Integrand size = 21, antiderivative size = 546 \[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac {2 a e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1-\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (1+\frac {1}{c x}\right )}{\sqrt {g}}+\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {g}}-\frac {b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b \log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac {b e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-\frac {i b e \sqrt {f} \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {g}}+\frac {i b e \sqrt {f} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (1+c x)}{\left (i c \sqrt {f}+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {g}} \]
-2*a*e*x-2*b*e*x*arccoth(c*x)-b*e*ln(-c^2*x^2+1)/c+x*(a+b*arccoth(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+2*a*e*arctan(x*g^(1/2)/f^(1/2))* f^(1/2)/g^(1/2)-b*e*arctan(x*g^(1/2)/f^(1/2))*ln(1-1/c/x)*f^(1/2)/g^(1/2)+ b*e*arctan(x*g^(1/2)/f^(1/2))*ln(1+1/c/x)*f^(1/2)/g^(1/2)+b*e*arctan(x*g^( 1/2)/f^(1/2))*ln(-2*(-c*x+1)*f^(1/2)*g^(1/2)/(I*c*f^(1/2)-g^(1/2))/(f^(1/2 )-I*x*g^(1/2)))*f^(1/2)/g^(1/2)-b*e*arctan(x*g^(1/2)/f^(1/2))*ln(2*(c*x+1) *f^(1/2)*g^(1/2)/(I*c*f^(1/2)+g^(1/2))/(f^(1/2)-I*x*g^(1/2)))*f^(1/2)/g^(1 /2)-1/2*I*b*e*polylog(2,1+2*(-c*x+1)*f^(1/2)*g^(1/2)/(I*c*f^(1/2)-g^(1/2)) /(f^(1/2)-I*x*g^(1/2)))*f^(1/2)/g^(1/2)+1/2*I*b*e*polylog(2,1-2*(c*x+1)*f^ (1/2)*g^(1/2)/(I*c*f^(1/2)+g^(1/2))/(f^(1/2)-I*x*g^(1/2)))*f^(1/2)/g^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1287\) vs. \(2(546)=1092\).
Time = 2.39 (sec) , antiderivative size = 1287, normalized size of antiderivative = 2.36 \[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx =\text {Too large to display} \]
a*d*x - 2*a*e*x + b*d*x*ArcCoth[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/S qrt[f]])/Sqrt[g] + (b*d*Log[1 - c^2*x^2])/(2*c) + a*e*x*Log[f + g*x^2] + b *e*(x*ArcCoth[c*x] + Log[1 - c^2*x^2]/(2*c))*Log[f + g*x^2] + (b*e*(-4*c*x *ArcCoth[c*x] + 4*Log[1/(c*Sqrt[1 - 1/(c^2*x^2)]*x)] + (Sqrt[c^2*f*g]*((-2 *I)*ArcCos[(c^2*f - g)/(c^2*f + g)]*ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + 4*ArcC oth[c*x]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - (ArcCos[(c^2*f - g)/(c^2*f + g)] + 2*ArcTan[Sqrt[c^2*f*g]/(c*g*x)])*Log[((2*I)*g*(I*c^2*f + Sqrt[c^2*f*g])* (-1 + 1/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x)))] - (ArcCos[(c^ 2*f - g)/(c^2*f + g)] - 2*ArcTan[Sqrt[c^2*f*g]/(c*g*x)])*Log[(2*g*(c^2*f + I*Sqrt[c^2*f*g])*(1 + 1/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x) ))] + (ArcCos[(c^2*f - g)/(c^2*f + g)] + 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*Sqrt[c^2*f*g])/(E^ArcCoth[c *x]*Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]]) ] + (ArcCos[(c^2*f - g)/(c^2*f + g)] - 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*E^ArcCoth[c*x]*Sqrt[c^2*f*g]) /(Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]])] + I*(-PolyLog[2, ((-(c^2*f) + g + (2*I)*Sqrt[c^2*f*g])*(g - (I*Sqrt[c^2*f* g])/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x)))] + PolyLog[2, ((c^ 2*f - g + (2*I)*Sqrt[c^2*f*g])*(I*g + Sqrt[c^2*f*g]/(c*x)))/((c^2*f + g)*( (-I)*g + Sqrt[c^2*f*g]/(c*x)))])))/g))/(2*c) - (b*e*g*(((-Log[-c^(-1) +...
Time = 2.21 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.44, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6636, 2925, 2841, 2840, 2838, 6543, 2009, 6537, 218, 6535, 2920, 27, 2005, 5411, 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.
| \(\displaystyle \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6636 |
| \(\displaystyle -2 e g \int \frac {x^2 \left (a+b \coth ^{-1}(c x)\right )}{g x^2+f}dx-b c \int \frac {x \left (d+e \log \left (g x^2+f\right )\right )}{1-c^2 x^2}dx+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle -2 e g \int \frac {x^2 \left (a+b \coth ^{-1}(c x)\right )}{g x^2+f}dx-\frac {1}{2} b c \int \frac {d+e \log \left (g x^2+f\right )}{1-c^2 x^2}dx^2+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )\) |
| \(\Big \downarrow \) 2841 |
| \(\displaystyle -2 e g \int \frac {x^2 \left (a+b \coth ^{-1}(c x)\right )}{g x^2+f}dx-\frac {1}{2} b c \left (\frac {e g \int \frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{f c^2+g}\right )}{g x^2+f}dx^2}{c^2}-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )\) |
| \(\Big \downarrow \) 2840 |
| \(\displaystyle -2 e g \int \frac {x^2 \left (a+b \coth ^{-1}(c x)\right )}{g x^2+f}dx-\frac {1}{2} b c \left (\frac {e \int \frac {\log \left (1-\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{x^2}d\left (g x^2+f\right )}{c^2}-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -2 e g \int \frac {x^2 \left (a+b \coth ^{-1}(c x)\right )}{g x^2+f}dx+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle -2 e g \left (\frac {\int \left (a+b \coth ^{-1}(c x)\right )dx}{g}-\frac {f \int \frac {a+b \coth ^{-1}(c x)}{g x^2+f}dx}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \int \frac {a+b \coth ^{-1}(c x)}{g x^2+f}dx}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6537 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \left (a \int \frac {1}{g x^2+f}dx+b \int \frac {\coth ^{-1}(c x)}{g x^2+f}dx\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \left (b \int \frac {\coth ^{-1}(c x)}{g x^2+f}dx+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 6535 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \left (b \left (\frac {1}{2} \int \frac {\log \left (1+\frac {1}{c x}\right )}{g x^2+f}dx-\frac {1}{2} \int \frac {\log \left (1-\frac {1}{c x}\right )}{g x^2+f}dx\right )+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2920 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \left (b \left (\frac {1}{2} \left (\frac {\int \frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (c-\frac {1}{x}\right ) x^2}dx}{c}-\frac {\log \left (1-\frac {1}{c x}\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )+\frac {1}{2} \left (\frac {\int \frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (c+\frac {1}{x}\right ) x^2}dx}{c}+\frac {\log \left (\frac {1}{c x}+1\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )\right )+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \left (b \left (\frac {1}{2} \left (\frac {\int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (c-\frac {1}{x}\right ) x^2}dx}{\sqrt {f} \sqrt {g}}-\frac {\log \left (1-\frac {1}{c x}\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )+\frac {1}{2} \left (\frac {\int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\left (c+\frac {1}{x}\right ) x^2}dx}{\sqrt {f} \sqrt {g}}+\frac {\log \left (\frac {1}{c x}+1\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )\right )+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \left (b \left (\frac {1}{2} \left (\frac {\int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (c x-1)}dx}{\sqrt {f} \sqrt {g}}-\frac {\log \left (1-\frac {1}{c x}\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )+\frac {1}{2} \left (\frac {\int \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x (c x+1)}dx}{\sqrt {f} \sqrt {g}}+\frac {\log \left (\frac {1}{c x}+1\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )\right )+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{g}-\frac {f \left (b \left (\frac {1}{2} \left (\frac {\int \left (\frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c x-1}-\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x}\right )dx}{\sqrt {f} \sqrt {g}}-\frac {\log \left (1-\frac {1}{c x}\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )+\frac {1}{2} \left (\frac {\int \left (\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{x}-\frac {c \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{c x+1}\right )dx}{\sqrt {f} \sqrt {g}}+\frac {\log \left (\frac {1}{c x}+1\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )\right )+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \left (-\frac {\log \left (\frac {g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{c^2}-\frac {e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{f c^2+g}\right )}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 e g \left (\frac {a x+\frac {b \log \left (1-c^2 x^2\right )}{2 c}+b x \coth ^{-1}(c x)}{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 {\log \left (1-\frac {1}{c x}\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (-\sqrt {g}+i c \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )-\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} (1-c x)}{\left (i c \sqrt {f}-\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}\right )+\frac {1}{2} \left (\frac {\log \left (\frac {1}{c x}+1\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+\frac {-\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} (c x+1)}{\left (\sqrt {g}+i c \sqrt {f}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )+\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} (c x+1)}{\left (i \sqrt {f} c+\sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}\right )\right )\right )}{g}\right )+x \left (a+b \coth ^{-1}(c x)\right ) \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*ArcCoth[c*x])*(d + e*Log[f + g*x^2]) - (b*c*(-((Log[(g*(1 - c^2*x ^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]))/c^2) - (e*PolyLog[2, (c^2*(f + g *x^2))/(c^2*f + g)])/c^2))/2 - 2*e*g*((a*x + b*x*ArcCoth[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*((-((ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 - 1/(c*x)])/(Sqrt[f]*Sqrt[g]) ) + (-(ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x) ]) + ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*c* Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))] - (I/2)*PolyLog[2, ((-I)*Sqrt [g]*x)/Sqrt[f]] + (I/2)*PolyLog[2, (I*Sqrt[g]*x)/Sqrt[f]] + (I/2)*PolyLog[ 2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)] - (I/2)*PolyLog[2, 1 + (2*Sqrt [f]*Sqrt[g]*(1 - c*x))/((I*c*Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))]) /(Sqrt[f]*Sqrt[g]))/2 + ((ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + 1/(c*x)])/(S qrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)] - ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(1 + c* x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))] + (I/2)*PolyLog[2, ((-I)*Sqrt[g]*x)/Sqrt[f]] - (I/2)*PolyLog[2, (I*Sqrt[g]*x)/Sqrt[f]] - (I/2 )*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)] + (I/2)*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(1 + c*x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqr t[g]*x))])/(Sqrt[f]*Sqrt[g]))/2)))/g)
3.3.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
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_))^(p_.)]*(b_.))/((f_) + (g_.) *(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p Int[u*(x^(n - 1)/(d + e*x^n)), x ], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
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[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Int[ArcCoth[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/2 Int [Log[1 + 1/(c*x)]/(d + e*x^2), x], x] - Simp[1/2 Int[Log[1 - 1/(c*x)]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]
Int[(ArcCoth[(c_.)*(x_)]*(b_.) + (a_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Simp[a Int[1/(d + e*x^2), x], x] + Simp[b Int[ArcCoth[c*x]/(d + e*x^2) , x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCoth[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcCoth[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_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]* (e_.)), x_Symbol] :> Simp[x*(d + e*Log[f + g*x^2])*(a + b*ArcCoth[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*ArcCoth[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 = 8.48 (sec) , antiderivative size = 3508, normalized size of antiderivative = 6.42
1/4*I*e*b/c*Pi*csgn(I*c^2)^3*ln(c*x+1)-1/4*I*e*b/c*Pi*csgn(I/c^2*(c^2*f+(( c*x+1)^2-2*c*x-1)*g))^3*ln(c*x+1)-1/4*I*e*b*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*(c^2*f+((c*x+1)^2-2*c*x-1)*g))^3*ln(c*x+1)*x-1/4*I*e*b*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*Pi*csgn(I*c^2)^ 3*ln(c*x+1)*x-1/4*I*e*b/c*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*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))^2-1/4*I*e*b/c*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*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+1/4*I*e*b/c*Pi*csgn( I*c^2)^3*ln(c*x-1)-1/4*I*e*b/c*Pi*csgn(I/c^2*(c^2*f+((c*x-1)^2+2*c*x-1)*g) )^3*ln(c*x-1)+1/4*I*e*b*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*Pi*csgn(I*c^2)^3*ln(c*x-1)*x+1/4*I*e*b*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*(c^2*f+((c*x-1)^2+2*c*x-1)*g))^3*ln(c*x-1)*x+1/2*d* b/c*ln(c*x+1)+1/2/c*ln(c*x-1)*b*d-1/2*b*d*ln(c*x-1)*x+b*e*x*ln(c*x-1)+a*e* x*ln(g*x^2+f)+a*d*x-b*d/c-b*e/c*ln(c*x-1)+1/2*d*b*ln(c*x+1)*x-b*x*e*ln(c*x +1)-1/4*I*e*b*Pi*csgn(I*c^2)*csgn(I*c)^2*ln(c*x-1)*x-1/4*I*e*b*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* ln(c*x-1)*x-1/4*I*e*b*Pi*csgn(I/c^2)*csgn(I*(c^2*f+((c*x-1)^2+2*c*x-1)*...
\[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} \,d x } \]
Timed out. \[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\text {Timed out} \]
\[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} \,d x } \]
(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*arccoth(c*x) + log(-c^2*x ^2 + 1))*b*d/c
\[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int { {\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )} \,d x } \]
Timed out. \[ \int \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx=\int \left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right ) \,d x \]