3.614 \(\int (1+x^4) (1-2 \log (x)+\log ^3(x)) \, dx\)

Optimal. Leaf size=60 \[ \frac{169 x^5}{625}+\frac{1}{5} x^5 \log ^3(x)-\frac{3}{25} x^5 \log ^2(x)-\frac{44}{125} x^5 \log (x)-3 x+x \log ^3(x)-3 x \log ^2(x)+4 x \log (x) \]

[Out]

-3*x + (169*x^5)/625 + 4*x*Log[x] - (44*x^5*Log[x])/125 - 3*x*Log[x]^2 - (3*x^5*Log[x]^2)/25 + x*Log[x]^3 + (x
^5*Log[x]^3)/5

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Rubi [A]  time = 0.0906383, antiderivative size = 73, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6742, 2313, 12, 2330, 2296, 2295, 2305, 2304} \[ \frac{169 x^5}{625}+\frac{1}{5} x^5 \log ^3(x)-\frac{3}{25} x^5 \log ^2(x)+\frac{6}{125} x^5 \log (x)-\frac{2}{5} \left (x^5+5 x\right ) \log (x)-3 x+x \log ^3(x)-3 x \log ^2(x)+6 x \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)*(1 - 2*Log[x] + Log[x]^3),x]

[Out]

-3*x + (169*x^5)/625 + 6*x*Log[x] + (6*x^5*Log[x])/125 - (2*(5*x + x^5)*Log[x])/5 - 3*x*Log[x]^2 - (3*x^5*Log[
x]^2)/25 + x*Log[x]^3 + (x^5*Log[x]^3)/5

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \left (1+x^4\right ) \left (1-2 \log (x)+\log ^3(x)\right ) \, dx &=\int \left (1+x^4-2 \left (1+x^4\right ) \log (x)+\left (1+x^4\right ) \log ^3(x)\right ) \, dx\\ &=x+\frac{x^5}{5}-2 \int \left (1+x^4\right ) \log (x) \, dx+\int \left (1+x^4\right ) \log ^3(x) \, dx\\ &=x+\frac{x^5}{5}-\frac{2}{5} \left (5 x+x^5\right ) \log (x)+2 \int \frac{1}{5} \left (5+x^4\right ) \, dx+\int \left (\log ^3(x)+x^4 \log ^3(x)\right ) \, dx\\ &=x+\frac{x^5}{5}-\frac{2}{5} \left (5 x+x^5\right ) \log (x)+\frac{2}{5} \int \left (5+x^4\right ) \, dx+\int \log ^3(x) \, dx+\int x^4 \log ^3(x) \, dx\\ &=3 x+\frac{7 x^5}{25}-\frac{2}{5} \left (5 x+x^5\right ) \log (x)+x \log ^3(x)+\frac{1}{5} x^5 \log ^3(x)-\frac{3}{5} \int x^4 \log ^2(x) \, dx-3 \int \log ^2(x) \, dx\\ &=3 x+\frac{7 x^5}{25}-\frac{2}{5} \left (5 x+x^5\right ) \log (x)-3 x \log ^2(x)-\frac{3}{25} x^5 \log ^2(x)+x \log ^3(x)+\frac{1}{5} x^5 \log ^3(x)+\frac{6}{25} \int x^4 \log (x) \, dx+6 \int \log (x) \, dx\\ &=-3 x+\frac{169 x^5}{625}+6 x \log (x)+\frac{6}{125} x^5 \log (x)-\frac{2}{5} \left (5 x+x^5\right ) \log (x)-3 x \log ^2(x)-\frac{3}{25} x^5 \log ^2(x)+x \log ^3(x)+\frac{1}{5} x^5 \log ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0036325, size = 60, normalized size = 1. \[ \frac{169 x^5}{625}+\frac{1}{5} x^5 \log ^3(x)-\frac{3}{25} x^5 \log ^2(x)-\frac{44}{125} x^5 \log (x)-3 x+x \log ^3(x)-3 x \log ^2(x)+4 x \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3*x + (169*x^5)/625 + 4*x*Log[x] - (44*x^5*Log[x])/125 - 3*x*Log[x]^2 - (3*x^5*Log[x]^2)/25 + x*Log[x]^3 + (x
^5*Log[x]^3)/5

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Maple [A]  time = 0.002, size = 53, normalized size = 0.9 \begin{align*} -3\,x+{\frac{169\,{x}^{5}}{625}}+4\,x\ln \left ( x \right ) -{\frac{44\,{x}^{5}\ln \left ( x \right ) }{125}}-3\,x \left ( \ln \left ( x \right ) \right ) ^{2}-{\frac{3\,{x}^{5} \left ( \ln \left ( x \right ) \right ) ^{2}}{25}}+x \left ( \ln \left ( x \right ) \right ) ^{3}+{\frac{{x}^{5} \left ( \ln \left ( x \right ) \right ) ^{3}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x+169/625*x^5+4*x*ln(x)-44/125*x^5*ln(x)-3*x*ln(x)^2-3/25*x^5*ln(x)^2+x*ln(x)^3+1/5*x^5*ln(x)^3

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Maxima [A]  time = 0.95742, size = 89, normalized size = 1.48 \begin{align*} \frac{1}{625} \,{\left (125 \, \log \left (x\right )^{3} - 75 \, \log \left (x\right )^{2} + 30 \, \log \left (x\right ) - 6\right )} x^{5} - \frac{2}{25} \, x^{5}{\left (5 \, \log \left (x\right ) - 1\right )} + \frac{1}{5} \, x^{5} +{\left (\log \left (x\right )^{3} - 3 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 6\right )} x - 2 \, x{\left (\log \left (x\right ) - 1\right )} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/625*(125*log(x)^3 - 75*log(x)^2 + 30*log(x) - 6)*x^5 - 2/25*x^5*(5*log(x) - 1) + 1/5*x^5 + (log(x)^3 - 3*log
(x)^2 + 6*log(x) - 6)*x - 2*x*(log(x) - 1) + x

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Fricas [A]  time = 2.06399, size = 144, normalized size = 2.4 \begin{align*} \frac{169}{625} \, x^{5} + \frac{1}{5} \,{\left (x^{5} + 5 \, x\right )} \log \left (x\right )^{3} - \frac{3}{25} \,{\left (x^{5} + 25 \, x\right )} \log \left (x\right )^{2} - \frac{4}{125} \,{\left (11 \, x^{5} - 125 \, x\right )} \log \left (x\right ) - 3 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

169/625*x^5 + 1/5*(x^5 + 5*x)*log(x)^3 - 3/25*(x^5 + 25*x)*log(x)^2 - 4/125*(11*x^5 - 125*x)*log(x) - 3*x

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Sympy [A]  time = 0.130412, size = 51, normalized size = 0.85 \begin{align*} \frac{169 x^{5}}{625} - 3 x + \left (- \frac{44 x^{5}}{125} + 4 x\right ) \log{\left (x \right )} + \left (- \frac{3 x^{5}}{25} - 3 x\right ) \log{\left (x \right )}^{2} + \left (\frac{x^{5}}{5} + x\right ) \log{\left (x \right )}^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

169*x**5/625 - 3*x + (-44*x**5/125 + 4*x)*log(x) + (-3*x**5/25 - 3*x)*log(x)**2 + (x**5/5 + x)*log(x)**3

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Giac [A]  time = 1.13155, size = 70, normalized size = 1.17 \begin{align*} \frac{1}{5} \, x^{5} \log \left (x\right )^{3} - \frac{3}{25} \, x^{5} \log \left (x\right )^{2} - \frac{44}{125} \, x^{5} \log \left (x\right ) + \frac{169}{625} \, x^{5} + x \log \left (x\right )^{3} - 3 \, x \log \left (x\right )^{2} + 4 \, x \log \left (x\right ) - 3 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*log(x)^3 - 3/25*x^5*log(x)^2 - 44/125*x^5*log(x) + 169/625*x^5 + x*log(x)^3 - 3*x*log(x)^2 + 4*x*log(x
) - 3*x