3.5.35 \(\int \frac {x^5-x^6+e^x (-x+x^2)+(6 x^5-3 x^6+e^x (-2 x+2 x^2)) \log (x)+(-e^x x+x^5+(-2 e^x x+2 x^5) \log (x)) \log (-e^x+x^4)}{e^x-x^4} \, dx\)

Optimal. Leaf size=23 \[ 5+x^2 \log (x) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \]

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Rubi [B]  time = 3.50, antiderivative size = 62, normalized size of antiderivative = 2.70, number of steps used = 33, number of rules used = 12, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.126, Rules used = {6688, 14, 6742, 2554, 43, 2334, 12, 2303, 2551, 2304, 30, 2557} \begin {gather*} -\frac {x^3}{3}+\frac {1}{3} x^3 \log (x)+\frac {x^2}{2}-x^2 \log (x) \log \left (x^4-e^x\right )-\frac {1}{6} \left (3 x^2-2 x^3\right ) (2 \log (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5 - x^6 + E^x*(-x + x^2) + (6*x^5 - 3*x^6 + E^x*(-2*x + 2*x^2))*Log[x] + (-(E^x*x) + x^5 + (-2*E^x*x +
2*x^5)*Log[x])*Log[-E^x + x^4])/(E^x - x^4),x]

[Out]

x^2/2 - x^3/3 + (x^3*Log[x])/3 - ((3*x^2 - 2*x^3)*(1 + 2*Log[x]))/6 - x^2*Log[x]*Log[-E^x + x^4]

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2303

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

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(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] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int x \left (-1+x-\log \left (-e^x+x^4\right )+\frac {\log (x) \left (2 e^x (-1+x)-3 (-2+x) x^4-2 \left (e^x-x^4\right ) \log \left (-e^x+x^4\right )\right )}{e^x-x^4}\right ) \, dx\\ &=\int \left (\frac {(-4+x) x^5 \log (x)}{-e^x+x^4}+x (1+2 \log (x)) \left (-1+x-\log \left (-e^x+x^4\right )\right )\right ) \, dx\\ &=\int \frac {(-4+x) x^5 \log (x)}{-e^x+x^4} \, dx+\int x (1+2 \log (x)) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \, dx\\ &=\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx+\int \left ((-1+x) x (1+2 \log (x))-x (1+2 \log (x)) \log \left (-e^x+x^4\right )\right ) \, dx-\int \frac {-4 \int \frac {x^5}{-e^x+x^4} \, dx+\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx+\int (-1+x) x (1+2 \log (x)) \, dx-\int x (1+2 \log (x)) \log \left (-e^x+x^4\right ) \, dx-\int \left (-\frac {4 \int \frac {x^5}{-e^x+x^4} \, dx}{x}+\frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x}\right ) \, dx\\ &=-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-2 \int \frac {1}{6} x (-3+2 x) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx-\int \left (x \log \left (-e^x+x^4\right )+2 x \log (x) \log \left (-e^x+x^4\right )\right ) \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-\frac {1}{3} \int x (-3+2 x) \, dx-2 \int x \log (x) \log \left (-e^x+x^4\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx-\int x \log \left (-e^x+x^4\right ) \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-\frac {1}{2} x^2 \log \left (-e^x+x^4\right )-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \left (-3 x+2 x^2\right ) \, dx+\frac {1}{2} \int \frac {x^2 \left (-e^x+4 x^3\right )}{-e^x+x^4} \, dx+2 \int \frac {x^2 \left (-e^x+4 x^3\right ) \log (x)}{2 \left (-e^x+x^4\right )} \, dx+2 \int \frac {1}{2} x \log \left (-e^x+x^4\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-\frac {1}{2} x^2 \log \left (-e^x+x^4\right )-x^2 \log (x) \log \left (-e^x+x^4\right )+\frac {1}{2} \int \left (x^2-\frac {(-4+x) x^5}{-e^x+x^4}\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx+\int \frac {x^2 \left (-e^x+4 x^3\right ) \log (x)}{-e^x+x^4} \, dx+\int x \log \left (-e^x+x^4\right ) \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {x^3}{18}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{2} \int \frac {(-4+x) x^5}{-e^x+x^4} \, dx-\frac {1}{2} \int \frac {x^2 \left (-e^x+4 x^3\right )}{-e^x+x^4} \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\int \frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx-3 \int \frac {x^6}{-e^x+x^4} \, dx}{3 x} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {x^3}{18}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx-3 \int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx-\frac {1}{2} \int \left (x^2-\frac {(-4+x) x^5}{-e^x+x^4}\right ) \, dx-\frac {1}{2} \int \left (-\frac {4 x^5}{-e^x+x^4}+\frac {x^6}{-e^x+x^4}\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \left (\frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx}{x}-\frac {3 \int \frac {x^6}{-e^x+x^4} \, dx}{x}\right ) \, dx+\frac {1}{2} \int \frac {(-4+x) x^5}{-e^x+x^4} \, dx-\frac {1}{2} \int \frac {x^6}{-e^x+x^4} \, dx+2 \int \frac {x^5}{-e^x+x^4} \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\frac {1}{2} \int \frac {x^6}{-e^x+x^4} \, dx+\frac {1}{2} \int \left (-\frac {4 x^5}{-e^x+x^4}+\frac {x^6}{-e^x+x^4}\right ) \, dx+2 \int \frac {x^5}{-e^x+x^4} \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \left (x^2+\frac {12 \int \frac {x^5}{-e^x+x^4} \, dx}{x}\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {x^3}{3}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 21, normalized size = 0.91 \begin {gather*} x^2 \log (x) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5 - x^6 + E^x*(-x + x^2) + (6*x^5 - 3*x^6 + E^x*(-2*x + 2*x^2))*Log[x] + (-(E^x*x) + x^5 + (-2*E^
x*x + 2*x^5)*Log[x])*Log[-E^x + x^4])/(E^x - x^4),x]

[Out]

x^2*Log[x]*(-1 + x - Log[-E^x + x^4])

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fricas [A]  time = 0.89, size = 29, normalized size = 1.26 \begin {gather*} -x^{2} \log \left (x^{4} - e^{x}\right ) \log \relax (x) + {\left (x^{3} - x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(x)*x+2*x^5)*log(x)-exp(x)*x+x^5)*log(-exp(x)+x^4)+((2*x^2-2*x)*exp(x)-3*x^6+6*x^5)*log(x)+
(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x, algorithm="fricas")

[Out]

-x^2*log(x^4 - e^x)*log(x) + (x^3 - x^2)*log(x)

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giac [A]  time = 0.45, size = 30, normalized size = 1.30 \begin {gather*} x^{3} \log \relax (x) - x^{2} \log \left (x^{4} - e^{x}\right ) \log \relax (x) - x^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(x)*x+2*x^5)*log(x)-exp(x)*x+x^5)*log(-exp(x)+x^4)+((2*x^2-2*x)*exp(x)-3*x^6+6*x^5)*log(x)+
(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x, algorithm="giac")

[Out]

x^3*log(x) - x^2*log(x^4 - e^x)*log(x) - x^2*log(x)

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maple [A]  time = 0.03, size = 27, normalized size = 1.17

method	result	size
risch	\(-x^{2} \ln \relax (x ) \ln \left (-{\mathrm e}^{x}+x^{4}\right )+x^{2} \left (x -1\right ) \ln \relax (x )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*exp(x)*x+2*x^5)*ln(x)-exp(x)*x+x^5)*ln(-exp(x)+x^4)+((2*x^2-2*x)*exp(x)-3*x^6+6*x^5)*ln(x)+(x^2-x)*e
xp(x)-x^6+x^5)/(exp(x)-x^4),x,method=_RETURNVERBOSE)

[Out]

-x^2*ln(x)*ln(-exp(x)+x^4)+x^2*(x-1)*ln(x)

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maxima [A]  time = 0.57, size = 29, normalized size = 1.26 \begin {gather*} -x^{2} \log \left (x^{4} - e^{x}\right ) \log \relax (x) + {\left (x^{3} - x^{2}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(x)*x+2*x^5)*log(x)-exp(x)*x+x^5)*log(-exp(x)+x^4)+((2*x^2-2*x)*exp(x)-3*x^6+6*x^5)*log(x)+
(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x, algorithm="maxima")

[Out]

-x^2*log(x^4 - e^x)*log(x) + (x^3 - x^2)*log(x)

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mupad [B]  time = 0.63, size = 21, normalized size = 0.91 \begin {gather*} -x^2\,\ln \relax (x)\,\left (\ln \left (x^4-{\mathrm {e}}^x\right )-x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^4 - exp(x))*(log(x)*(2*x*exp(x) - 2*x^5) + x*exp(x) - x^5) + exp(x)*(x - x^2) + log(x)*(exp(x)*(2*
x - 2*x^2) - 6*x^5 + 3*x^6) - x^5 + x^6)/(exp(x) - x^4),x)

[Out]

-x^2*log(x)*(log(x^4 - exp(x)) - x + 1)

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sympy [A]  time = 0.58, size = 24, normalized size = 1.04 \begin {gather*} - x^{2} \log {\relax (x )} \log {\left (x^{4} - e^{x} \right )} + \left (x^{3} - x^{2}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*exp(x)*x+2*x**5)*ln(x)-exp(x)*x+x**5)*ln(-exp(x)+x**4)+((2*x**2-2*x)*exp(x)-3*x**6+6*x**5)*ln(
x)+(x**2-x)*exp(x)-x**6+x**5)/(exp(x)-x**4),x)

[Out]

-x**2*log(x)*log(x**4 - exp(x)) + (x**3 - x**2)*log(x)

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