2 + c*f), x)) + 1/2*(2*c*x*arctanh(c*x) + log(-c^ 2*x^2 + 1))*b*d/c
\[ \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} \,d x } \]
Timed out. \[ \int (a+b \text {arctanh}(c x)) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]