3.85.36 \(\int \frac {-4-4 x+2 e x+(-400 x^3-400 x^4+200 e x^4) \log (x^2)+(-400 x^3-500 x^4+250 e x^4) \log ^2(x^2)+(-10000 x^7+5000 e x^7) \log ^3(x^2)+(-10000 x^7+5000 e x^7) \log ^4(x^2)}{-2+e} \, dx\)

Optimal. Leaf size=26 \[ -2+\left (\frac {2}{2-e}+x+25 x^4 \log ^2\left (x^2\right )\right )^2 \]

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Rubi [B]  time = 0.30, antiderivative size = 108, normalized size of antiderivative = 4.15, number of steps used = 28, number of rules used = 8, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6, 12, 1593, 43, 2334, 2353, 2305, 2304} \begin {gather*} x^2+625 x^8 \log ^4\left (x^2\right )+50 x^5 \log ^2\left (x^2\right )-40 x^5 \log \left (x^2\right )+\frac {100 x^4 \log ^2\left (x^2\right )}{2-e}-\frac {100 x^4 \log \left (x^2\right )}{2-e}+\frac {20 \left (2 (2-e) x^5+5 x^4\right ) \log \left (x^2\right )}{2-e}+\frac {4 x}{2-e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 4*x + 2*E*x + (-400*x^3 - 400*x^4 + 200*E*x^4)*Log[x^2] + (-400*x^3 - 500*x^4 + 250*E*x^4)*Log[x^2]^
2 + (-10000*x^7 + 5000*E*x^7)*Log[x^2]^3 + (-10000*x^7 + 5000*E*x^7)*Log[x^2]^4)/(-2 + E),x]

[Out]

(4*x)/(2 - E) + x^2 - (100*x^4*Log[x^2])/(2 - E) - 40*x^5*Log[x^2] + (20*(5*x^4 + 2*(2 - E)*x^5)*Log[x^2])/(2
- E) + (100*x^4*Log[x^2]^2)/(2 - E) + 50*x^5*Log[x^2]^2 + 625*x^8*Log[x^2]^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 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 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 2353

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+(-4+2 e) x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx\\ &=\frac {\int \left (-4+(-4+2 e) x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {\int \left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {\int \left (-400 x^3+(-400+200 e) x^4\right ) \log \left (x^2\right ) \, dx}{-2+e}+\frac {\int \left (-400 x^3+(-500+250 e) x^4\right ) \log ^2\left (x^2\right ) \, dx}{-2+e}+\frac {\int (-10000+5000 e) x^7 \log ^3\left (x^2\right ) \, dx}{-2+e}+\frac {\int (-10000+5000 e) x^7 \log ^4\left (x^2\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+5000 \int x^7 \log ^3\left (x^2\right ) \, dx+5000 \int x^7 \log ^4\left (x^2\right ) \, dx+\frac {\int x^3 (-400+(-400+200 e) x) \log \left (x^2\right ) \, dx}{-2+e}+\frac {\int x^3 (-400+(-500+250 e) x) \log ^2\left (x^2\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}+625 x^8 \log ^3\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )-3750 \int x^7 \log ^2\left (x^2\right ) \, dx-5000 \int x^7 \log ^3\left (x^2\right ) \, dx+\frac {2 \int 20 x^3 (-5+2 (-2+e) x) \, dx}{2-e}+\frac {\int \left (-400 x^3 \log ^2\left (x^2\right )+250 (-2+e) x^4 \log ^2\left (x^2\right )\right ) \, dx}{-2+e}\\ &=\frac {4 x}{2-e}+x^2+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}-\frac {1875}{4} x^8 \log ^2\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )+250 \int x^4 \log ^2\left (x^2\right ) \, dx+1875 \int x^7 \log \left (x^2\right ) \, dx+3750 \int x^7 \log ^2\left (x^2\right ) \, dx+\frac {40 \int x^3 (-5+2 (-2+e) x) \, dx}{2-e}+\frac {400 \int x^3 \log ^2\left (x^2\right ) \, dx}{2-e}\\ &=\frac {4 x}{2-e}+x^2-\frac {1875 x^8}{32}+\frac {1875}{8} x^8 \log \left (x^2\right )+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}+\frac {100 x^4 \log ^2\left (x^2\right )}{2-e}+50 x^5 \log ^2\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )-200 \int x^4 \log \left (x^2\right ) \, dx-1875 \int x^7 \log \left (x^2\right ) \, dx+\frac {40 \int \left (-5 x^3+2 (-2+e) x^4\right ) \, dx}{2-e}-\frac {400 \int x^3 \log \left (x^2\right ) \, dx}{2-e}\\ &=\frac {4 x}{2-e}+x^2-\frac {100 x^4 \log \left (x^2\right )}{2-e}-40 x^5 \log \left (x^2\right )+\frac {20 \left (5 x^4+2 (2-e) x^5\right ) \log \left (x^2\right )}{2-e}+\frac {100 x^4 \log ^2\left (x^2\right )}{2-e}+50 x^5 \log ^2\left (x^2\right )+625 x^8 \log ^4\left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 29, normalized size = 1.12 \begin {gather*} \frac {\left (-2+(-2+e) x+25 (-2+e) x^4 \log ^2\left (x^2\right )\right )^2}{(-2+e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 4*x + 2*E*x + (-400*x^3 - 400*x^4 + 200*E*x^4)*Log[x^2] + (-400*x^3 - 500*x^4 + 250*E*x^4)*Log
[x^2]^2 + (-10000*x^7 + 5000*E*x^7)*Log[x^2]^3 + (-10000*x^7 + 5000*E*x^7)*Log[x^2]^4)/(-2 + E),x]

[Out]

(-2 + (-2 + E)*x + 25*(-2 + E)*x^4*Log[x^2]^2)^2/(-2 + E)^2

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fricas [B]  time = 0.61, size = 67, normalized size = 2.58 \begin {gather*} \frac {625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{4} + x^{2} e + 50 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} - 2 \, x^{2} - 4 \, x}{e - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5000*x^7*exp(1)-10000*x^7)*log(x^2)^4+(5000*x^7*exp(1)-10000*x^7)*log(x^2)^3+(250*x^4*exp(1)-500*x
^4-400*x^3)*log(x^2)^2+(200*x^4*exp(1)-400*x^4-400*x^3)*log(x^2)+2*x*exp(1)-4*x-4)/(exp(1)-2),x, algorithm="fr
icas")

[Out]

(625*(x^8*e - 2*x^8)*log(x^2)^4 + x^2*e + 50*(x^5*e - 2*x^5 - 2*x^4)*log(x^2)^2 - 2*x^2 - 4*x)/(e - 2)

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giac [B]  time = 0.16, size = 221, normalized size = 8.50 \begin {gather*} -\frac {40000 \, x^{8} \log \left (x^{2}\right )^{4} + 512 \, x^{5} e - 2560 \, x^{5} \log \left (x^{2}\right ) - 3200 \, x^{4} \log \left (x^{2}\right ) - 32 \, x^{2} e - 1600 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} + 64 \, x^{2} - 625 \, {\left (32 \, x^{8} \log \left (x^{2}\right )^{4} - 32 \, x^{8} \log \left (x^{2}\right )^{3} + 24 \, x^{8} \log \left (x^{2}\right )^{2} - 12 \, x^{8} \log \left (x^{2}\right ) + 3 \, x^{8}\right )} e - 625 \, {\left (32 \, x^{8} \log \left (x^{2}\right )^{3} - 24 \, x^{8} \log \left (x^{2}\right )^{2} + 12 \, x^{8} \log \left (x^{2}\right ) - 3 \, x^{8}\right )} e + 256 \, {\left (5 \, x^{5} \log \left (x^{2}\right ) - 2 \, x^{5}\right )} e - 640 \, {\left (2 \, x^{5} e - 4 \, x^{5} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) + 128 \, x}{32 \, {\left (e - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5000*x^7*exp(1)-10000*x^7)*log(x^2)^4+(5000*x^7*exp(1)-10000*x^7)*log(x^2)^3+(250*x^4*exp(1)-500*x
^4-400*x^3)*log(x^2)^2+(200*x^4*exp(1)-400*x^4-400*x^3)*log(x^2)+2*x*exp(1)-4*x-4)/(exp(1)-2),x, algorithm="gi
ac")

[Out]

-1/32*(40000*x^8*log(x^2)^4 + 512*x^5*e - 2560*x^5*log(x^2) - 3200*x^4*log(x^2) - 32*x^2*e - 1600*(x^5*e - 2*x
^5 - 2*x^4)*log(x^2)^2 + 64*x^2 - 625*(32*x^8*log(x^2)^4 - 32*x^8*log(x^2)^3 + 24*x^8*log(x^2)^2 - 12*x^8*log(
x^2) + 3*x^8)*e - 625*(32*x^8*log(x^2)^3 - 24*x^8*log(x^2)^2 + 12*x^8*log(x^2) - 3*x^8)*e + 256*(5*x^5*log(x^2
) - 2*x^5)*e - 640*(2*x^5*e - 4*x^5 - 5*x^4)*log(x^2) + 128*x)/(e - 2)

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maple [B]  time = 0.06, size = 76, normalized size = 2.92

method	result	size
risch	\(625 x^{8} \ln \left (x^{2}\right )^{4}+\frac {\left (50 x^{5} {\mathrm e}-100 x^{5}-100 x^{4}\right ) \ln \left (x^{2}\right )^{2}}{{\mathrm e}-2}+\frac {x^{2} {\mathrm e}}{{\mathrm e}-2}-\frac {2 x^{2}}{{\mathrm e}-2}-\frac {4 x}{{\mathrm e}-2}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5000*x^7*exp(1)-10000*x^7)*ln(x^2)^4+(5000*x^7*exp(1)-10000*x^7)*ln(x^2)^3+(250*x^4*exp(1)-500*x^4-400*x
^3)*ln(x^2)^2+(200*x^4*exp(1)-400*x^4-400*x^3)*ln(x^2)+2*x*exp(1)-4*x-4)/(exp(1)-2),x,method=_RETURNVERBOSE)

[Out]

625*x^8*ln(x^2)^4+1/(exp(1)-2)*(50*x^5*exp(1)-100*x^5-100*x^4)*ln(x^2)^2+1/(exp(1)-2)*x^2*exp(1)-2/(exp(1)-2)*
x^2-4/(exp(1)-2)*x

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maxima [B]  time = 0.35, size = 149, normalized size = 5.73 \begin {gather*} -\frac {625 \, x^{8} {\left (e - 2\right )} \log \left (x^{2}\right )^{3} - 625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{4} - 625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{3} - x^{2} e - 50 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} + 2 \, x^{2} + 20 \, {\left (2 \, x^{5} {\left (e - 2\right )} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) - 20 \, {\left (2 \, x^{5} e - 4 \, x^{5} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) + 4 \, x}{e - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5000*x^7*exp(1)-10000*x^7)*log(x^2)^4+(5000*x^7*exp(1)-10000*x^7)*log(x^2)^3+(250*x^4*exp(1)-500*x
^4-400*x^3)*log(x^2)^2+(200*x^4*exp(1)-400*x^4-400*x^3)*log(x^2)+2*x*exp(1)-4*x-4)/(exp(1)-2),x, algorithm="ma
xima")

[Out]

-(625*x^8*(e - 2)*log(x^2)^3 - 625*(x^8*e - 2*x^8)*log(x^2)^4 - 625*(x^8*e - 2*x^8)*log(x^2)^3 - x^2*e - 50*(x
^5*e - 2*x^5 - 2*x^4)*log(x^2)^2 + 2*x^2 + 20*(2*x^5*(e - 2) - 5*x^4)*log(x^2) - 20*(2*x^5*e - 4*x^5 - 5*x^4)*
log(x^2) + 4*x)/(e - 2)

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mupad [B]  time = 5.25, size = 56, normalized size = 2.15 \begin {gather*} -\frac {x\,\left (25\,x^3\,{\ln \left (x^2\right )}^2+1\right )\,\left (2\,x-x\,\mathrm {e}+50\,x^4\,{\ln \left (x^2\right )}^2-25\,x^4\,{\ln \left (x^2\right )}^2\,\mathrm {e}+4\right )}{\mathrm {e}-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + log(x^2)*(400*x^3 - 200*x^4*exp(1) + 400*x^4) - log(x^2)^3*(5000*x^7*exp(1) - 10000*x^7) - log(x^2
)^4*(5000*x^7*exp(1) - 10000*x^7) - 2*x*exp(1) + log(x^2)^2*(400*x^3 - 250*x^4*exp(1) + 500*x^4) + 4)/(exp(1)
- 2),x)

[Out]

-(x*(25*x^3*log(x^2)^2 + 1)*(2*x - x*exp(1) + 50*x^4*log(x^2)^2 - 25*x^4*log(x^2)^2*exp(1) + 4))/(exp(1) - 2)

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sympy [B]  time = 0.22, size = 53, normalized size = 2.04 \begin {gather*} 625 x^{8} \log {\left (x^{2} \right )}^{4} + x^{2} - \frac {4 x}{-2 + e} + \frac {\left (- 100 x^{5} + 50 e x^{5} - 100 x^{4}\right ) \log {\left (x^{2} \right )}^{2}}{-2 + e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5000*x**7*exp(1)-10000*x**7)*ln(x**2)**4+(5000*x**7*exp(1)-10000*x**7)*ln(x**2)**3+(250*x**4*exp(1
)-500*x**4-400*x**3)*ln(x**2)**2+(200*x**4*exp(1)-400*x**4-400*x**3)*ln(x**2)+2*x*exp(1)-4*x-4)/(exp(1)-2),x)

[Out]

625*x**8*log(x**2)**4 + x**2 - 4*x/(-2 + E) + (-100*x**5 + 50*E*x**5 - 100*x**4)*log(x**2)**2/(-2 + E)

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