3.88.95 \(\int \frac {10 x}{125-75 x+15 x^2-x^3+(600 x-240 x^2+24 x^3) (i \pi +\log (\frac {e}{4}))^2+(960 x^2-192 x^3) (i \pi +\log (\frac {e}{4}))^4+512 x^3 (i \pi +\log (\frac {e}{4}))^6} \, dx\)

Optimal. Leaf size=28 \[ \frac {x^2}{\left (5-x+8 x \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2\right )^2} \]

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Rubi [B]  time = 0.56, antiderivative size = 133, normalized size of antiderivative = 4.75, number of steps used = 4, number of rules used = 3, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6, 12, 2074} \begin {gather*} \frac {25}{\left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )^2 \left (5+x \left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )\right )^2}-\frac {10}{\left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )^2 \left (5+x \left (7-8 \pi ^2+8 \log ^2(4)+16 i \pi (1-\log (4))-16 \log (4)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10*x)/(125 - 75*x + 15*x^2 - x^3 + (600*x - 240*x^2 + 24*x^3)*(I*Pi + Log[E/4])^2 + (960*x^2 - 192*x^3)*(
I*Pi + Log[E/4])^4 + 512*x^3*(I*Pi + Log[E/4])^6),x]

[Out]

25/((7 - 8*Pi^2 + (16*I)*Pi*(1 - Log[4]) - 16*Log[4] + 8*Log[4]^2)^2*(5 + x*(7 - 8*Pi^2 + (16*I)*Pi*(1 - Log[4
]) - 16*Log[4] + 8*Log[4]^2))^2) - 10/((7 - 8*Pi^2 + (16*I)*Pi*(1 - Log[4]) - 16*Log[4] + 8*Log[4]^2)^2*(5 + x
*(7 - 8*Pi^2 + (16*I)*Pi*(1 - Log[4]) - 16*Log[4] + 8*Log[4]^2)))

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 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 x}{125-75 x+15 x^2+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+x^3 \left (-1+512 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6\right )} \, dx\\ &=10 \int \frac {x}{125-75 x+15 x^2+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+x^3 \left (-1+512 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6\right )} \, dx\\ &=10 \int \left (\frac {5}{\left (-7+8 \pi ^2-16 i \pi (1-\log (4))+16 \log (4)-8 \log ^2(4)\right ) \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )^3}+\frac {1}{\left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right ) \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )^2}\right ) \, dx\\ &=\frac {25}{\left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )^2 \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )^2}-\frac {10}{\left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )^2 \left (5+x \left (7-8 \pi ^2+16 i \pi (1-\log (4))-16 \log (4)+8 \log ^2(4)\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 93, normalized size = 3.32 \begin {gather*} -\frac {5 \left (5-2 x \left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )\right )}{\left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )^2 \left (-5+x \left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x)/(125 - 75*x + 15*x^2 - x^3 + (600*x - 240*x^2 + 24*x^3)*(I*Pi + Log[E/4])^2 + (960*x^2 - 192*
x^3)*(I*Pi + Log[E/4])^4 + 512*x^3*(I*Pi + Log[E/4])^6),x]

[Out]

(-5*(5 - 2*x*(-7 + 8*Pi^2 + (16*I)*Pi*(-1 + Log[4]) + 16*Log[4] - 8*Log[4]^2)))/((-7 + 8*Pi^2 + (16*I)*Pi*(-1
+ Log[4]) + 16*Log[4] - 8*Log[4]^2)^2*(-5 + x*(-7 + 8*Pi^2 + (16*I)*Pi*(-1 + Log[4]) + 16*Log[4] - 8*Log[4]^2)
)^2)

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fricas [C]  time = 0.57, size = 199, normalized size = 7.11 \begin {gather*} -\frac {5 \, {\left (16 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{2} x - 32 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )} x + 14 \, x + 5\right )}}{4096 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{8} x^{2} - 32768 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{7} x^{2} + 5120 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{6} {\left (22 \, x^{2} + x\right )} - 2048 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{5} {\left (106 \, x^{2} + 15 \, x\right )} + 64 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{4} {\left (4006 \, x^{2} + 1170 \, x + 25\right )} - 256 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{3} {\left (742 \, x^{2} + 370 \, x + 25\right )} + 80 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )}^{2} {\left (1078 \, x^{2} + 819 \, x + 115\right )} - 224 \, {\left (i \, \pi + 2 \, \log \relax (2)\right )} {\left (98 \, x^{2} + 105 \, x + 25\right )} + 2401 \, x^{2} + 3430 \, x + 1225} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*x/(512*x^3*log(-1/4*exp(1))^6+(-192*x^3+960*x^2)*log(-1/4*exp(1))^4+(24*x^3-240*x^2+600*x)*log(-1
/4*exp(1))^2-x^3+15*x^2-75*x+125),x, algorithm="fricas")

[Out]

-5*(16*(I*pi + 2*log(2))^2*x - 32*(I*pi + 2*log(2))*x + 14*x + 5)/(4096*(I*pi + 2*log(2))^8*x^2 - 32768*(I*pi
+ 2*log(2))^7*x^2 + 5120*(I*pi + 2*log(2))^6*(22*x^2 + x) - 2048*(I*pi + 2*log(2))^5*(106*x^2 + 15*x) + 64*(I*
pi + 2*log(2))^4*(4006*x^2 + 1170*x + 25) - 256*(I*pi + 2*log(2))^3*(742*x^2 + 370*x + 25) + 80*(I*pi + 2*log(
2))^2*(1078*x^2 + 819*x + 115) - 224*(I*pi + 2*log(2))*(98*x^2 + 105*x + 25) + 2401*x^2 + 3430*x + 1225)

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giac [B]  time = 0.15, size = 56, normalized size = 2.00 \begin {gather*} -\frac {5 \, {\left (16 \, x \log \left (-\frac {1}{4} \, e\right )^{2} - 2 \, x + 5\right )}}{{\left (64 \, \log \left (-\frac {1}{4} \, e\right )^{4} - 16 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 1\right )} {\left (8 \, x \log \left (-\frac {1}{4} \, e\right )^{2} - x + 5\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*x/(512*x^3*log(-1/4*exp(1))^6+(-192*x^3+960*x^2)*log(-1/4*exp(1))^4+(24*x^3-240*x^2+600*x)*log(-1
/4*exp(1))^2-x^3+15*x^2-75*x+125),x, algorithm="giac")

[Out]

-5*(16*x*log(-1/4*e)^2 - 2*x + 5)/((64*log(-1/4*e)^4 - 16*log(-1/4*e)^2 + 1)*(8*x*log(-1/4*e)^2 - x + 5)^2)

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maple [B]  time = 1.34, size = 66, normalized size = 2.36

method	result	size
default	\(-\frac {10}{\left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2} \left (8 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+5-x \right )}+\frac {25}{\left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2} \left (8 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+5-x \right )^{2}}\)	\(66\)
gosper	\(-\frac {5 \left (16 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-2 x +5\right )}{\left (64 \ln \left (-\frac {{\mathrm e}}{4}\right )^{4} x^{2}-16 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2} x^{2}+80 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+x^{2}-10 x +25\right ) \left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2}}\)	\(75\)
risch	\(\frac {\frac {10 x}{-448+2048 \ln \relax (2)-1024 i \pi -2048 \ln \relax (2)^{2}+2048 i \ln \relax (2) \pi +512 \pi ^{2}}-\frac {25}{64 \left (-7+32 \ln \relax (2)-16 i \pi -32 \ln \relax (2)^{2}+32 i \ln \relax (2) \pi +8 \pi ^{2}\right )^{2}}}{-5 i \pi \ln \relax (2) x +\frac {5 i \pi x}{2}+\pi ^{4} x^{2}-23 i \pi \ln \relax (2) x^{2}-24 \pi ^{2} \ln \relax (2)^{2} x^{2}-32 i \pi \ln \relax (2)^{3} x^{2}+16 x^{2} \ln \relax (2)^{4}+24 \pi ^{2} \ln \relax (2) x^{2}+48 i \pi \ln \relax (2)^{2} x^{2}-32 x^{2} \ln \relax (2)^{3}-\frac {23 \pi ^{2} x^{2}}{4}+\frac {7 i \pi \,x^{2}}{2}+8 i \pi ^{3} \ln \relax (2) x^{2}+23 x^{2} \ln \relax (2)^{2}-\frac {5 x \,\pi ^{2}}{4}-4 i \pi ^{3} x^{2}+5 x \ln \relax (2)^{2}-7 x^{2} \ln \relax (2)-5 x \ln \relax (2)+\frac {49 x^{2}}{64}+\frac {35 x}{32}+\frac {25}{64}}\)	\(238\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*x/(512*x^3*ln(-1/4*exp(1))^6+(-192*x^3+960*x^2)*ln(-1/4*exp(1))^4+(24*x^3-240*x^2+600*x)*ln(-1/4*exp(1)
)^2-x^3+15*x^2-75*x+125),x,method=_RETURNVERBOSE)

[Out]

-10/(8*ln(-1/4*exp(1))^2-1)^2/(8*x*ln(-1/4*exp(1))^2+5-x)+25/(8*ln(-1/4*exp(1))^2-1)^2/(8*x*ln(-1/4*exp(1))^2+
5-x)^2

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maxima [B]  time = 0.35, size = 114, normalized size = 4.07 \begin {gather*} -\frac {5 \, {\left (2 \, {\left (8 \, \log \left (-\frac {1}{4} \, e\right )^{2} - 1\right )} x + 5\right )}}{1600 \, \log \left (-\frac {1}{4} \, e\right )^{4} + {\left (4096 \, \log \left (-\frac {1}{4} \, e\right )^{8} - 2048 \, \log \left (-\frac {1}{4} \, e\right )^{6} + 384 \, \log \left (-\frac {1}{4} \, e\right )^{4} - 32 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 1\right )} x^{2} + 10 \, {\left (512 \, \log \left (-\frac {1}{4} \, e\right )^{6} - 192 \, \log \left (-\frac {1}{4} \, e\right )^{4} + 24 \, \log \left (-\frac {1}{4} \, e\right )^{2} - 1\right )} x - 400 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*x/(512*x^3*log(-1/4*exp(1))^6+(-192*x^3+960*x^2)*log(-1/4*exp(1))^4+(24*x^3-240*x^2+600*x)*log(-1
/4*exp(1))^2-x^3+15*x^2-75*x+125),x, algorithm="maxima")

[Out]

-5*(2*(8*log(-1/4*e)^2 - 1)*x + 5)/(1600*log(-1/4*e)^4 + (4096*log(-1/4*e)^8 - 2048*log(-1/4*e)^6 + 384*log(-1
/4*e)^4 - 32*log(-1/4*e)^2 + 1)*x^2 + 10*(512*log(-1/4*e)^6 - 192*log(-1/4*e)^4 + 24*log(-1/4*e)^2 - 1)*x - 40
0*log(-1/4*e)^2 + 25)

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mupad [B]  time = 0.20, size = 47, normalized size = 1.68 \begin {gather*} -\frac {5\,\left (16\,x\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-2\,x+5\right )}{{\left (8\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-1\right )}^2\,{\left (8\,x\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-x+5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x)/(log(-exp(1)/4)^2*(600*x - 240*x^2 + 24*x^3) - 75*x + log(-exp(1)/4)^4*(960*x^2 - 192*x^3) + 512*x^
3*log(-exp(1)/4)^6 + 15*x^2 - x^3 + 125),x)

[Out]

-(5*(16*x*log(-exp(1)/4)^2 - 2*x + 5))/((8*log(-exp(1)/4)^2 - 1)^2*(8*x*log(-exp(1)/4)^2 - x + 5)^2)

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sympy [B]  time = 9.05, size = 765, normalized size = 27.32 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*x/(512*x**3*ln(-1/4*exp(1))**6+(-192*x**3+960*x**2)*ln(-1/4*exp(1))**4+(24*x**3-240*x**2+600*x)*l
n(-1/4*exp(1))**2-x**3+15*x**2-75*x+125),x)

[Out]

-10*(x*(-64*log(2)**2 - 14 + 64*log(2) + 16*pi**2 - 32*I*pi + 64*I*pi*log(2)) - 5)/(x**2*(-13516800*pi**4*log(
2)**2 - 917504*pi**6*log(2) - 9175040*pi**4*log(2)**4 - 34734080*pi**2*log(2)**3 - 8192*pi**8 - 44040192*pi**2
*log(2)**5 - 512768*pi**4 - 2279424*pi**2*log(2) - 8204288*log(2)**4 - 14417920*log(2)**6 - 689920*log(2)**2 -
2097152*log(2)**8 - 4802 + 87808*log(2) + 8388608*log(2)**7 + 3039232*log(2)**3 + 172480*pi**2 + 13893632*log
(2)**5 + 14680064*pi**2*log(2)**6 + 12306432*pi**2*log(2)**2 + 54067200*pi**2*log(2)**4 + 225280*pi**6 + 43417
60*pi**4*log(2) + 917504*pi**6*log(2)**2 + 18350080*pi**4*log(2)**3 - 5505024*I*pi**5*log(2)**2 - 36044800*I*p
i**3*log(2)**3 - 131072*I*pi**7*log(2) - 434176*I*pi**5 - 4102144*I*pi**3*log(2) - 14680064*I*pi**3*log(2)**5
- 34734080*I*pi*log(2)**4 - 29360128*I*pi*log(2)**6 - 4558848*I*pi*log(2)**2 - 43904*I*pi + 689920*I*pi*log(2)
+ 8388608*I*pi*log(2)**7 + 379904*I*pi**3 + 16408576*I*pi*log(2)**3 + 43253760*I*pi*log(2)**5 + 65536*I*pi**7
+ 17367040*I*pi**3*log(2)**2 + 36700160*I*pi**3*log(2)**4 + 3670016*I*pi**5*log(2)**3 + 2703360*I*pi**5*log(2
)) + x*(-614400*pi**4*log(2)**2 - 4915200*pi**2*log(2)**3 - 149760*pi**4 - 1136640*pi**2*log(2) - 2396160*log(
2)**4 - 524160*log(2)**2 - 655360*log(2)**6 - 6860 + 94080*log(2) + 1966080*log(2)**5 + 1515520*log(2)**3 + 13
1040*pi**2 + 2457600*pi**2*log(2)**4 + 10240*pi**6 + 3594240*pi**2*log(2)**2 + 614400*pi**4*log(2) - 1198080*I
*pi**3*log(2) - 61440*I*pi**5 - 1638400*I*pi**3*log(2)**3 - 4915200*I*pi*log(2)**4 - 2273280*I*pi*log(2)**2 -
47040*I*pi + 1966080*I*pi*log(2)**5 + 524160*I*pi*log(2) + 4792320*I*pi*log(2)**3 + 189440*I*pi**3 + 122880*I*
pi**5*log(2) + 2457600*I*pi**3*log(2)**2) - 76800*pi**2*log(2) - 3200*pi**4 - 73600*log(2)**2 - 51200*log(2)**
4 - 2450 + 22400*log(2) + 102400*log(2)**3 + 18400*pi**2 + 76800*pi**2*log(2)**2 - 25600*I*pi**3*log(2) - 1536
00*I*pi*log(2)**2 - 11200*I*pi + 102400*I*pi*log(2)**3 + 73600*I*pi*log(2) + 12800*I*pi**3)

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