Integrand size = 15, antiderivative size = 227 \[ \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx=\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{16} \left (4+(1-i) 2^{3/4}\right ) \log (x) \log \left (1+\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{16} \left (4+(1-i) 2^{3/4}\right ) \operatorname {PolyLog}\left (2,-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right ) \]
1/8*(2+I*(-2)^(1/4))*ln(x)*ln(1-(1/2+1/2*I)*x*2^(1/4))+1/16*(4+(1-I)*2^(3/ 4))*ln(x)*ln(1+(1/2+1/2*I)*x*2^(1/4))+1/8*(2+(-2)^(1/4))*ln(x)*ln(1-1/2*(- 1)^(3/4)*x*2^(3/4))+1/8*(2-(-2)^(1/4))*ln(x)*ln(1+1/2*(-1)^(3/4)*x*2^(3/4) )+1/16*(4+(1-I)*2^(3/4))*polylog(2,(-1/2-1/2*I)*x*2^(1/4))+1/8*(2+I*(-2)^( 1/4))*polylog(2,(1/2+1/2*I)*x*2^(1/4))+1/8*(2-(-2)^(1/4))*polylog(2,-1/2*( -1)^(3/4)*x*2^(3/4))+1/8*(2+(-2)^(1/4))*polylog(2,1/2*(-1)^(3/4)*x*2^(3/4) )
Time = 0.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.85 \[ \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx=\frac {1}{8} \left (\left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\sqrt [4]{-\frac {1}{2}} x\right )+\left (2+\frac {1-i}{\sqrt [4]{2}}\right ) \log (x) \log \left (1+\sqrt [4]{-\frac {1}{2}} x\right )-\left (-2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(1-i) x}{2^{3/4}}\right )+\left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(1-i) x}{2^{3/4}}\right )+\left (2+\frac {1-i}{\sqrt [4]{2}}\right ) \operatorname {PolyLog}\left (2,-\frac {(1+i) x}{2^{3/4}}\right )+\left (2+\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,-\frac {(1-i) x}{2^{3/4}}\right )-\left (-2+\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(1-i) x}{2^{3/4}}\right )+\left (2+i \sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(1+i) x}{2^{3/4}}\right )\right ) \]
((2 + I*(-2)^(1/4))*Log[x]*Log[1 - (-1/2)^(1/4)*x] + (2 + (1 - I)/2^(1/4)) *Log[x]*Log[1 + (-1/2)^(1/4)*x] - (-2 + (-2)^(1/4))*Log[x]*Log[1 - ((1 - I )*x)/2^(3/4)] + (2 + (-2)^(1/4))*Log[x]*Log[1 + ((1 - I)*x)/2^(3/4)] + (2 + (1 - I)/2^(1/4))*PolyLog[2, ((-1 - I)*x)/2^(3/4)] + (2 + (-2)^(1/4))*Pol yLog[2, ((-1 + I)*x)/2^(3/4)] - (-2 + (-2)^(1/4))*PolyLog[2, ((1 - I)*x)/2 ^(3/4)] + (2 + I*(-2)^(1/4))*PolyLog[2, ((1 + I)*x)/2^(3/4)])/8
Time = 0.38 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2804, 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.
| \(\displaystyle \int \frac {\left (x^3+1\right ) \log (x)}{x^4+2} \, dx\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle \int \left (\frac {\left (\sqrt [4]{-2}-2\right ) \log (x)}{8 \left (\sqrt [4]{-2}-x\right )}+\frac {\left (\sqrt [4]{-2}-2 i\right ) \log (x)}{8 \left (\sqrt [4]{-2}-i x\right )}+\frac {\left (\sqrt [4]{-2}+2 i\right ) \log (x)}{8 \left (\sqrt [4]{-2}+i x\right )}+\frac {\left (2+\sqrt [4]{-2}\right ) \log (x)}{8 \left (x+\sqrt [4]{-2}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (2-i \sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \operatorname {PolyLog}\left (2,\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2-i \sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac {(1+i) x}{2^{3/4}}\right )+\frac {1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac {(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac {1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (\frac {(-1)^{3/4} x}{\sqrt [4]{2}}+1\right )\) |
((2 + I*(-2)^(1/4))*Log[x]*Log[1 - ((1 + I)*x)/2^(3/4)])/8 + ((2 - I*(-2)^ (1/4))*Log[x]*Log[1 + ((1 + I)*x)/2^(3/4)])/8 + ((2 + (-2)^(1/4))*Log[x]*L og[1 - ((-1)^(3/4)*x)/2^(1/4)])/8 + ((2 - (-2)^(1/4))*Log[x]*Log[1 + ((-1) ^(3/4)*x)/2^(1/4)])/8 + ((2 - I*(-2)^(1/4))*PolyLog[2, ((-1 - I)*x)/2^(3/4 )])/8 + ((2 + I*(-2)^(1/4))*PolyLog[2, ((1 + I)*x)/2^(3/4)])/8 + ((2 - (-2 )^(1/4))*PolyLog[2, -(((-1)^(3/4)*x)/2^(1/4))])/8 + ((2 + (-2)^(1/4))*Poly Log[2, ((-1)^(3/4)*x)/2^(1/4)])/8
3.3.57.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (171 ) = 342\).
Time = 0.37 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(\frac {\left (\left (\frac {2^{\frac {3}{4}}}{2}+\frac {i 2^{\frac {3}{4}}}{2}\right )^{3}+1\right ) \left (\ln \left (x \right ) \ln \left (\frac {\frac {2^{\frac {3}{4}}}{2}+\frac {i 2^{\frac {3}{4}}}{2}-x}{\frac {2^{\frac {3}{4}}}{2}+\frac {i 2^{\frac {3}{4}}}{2}}\right )+\operatorname {dilog}\left (\frac {\frac {2^{\frac {3}{4}}}{2}+\frac {i 2^{\frac {3}{4}}}{2}-x}{\frac {2^{\frac {3}{4}}}{2}+\frac {i 2^{\frac {3}{4}}}{2}}\right )\right )}{4 \left (\frac {2^{\frac {3}{4}}}{2}+\frac {i 2^{\frac {3}{4}}}{2}\right )^{3}}+\frac {\left (\left (\frac {i 2^{\frac {3}{4}}}{2}-\frac {2^{\frac {3}{4}}}{2}\right )^{3}+1\right ) \left (\ln \left (x \right ) \ln \left (\frac {\frac {i 2^{\frac {3}{4}}}{2}-\frac {2^{\frac {3}{4}}}{2}-x}{\frac {i 2^{\frac {3}{4}}}{2}-\frac {2^{\frac {3}{4}}}{2}}\right )+\operatorname {dilog}\left (\frac {\frac {i 2^{\frac {3}{4}}}{2}-\frac {2^{\frac {3}{4}}}{2}-x}{\frac {i 2^{\frac {3}{4}}}{2}-\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{4 \left (\frac {i 2^{\frac {3}{4}}}{2}-\frac {2^{\frac {3}{4}}}{2}\right )^{3}}+\frac {\left (\left (-\frac {2^{\frac {3}{4}}}{2}-\frac {i 2^{\frac {3}{4}}}{2}\right )^{3}+1\right ) \left (\ln \left (x \right ) \ln \left (\frac {-\frac {2^{\frac {3}{4}}}{2}-\frac {i 2^{\frac {3}{4}}}{2}-x}{-\frac {2^{\frac {3}{4}}}{2}-\frac {i 2^{\frac {3}{4}}}{2}}\right )+\operatorname {dilog}\left (\frac {-\frac {2^{\frac {3}{4}}}{2}-\frac {i 2^{\frac {3}{4}}}{2}-x}{-\frac {2^{\frac {3}{4}}}{2}-\frac {i 2^{\frac {3}{4}}}{2}}\right )\right )}{4 \left (-\frac {2^{\frac {3}{4}}}{2}-\frac {i 2^{\frac {3}{4}}}{2}\right )^{3}}+\frac {\left (\left (-\frac {i 2^{\frac {3}{4}}}{2}+\frac {2^{\frac {3}{4}}}{2}\right )^{3}+1\right ) \left (\ln \left (x \right ) \ln \left (\frac {-\frac {i 2^{\frac {3}{4}}}{2}+\frac {2^{\frac {3}{4}}}{2}-x}{-\frac {i 2^{\frac {3}{4}}}{2}+\frac {2^{\frac {3}{4}}}{2}}\right )+\operatorname {dilog}\left (\frac {-\frac {i 2^{\frac {3}{4}}}{2}+\frac {2^{\frac {3}{4}}}{2}-x}{-\frac {i 2^{\frac {3}{4}}}{2}+\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{4 \left (-\frac {i 2^{\frac {3}{4}}}{2}+\frac {2^{\frac {3}{4}}}{2}\right )^{3}}\) | \(394\) |
| parts | \(\text {Expression too large to display}\) | \(790\) |
| risch | \(\text {Expression too large to display}\) | \(1210\) |
1/4*((1/2*2^(3/4)+1/2*I*2^(3/4))^3+1)/(1/2*2^(3/4)+1/2*I*2^(3/4))^3*(ln(x) *ln((1/2*2^(3/4)+1/2*I*2^(3/4)-x)/(1/2*2^(3/4)+1/2*I*2^(3/4)))+dilog((1/2* 2^(3/4)+1/2*I*2^(3/4)-x)/(1/2*2^(3/4)+1/2*I*2^(3/4))))+1/4*((1/2*I*2^(3/4) -1/2*2^(3/4))^3+1)/(1/2*I*2^(3/4)-1/2*2^(3/4))^3*(ln(x)*ln((1/2*I*2^(3/4)- 1/2*2^(3/4)-x)/(1/2*I*2^(3/4)-1/2*2^(3/4)))+dilog((1/2*I*2^(3/4)-1/2*2^(3/ 4)-x)/(1/2*I*2^(3/4)-1/2*2^(3/4))))+1/4*((-1/2*2^(3/4)-1/2*I*2^(3/4))^3+1) /(-1/2*2^(3/4)-1/2*I*2^(3/4))^3*(ln(x)*ln((-1/2*2^(3/4)-1/2*I*2^(3/4)-x)/( -1/2*2^(3/4)-1/2*I*2^(3/4)))+dilog((-1/2*2^(3/4)-1/2*I*2^(3/4)-x)/(-1/2*2^ (3/4)-1/2*I*2^(3/4))))+1/4*((-1/2*I*2^(3/4)+1/2*2^(3/4))^3+1)/(-1/2*I*2^(3 /4)+1/2*2^(3/4))^3*(ln(x)*ln((-1/2*I*2^(3/4)+1/2*2^(3/4)-x)/(-1/2*I*2^(3/4 )+1/2*2^(3/4)))+dilog((-1/2*I*2^(3/4)+1/2*2^(3/4)-x)/(-1/2*I*2^(3/4)+1/2*2 ^(3/4))))
\[ \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx=\int { \frac {{\left (x^{3} + 1\right )} \log \left (x\right )}{x^{4} + 2} \,d x } \]
\[ \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - x + 1\right ) \log {\left (x \right )}}{x^{4} + 2}\, dx \]
\[ \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx=\int { \frac {{\left (x^{3} + 1\right )} \log \left (x\right )}{x^{4} + 2} \,d x } \]
\[ \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx=\int { \frac {{\left (x^{3} + 1\right )} \log \left (x\right )}{x^{4} + 2} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^3\right ) \log (x)}{2+x^4} \, dx=\int \frac {\ln \left (x\right )\,\left (x^3+1\right )}{x^4+2} \,d x \]