3.257 \(\int \frac{(1+x^3) \log (x)}{2+x^4} \, dx\)

Optimal. Leaf size=227 \[ \frac{1}{16} \left (4+(1-i) 2^{3/4}\right ) \text{PolyLog}\left (2,-\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2+i \sqrt [4]{-2}\right ) \text{PolyLog}\left (2,\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2-\sqrt [4]{-2}\right ) \text{PolyLog}\left (2,-\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (2+\sqrt [4]{-2}\right ) \text{PolyLog}\left (2,\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{16} \left (4+(1-i) 2^{3/4}\right ) \log (x) \log \left (1+\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (\frac{(-1)^{3/4} x}{\sqrt [4]{2}}+1\right ) \]

[Out]

((2 + I*(-2)^(1/4))*Log[x]*Log[1 - ((1 + I)*x)/2^(3/4)])/8 + ((4 + (1 - I)*2^(3/4))*Log[x]*Log[1 + ((1 + I)*x)
/2^(3/4)])/16 + ((2 + (-2)^(1/4))*Log[x]*Log[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 + ((4 + (1 - I)*2^(3/4))*PolyLog[2, ((-1 - I)*x)/2^(3/4)])/16 + ((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))*PolyLog[2, ((-1)^(3/4)*x)/2^(1/4)])/8

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Rubi [A]  time = 0.197169, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2357, 2317, 2391} \[ \frac{1}{16} \left (4+(1-i) 2^{3/4}\right ) \text{PolyLog}\left (2,-\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2+i \sqrt [4]{-2}\right ) \text{PolyLog}\left (2,\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2-\sqrt [4]{-2}\right ) \text{PolyLog}\left (2,-\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (2+\sqrt [4]{-2}\right ) \text{PolyLog}\left (2,\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{16} \left (4+(1-i) 2^{3/4}\right ) \log (x) \log \left (1+\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1-\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (2-\sqrt [4]{-2}\right ) \log (x) \log \left (\frac{(-1)^{3/4} x}{\sqrt [4]{2}}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[((1 + x^3)*Log[x])/(2 + x^4),x]

[Out]

((2 + I*(-2)^(1/4))*Log[x]*Log[1 - ((1 + I)*x)/2^(3/4)])/8 + ((4 + (1 - I)*2^(3/4))*Log[x]*Log[1 + ((1 + I)*x)
/2^(3/4)])/16 + ((2 + (-2)^(1/4))*Log[x]*Log[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 + ((4 + (1 - I)*2^(3/4))*PolyLog[2, ((-1 - I)*x)/2^(3/4)])/16 + ((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))*PolyLog[2, ((-1)^(3/4)*x)/2^(1/4)])/8

Rule 2357

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]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2391

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]

Rubi steps

\begin{align*} \int \frac{\left (1+x^3\right ) \log (x)}{2+x^4} \, dx &=\int \left (\frac{\left (-2+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}-x\right )}+\frac{\left (-2 i+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}-i x\right )}+\frac{\left (2 i+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}+i x\right )}+\frac{\left (2+\sqrt [4]{-2}\right ) \log (x)}{8 \left (\sqrt [4]{-2}+x\right )}\right ) \, dx\\ &=\frac{1}{8} \left (-2+\sqrt [4]{-2}\right ) \int \frac{\log (x)}{\sqrt [4]{-2}-x} \, dx+\frac{1}{8} \left (-2 i+\sqrt [4]{-2}\right ) \int \frac{\log (x)}{\sqrt [4]{-2}-i x} \, dx+\frac{1}{8} \left (2 i+\sqrt [4]{-2}\right ) \int \frac{\log (x)}{\sqrt [4]{-2}+i x} \, dx+\frac{1}{8} \left (2+\sqrt [4]{-2}\right ) \int \frac{\log (x)}{\sqrt [4]{-2}+x} \, 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}{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 (1+\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (-2-\sqrt [4]{-2}\right ) \int \frac{\log \left (1-\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )}{x} \, dx+\frac{1}{8} \left (i \left (2 i-\sqrt [4]{-2}\right )\right ) \int \frac{\log \left (1-\sqrt [4]{-\frac{1}{2}} x\right )}{x} \, dx+\frac{1}{8} \left (-2+i \sqrt [4]{-2}\right ) \int \frac{\log \left (1+\sqrt [4]{-\frac{1}{2}} x\right )}{x} \, dx+\frac{1}{8} \left (-2+\sqrt [4]{-2}\right ) \int \frac{\log \left (1+\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )}{x} \, 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}{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 (1+\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{16} \left (4+(1-i) 2^{3/4}\right ) \text{Li}_2\left (-\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2+i \sqrt [4]{-2}\right ) \text{Li}_2\left (\frac{(1+i) x}{2^{3/4}}\right )+\frac{1}{8} \left (2-\sqrt [4]{-2}\right ) \text{Li}_2\left (-\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )+\frac{1}{8} \left (2+\sqrt [4]{-2}\right ) \text{Li}_2\left (\frac{(-1)^{3/4} x}{\sqrt [4]{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.251529, size = 194, normalized size = 0.85 \[ \frac{1}{8} \left (\left (2+\frac{1-i}{\sqrt [4]{2}}\right ) \text{PolyLog}\left (2,-\frac{(1+i) x}{2^{3/4}}\right )+\left (2+\sqrt [4]{-2}\right ) \text{PolyLog}\left (2,-\frac{(1-i) x}{2^{3/4}}\right )-\left (\sqrt [4]{-2}-2\right ) \text{PolyLog}\left (2,\frac{(1-i) x}{2^{3/4}}\right )+\left (2+i \sqrt [4]{-2}\right ) \text{PolyLog}\left (2,\frac{(1+i) x}{2^{3/4}}\right )+\left (2+i \sqrt [4]{-2}\right ) \log (x) \log \left (1-\sqrt [4]{-\frac{1}{2}} x\right )+\left (2+\frac{1-i}{\sqrt [4]{2}}\right ) \log (x) \log \left (\sqrt [4]{-\frac{1}{2}} x+1\right )-\left (\sqrt [4]{-2}-2\right ) \log (x) \log \left (1-\frac{(1-i) x}{2^{3/4}}\right )+\left (2+\sqrt [4]{-2}\right ) \log (x) \log \left (1+\frac{(1-i) x}{2^{3/4}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)*Log[x])/(2 + x^4),x]

[Out]

((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))*PolyLog[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

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 1210, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)*ln(x)/(x^4+2),x)

[Out]

-1/4/(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)))*2^(1/4)
-1/4/(1/2*2^(3/4)+1/2*I*2^(3/4))^3*dilog((1/2*2^(3/4)+1/2*I*2^(3/4)-x)/(1/2*2^(3/4)+1/2*I*2^(3/4)))*2^(1/4)+1/
4*I/(1/2*I*2^(3/4)-1/2*2^(3/4))^3*dilog((1/2*I*2^(3/4)-1/2*2^(3/4)-x)/(1/2*I*2^(3/4)-1/2*2^(3/4)))*2^(1/4)-1/4
*I/(-1/2*2^(3/4)-1/2*I*2^(3/4))^3*dilog((-1/2*2^(3/4)-1/2*I*2^(3/4)-x)/(-1/2*2^(3/4)-1/2*I*2^(3/4)))*2^(1/4)+1
/4/(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)))+1/4/(1/2*
2^(3/4)+1/2*I*2^(3/4))^3*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*I/(-1/2*I*2^(3/4
)+1/2*2^(3/4))^3*dilog((-1/2*I*2^(3/4)+1/2*2^(3/4)-x)/(-1/2*I*2^(3/4)+1/2*2^(3/4)))*2^(1/4)-1/4*I/(-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)))*2^(1/4)+1/4/(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)))*2^(1/4)+1/4/(1/2*I*2^
(3/4)-1/2*2^(3/4))^3*dilog((1/2*I*2^(3/4)-1/2*2^(3/4)-x)/(1/2*I*2^(3/4)-1/2*2^(3/4)))*2^(1/4)+1/4/(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)))+1/4/(1/2*I*2^(3/4)-1/2*2
^(3/4))^3*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*
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)))*2^(1/4)+1/4/(-1/2*2^(3/4)-1/2*I*2^(3/4))
^3*dilog((-1/2*2^(3/4)-1/2*I*2^(3/4)-x)/(-1/2*2^(3/4)-1/2*I*2^(3/4)))*2^(1/4)+1/4*I/(1/2*2^(3/4)+1/2*I*2^(3/4)
)^3*dilog((1/2*2^(3/4)+1/2*I*2^(3/4)-x)/(1/2*2^(3/4)+1/2*I*2^(3/4)))*2^(1/4)+1/4*I/(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)))*2^(1/4)+1/4/(-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)))+1/4/(-1/2*2^(3/4)-1/2*I*2^(3/4))^3*d
ilog((-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*I/(-1/2*2^(3/4)-1/2*I*2^(3/4))^3*ln(x)*l
n((-1/2*2^(3/4)-1/2*I*2^(3/4)-x)/(-1/2*2^(3/4)-1/2*I*2^(3/4)))*2^(1/4)+1/4*I/(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)))*2^(1/4)-1/4/(-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)))*2^(1/4)-1/4/(-1/2*I*2^(3/4)+1/2*2^(3/4))^3
*dilog((-1/2*I*2^(3/4)+1/2*2^(3/4)-x)/(-1/2*I*2^(3/4)+1/2*2^(3/4)))*2^(1/4)+1/4/(-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)))+1/4/(-1/2*I*2^(3/4)+1/2*2^(3/4))^3*dilo
g((-1/2*I*2^(3/4)+1/2*2^(3/4)-x)/(-1/2*I*2^(3/4)+1/2*2^(3/4)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{3} + 1\right )} \log \left (x\right )}{x^{4} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*log(x)/(x^4+2),x, algorithm="maxima")

[Out]

integrate((x^3 + 1)*log(x)/(x^4 + 2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (x^{3} + 1\right )} \log \left (x\right )}{x^{4} + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*log(x)/(x^4+2),x, algorithm="fricas")

[Out]

integral((x^3 + 1)*log(x)/(x^4 + 2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x + 1\right ) \left (x^{2} - x + 1\right ) \log{\left (x \right )}}{x^{4} + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)*ln(x)/(x**4+2),x)

[Out]

Integral((x + 1)*(x**2 - x + 1)*log(x)/(x**4 + 2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{3} + 1\right )} \log \left (x\right )}{x^{4} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*log(x)/(x^4+2),x, algorithm="giac")

[Out]

integrate((x^3 + 1)*log(x)/(x^4 + 2), x)