3.7.14 \(\int (1+x^4) (1-2 \log (x)+\log ^3(x)) \, dx\) [614]

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

[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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Rubi [A]
time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6874, 2350, 12, 2367, 2333, 2332, 2342, 2341} \begin {gather*} \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) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(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

Rule 12

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

Rule 2332

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

Rule 2333

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 2341

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
]

Rule 2342

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 2350

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

Rule 2367

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 6874

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

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*}

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Mathematica [A]
time = 0.01, size = 60, normalized size = 1.00 \begin {gather*} -3 x+\frac {169 x^5}{625}+4 x \log (x)-\frac {44}{125} x^5 \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 {gather*}

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 = 0.01, size = 53, normalized size = 0.88

method	result	size
risch	\(\left (\frac {1}{5} x^{5}+x \right ) \ln \left (x \right )^{3}+\left (-\frac {3}{25} x^{5}-3 x \right ) \ln \left (x \right )^{2}+\left (-\frac {44}{125} x^{5}+4 x \right ) \ln \left (x \right )+\frac {169 x^{5}}{625}-3 x\)	\(48\)
default	\(-3 x +\frac {169 x^{5}}{625}+4 x \ln \left (x \right )-\frac {44 x^{5} \ln \left (x \right )}{125}-3 x \ln \left (x \right )^{2}-\frac {3 x^{5} \ln \left (x \right )^{2}}{25}+x \ln \left (x \right )^{3}+\frac {x^{5} \ln \left (x \right )^{3}}{5}\)	\(53\)
norman	\(-3 x +\frac {169 x^{5}}{625}+4 x \ln \left (x \right )-\frac {44 x^{5} \ln \left (x \right )}{125}-3 x \ln \left (x \right )^{2}-\frac {3 x^{5} \ln \left (x \right )^{2}}{25}+x \ln \left (x \right )^{3}+\frac {x^{5} \ln \left (x \right )^{3}}{5}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.51, size = 66, normalized size = 1.10 \begin {gather*} \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 {gather*}

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 = 0.63, size = 48, normalized size = 0.80 \begin {gather*} \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 {gather*}

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 = 0.06, size = 51, normalized size = 0.85 \begin {gather*} \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 {gather*}

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 = 1.46, size = 52, normalized size = 0.87 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.37, size = 51, normalized size = 0.85 \begin {gather*} \frac {x\,\left (125\,x^4\,{\ln \left (x\right )}^3-75\,x^4\,{\ln \left (x\right )}^2-220\,x^4\,\ln \left (x\right )+169\,x^4+625\,{\ln \left (x\right )}^3-1875\,{\ln \left (x\right )}^2+2500\,\ln \left (x\right )-1875\right )}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(2500*log(x) - 220*x^4*log(x) - 1875*log(x)^2 + 625*log(x)^3 - 75*x^4*log(x)^2 + 125*x^4*log(x)^3 + 169*x^4
- 1875))/625

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