Integrand size = 22, antiderivative size = 749 \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\frac {3 p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (i \sqrt [3]{e} \sqrt {f}+\sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 i \sqrt {f} \sqrt {g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt {f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 (-1)^{5/6} \sqrt {f} \sqrt {g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt {f}+(-1)^{5/6} \sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {3 i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (i \sqrt [3]{e} \sqrt {f}+\sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1+\frac {2 i \sqrt {f} \sqrt {g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt {f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 (-1)^{5/6} \sqrt {f} \sqrt {g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt [3]{e} \sqrt {f}+(-1)^{5/6} \sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}} \]
arctan(x*g^(1/2)/f^(1/2))*ln(c*(e*x^3+d)^p)/f^(1/2)/g^(1/2)+3*p*arctan(x*g ^(1/2)/f^(1/2))*ln(2*f^(1/2)/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)-p*arct an(x*g^(1/2)/f^(1/2))*ln(2*(d^(1/3)+e^(1/3)*x)*f^(1/2)*g^(1/2)/(I*e^(1/3)* f^(1/2)+d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)-p*arctan(x *g^(1/2)/f^(1/2))*ln(-2*I*((-1)^(2/3)*d^(1/3)+e^(1/3)*x)*f^(1/2)*g^(1/2)/( e^(1/3)*f^(1/2)+(-1)^(1/6)*d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/f^(1/2) /g^(1/2)-p*arctan(x*g^(1/2)/f^(1/2))*ln(2*(-1)^(5/6)*(d^(1/3)+(-1)^(2/3)*e ^(1/3)*x)*f^(1/2)*g^(1/2)/(e^(1/3)*f^(1/2)+(-1)^(5/6)*d^(1/3)*g^(1/2))/(f^ (1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)-3/2*I*p*polylog(2,1-2*f^(1/2)/(f^(1/2) -I*x*g^(1/2)))/f^(1/2)/g^(1/2)+1/2*I*p*polylog(2,1-2*(d^(1/3)+e^(1/3)*x)*f ^(1/2)*g^(1/2)/(I*e^(1/3)*f^(1/2)+d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^(1/2)))/ f^(1/2)/g^(1/2)+1/2*I*p*polylog(2,1+2*I*((-1)^(2/3)*d^(1/3)+e^(1/3)*x)*f^( 1/2)*g^(1/2)/(e^(1/3)*f^(1/2)+(-1)^(1/6)*d^(1/3)*g^(1/2))/(f^(1/2)-I*x*g^( 1/2)))/f^(1/2)/g^(1/2)+1/2*I*p*polylog(2,1-2*(-1)^(5/6)*(d^(1/3)+(-1)^(2/3 )*e^(1/3)*x)*f^(1/2)*g^(1/2)/(e^(1/3)*f^(1/2)+(-1)^(5/6)*d^(1/3)*g^(1/2))/ (f^(1/2)-I*x*g^(1/2)))/f^(1/2)/g^(1/2)
Time = 0.50 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.16 \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\frac {-p \log \left (\frac {\sqrt {g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e} \sqrt {-f}+\sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e} \sqrt {-f}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )-p \log \left (\frac {\sqrt {g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e} \sqrt {-f}+(-1)^{2/3} \sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} x\right )+p \log \left (-\frac {\sqrt {g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e} \sqrt {-f}-\sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (\frac {\sqrt {g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{e} \sqrt {-f}+(-1)^{2/3} \sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+p \log \left (\frac {\sqrt [3]{-1} \sqrt {g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{e} \sqrt {-f}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} x\right )+\log \left (\sqrt {-f}-\sqrt {g} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-\log \left (\sqrt {-f}+\sqrt {g} x\right ) \log \left (c \left (d+e x^3\right )^p\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt [3]{e} \sqrt {-f}+\sqrt [3]{d} \sqrt {g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt [3]{e} \sqrt {-f}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt {g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt [3]{e} \sqrt {-f}+(-1)^{2/3} \sqrt [3]{d} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt [3]{e} \sqrt {-f}-\sqrt [3]{d} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt [3]{e} \sqrt {-f}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt {g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{e} \left (\sqrt {-f}+\sqrt {g} x\right )}{\sqrt [3]{e} \sqrt {-f}-(-1)^{2/3} \sqrt [3]{d} \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
(-(p*Log[(Sqrt[g]*(d^(1/3) + e^(1/3)*x))/(e^(1/3)*Sqrt[-f] + d^(1/3)*Sqrt[ g])]*Log[Sqrt[-f] - Sqrt[g]*x]) - p*Log[(Sqrt[g]*(-((-1)^(1/3)*d^(1/3)) + e^(1/3)*x))/(e^(1/3)*Sqrt[-f] - (-1)^(1/3)*d^(1/3)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*x] - p*Log[(Sqrt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/(e^(1/3)*S qrt[-f] + (-1)^(2/3)*d^(1/3)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*x] + p*Log[- ((Sqrt[g]*(d^(1/3) + e^(1/3)*x))/(e^(1/3)*Sqrt[-f] - d^(1/3)*Sqrt[g]))]*Lo g[Sqrt[-f] + Sqrt[g]*x] + p*Log[(Sqrt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) /(-(e^(1/3)*Sqrt[-f]) + (-1)^(2/3)*d^(1/3)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g ]*x] + p*Log[((-1)^(1/3)*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/(e^(1/3 )*Sqrt[-f] + (-1)^(1/3)*d^(1/3)*Sqrt[g])]*Log[Sqrt[-f] + Sqrt[g]*x] + Log[ Sqrt[-f] - Sqrt[g]*x]*Log[c*(d + e*x^3)^p] - Log[Sqrt[-f] + Sqrt[g]*x]*Log [c*(d + e*x^3)^p] - p*PolyLog[2, (e^(1/3)*(Sqrt[-f] - Sqrt[g]*x))/(e^(1/3) *Sqrt[-f] + d^(1/3)*Sqrt[g])] - p*PolyLog[2, (e^(1/3)*(Sqrt[-f] - Sqrt[g]* x))/(e^(1/3)*Sqrt[-f] - (-1)^(1/3)*d^(1/3)*Sqrt[g])] - p*PolyLog[2, (e^(1/ 3)*(Sqrt[-f] - Sqrt[g]*x))/(e^(1/3)*Sqrt[-f] + (-1)^(2/3)*d^(1/3)*Sqrt[g]) ] + p*PolyLog[2, (e^(1/3)*(Sqrt[-f] + Sqrt[g]*x))/(e^(1/3)*Sqrt[-f] - d^(1 /3)*Sqrt[g])] + p*PolyLog[2, (e^(1/3)*(Sqrt[-f] + Sqrt[g]*x))/(e^(1/3)*Sqr t[-f] + (-1)^(1/3)*d^(1/3)*Sqrt[g])] + p*PolyLog[2, (e^(1/3)*(Sqrt[-f] + S qrt[g]*x))/(e^(1/3)*Sqrt[-f] - (-1)^(2/3)*d^(1/3)*Sqrt[g])])/(2*Sqrt[-f]*S qrt[g])
Time = 1.08 (sec) , antiderivative size = 706, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2920, 27, 7276, 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.
| \(\displaystyle \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2920 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt {f} \sqrt {g}}-3 e p \int \frac {x^2 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (e x^3+d\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {3 e p \int \frac {x^2 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{e x^3+d}dx}{\sqrt {f} \sqrt {g}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {3 e p \int \left (\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{3 e^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{3 e^{2/3} \left (\sqrt [3]{e} x-\sqrt [3]{-1} \sqrt [3]{d}\right )}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{3 e^{2/3} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}\right )dx}{\sqrt {f} \sqrt {g}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^3\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {3 e p \left (\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt [3]{d} \sqrt {g}+i \sqrt [3]{e} \sqrt {f}\right )}\right )}{3 e}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 i \sqrt {f} \sqrt {g} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt [6]{-1} \sqrt [3]{d} \sqrt {g}+\sqrt [3]{e} \sqrt {f}\right )}\right )}{3 e}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 (-1)^{5/6} \sqrt {f} \sqrt {g} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left ((-1)^{5/6} \sqrt [3]{d} \sqrt {g}+\sqrt [3]{e} \sqrt {f}\right )}\right )}{3 e}-\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{e}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (i \sqrt [3]{e} \sqrt {f}+\sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{6 e}-\frac {i \operatorname {PolyLog}\left (2,\frac {2 i \sqrt {f} \sqrt {g} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{\left (\sqrt [3]{e} \sqrt {f}+\sqrt [6]{-1} \sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{6 e}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 (-1)^{5/6} \sqrt {f} \sqrt {g} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (\sqrt [3]{e} \sqrt {f}+(-1)^{5/6} \sqrt [3]{d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{6 e}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 e}\right )}{\sqrt {f} \sqrt {g}}\) |
(ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^3)^p])/(Sqrt[f]*Sqrt[g]) - (3* e*p*(-((ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x )])/e) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^ (1/3)*x))/((I*e^(1/3)*Sqrt[f] + d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))] )/(3*e) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[((-2*I)*Sqrt[f]*Sqrt[g]*((-1)^( 2/3)*d^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(1/6)*d^(1/3)*Sqrt[g]) *(Sqrt[f] - I*Sqrt[g]*x))])/(3*e) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*(- 1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((e^(1/3)*Sqrt[ f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(3*e) + ((I/2) *PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/e - ((I/6)*PolyLog[2 , 1 - (2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^(1/3)*x))/((I*e^(1/3)*Sqrt[f] + d^(1 /3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/e - ((I/6)*PolyLog[2, 1 + ((2*I)*S qrt[f]*Sqrt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^ (1/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/e - ((I/6)*PolyLog[2, 1 - (2*(-1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((e^(1/3 )*Sqrt[f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/e))/(Sq rt[f]*Sqrt[g])
3.3.61.3.1 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_.) + 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[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.45 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(\frac {\left (\ln \left (\left (e \,x^{3}+d \right )^{p}\right )-p \ln \left (e \,x^{3}+d \right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}}+\frac {p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (g \,\textit {\_Z}^{2}+f \right )}{\sum }\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (e \,x^{3}+d \right )-\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} e g +3 \underline {\hspace {1.25 ex}}\alpha \,\textit {\_Z}^{2} e g -3 \textit {\_Z} e f -e f \underline {\hspace {1.25 ex}}\alpha +d g \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\textit {\_R1} -x +\underline {\hspace {1.25 ex}}\alpha }{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x +\underline {\hspace {1.25 ex}}\alpha }{\textit {\_R1}}\right )\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2 g}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{3}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}}\) | \(272\) |
(ln((e*x^3+d)^p)-p*ln(e*x^3+d))/(f*g)^(1/2)*arctan(g*x/(f*g)^(1/2))+1/2*p/ g*sum(1/_alpha*(ln(x-_alpha)*ln(e*x^3+d)-sum(ln(x-_alpha)*ln((_R1-x+_alpha )/_R1)+dilog((_R1-x+_alpha)/_R1),_R1=RootOf(_Z^3*e*g+3*_Z^2*_alpha*e*g-3*_ Z*e*f-_alpha*e*f+d*g))),_alpha=RootOf(_Z^2*g+f))+(1/2*I*Pi*csgn(I*(e*x^3+d )^p)*csgn(I*c*(e*x^3+d)^p)^2-1/2*I*Pi*csgn(I*(e*x^3+d)^p)*csgn(I*c*(e*x^3+ d)^p)*csgn(I*c)-1/2*I*Pi*csgn(I*c*(e*x^3+d)^p)^3+1/2*I*Pi*csgn(I*c*(e*x^3+ d)^p)^2*csgn(I*c)+ln(c))/(f*g)^(1/2)*arctan(g*x/(f*g)^(1/2))
\[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
\[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}{g\,x^2+f} \,d x \]