3.20.34 \(\int \frac {2^{-2 x} (-800-400 x+(-800 x-200 x^2) \log (2)+(-200-200 x \log (2)) \log (5)+2^x (10240+6400 x+960 x^2+(5120 x+2560 x^2+320 x^3) \log (2)+(5120+1600 x+(2560 x+640 x^2) \log (2)) \log (5)+(640+320 x \log (2)) \log ^2(5))+2^{2 x} (-32768-24576 x-6144 x^2-512 x^3+(-24576-12288 x-1536 x^2) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)))}{64 x^3+48 x^4+12 x^5+x^6+(48 x^3+24 x^4+3 x^5) \log (5)+(12 x^3+3 x^4) \log ^2(5)+x^3 \log ^3(5)} \, dx\)
Optimal. Leaf size=24 \[ \frac {\left (16-\frac {5\ 2^{1-x}}{4+x+\log (5)}\right )^2}{x^2} \]
________________________________________________________________________________________
Rubi [C] time = 12.94, antiderivative size = 839, normalized size of antiderivative = 34.96,
number of steps used = 49, number of rules used = 5, integrand size = 210, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used
= {6, 6688, 6742, 2178, 2177} \begin {gather*} \frac {400 \log (2) (2+\log (2) (4+\log (5))) \text {Ei}(-2 x \log (2))}{(4+\log (5))^3}+\frac {600 (2+\log (2) (4+\log (5))) \text {Ei}(-2 x \log (2))}{(4+\log (5))^4}-\frac {400 \log ^2(2) \text {Ei}(-2 x \log (2))}{(4+\log (5))^2}-\frac {1400 \log (2) \text {Ei}(-2 x \log (2))}{(4+\log (5))^3}-\frac {1200 \text {Ei}(-2 x \log (2))}{(4+\log (5))^4}+\frac {25\ 2^{3-2 x} \log ^2(2)}{(4+\log (5)) (x+\log (5)+4)}-\frac {5\ 2^{6-x}}{(4+\log (5))^2 (x+\log (5)+4)}-\frac {75\ 2^{3-2 x}}{(4+\log (5))^3 (x+\log (5)+4)}+\frac {256}{x^2}-\frac {25\ 2^{2-2 x} \log (2)}{(4+\log (5)) (x+\log (5)+4)^2}-\frac {25\ 2^{2-2 x}}{(4+\log (5))^2 (x+\log (5)+4)^2}+\frac {5\ 2^{6-x} \left (4-\log (2) \log ^2(5)-\log (16) \log (25)-\log \left (\frac {65536}{5}\right )\right )}{x (4+\log (5))^3}+\frac {320 \text {Ei}(-x \log (2)) \log (2) \left (4-\log (2) \log ^2(5)-\log (16) \log (25)-\log \left (\frac {65536}{5}\right )\right )}{(4+\log (5))^3}-\frac {320 \text {Ei}(-x \log (2)) \log (2) \left (16+\log ^2(5)+4 \log (25)\right )}{(4+\log (5))^4}+\frac {5\ 2^{10+\log (5)} \text {Ei}(-\log (2) (x+\log (5)+4)) \log (2) \left (16+\log ^2(5)+4 \log (25)\right )}{(4+\log (5))^4}-\frac {5\ 2^{5-x} (8+\log (25))}{x^2 (4+\log (5))^2}+\frac {160 \text {Ei}(-x \log (2)) \log ^2(2) (8+\log (25))}{(4+\log (5))^2}+\frac {5\ 2^{5-x} \log (2) (8+\log (25))}{x (4+\log (5))^2}-\frac {25\ 2^{3-2 x} \log (2) (2+\log (2) (4+\log (5)))}{(4+\log (5))^2 (x+\log (5)+4)}+\frac {25\ 2^{4-2 x} (2+\log (2) (4+\log (5)))}{(4+\log (5))^3 (x+\log (5)+4)}+\frac {25\ 2^{2-2 x} (2+\log (2) (4+\log (5)))}{(4+\log (5))^2 (x+\log (5)+4)^2}-\frac {25\ 2^{12+\log (25)} \text {Ei}(-2 \log (2) (x+\log (5)+4)) \log ^2(2) (2+\log (2) (4+\log (5)))}{(4+\log (5))^2}+\frac {25\ 2^{3-2 x} (2+\log (2) (4+\log (5)))}{x (4+\log (5))^3}+\frac {25\ 2^{13+\log (25)} \text {Ei}(-2 \log (2) (x+\log (5)+4)) \log (2) (2+\log (2) (4+\log (5)))}{(4+\log (5))^3}-\frac {75\ 2^{11+\log (25)} \text {Ei}(-2 \log (2) (x+\log (5)+4)) (2+\log (2) (4+\log (5)))}{(4+\log (5))^4}+\frac {25\ 2^{12+\log (25)} \text {Ei}(-2 \log (2) (x+\log (5)+4)) \log ^3(2)}{4+\log (5)}+\frac {25\ 2^{2-2 x}}{x^2 (4+\log (5))^2}-\frac {5\ 2^{10+\log (5)} \text {Ei}(-\log (2) (x+\log (5)+4)) \log (2)}{(4+\log (5))^2}-\frac {25\ 2^{3-2 x} \log (2)}{x (4+\log (5))^2}-\frac {75\ 2^{3-2 x}}{x (4+\log (5))^3}-\frac {75\ 2^{12+\log (25)} \text {Ei}(-2 \log (2) (x+\log (5)+4)) \log (2)}{(4+\log (5))^3}+\frac {25\ 2^{11+\log (25)} \text {Ei}(-2 \log (2) (x+\log (5)+4)) \log (2)}{(4+\log (5))^3}+\frac {75\ 2^{12+\log (25)} \text {Ei}(-2 \log (2) (x+\log (5)+4))}{(4+\log (5))^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-800 - 400*x + (-800*x - 200*x^2)*Log[2] + (-200 - 200*x*Log[2])*Log[5] + 2^x*(10240 + 6400*x + 960*x^2 +
(5120*x + 2560*x^2 + 320*x^3)*Log[2] + (5120 + 1600*x + (2560*x + 640*x^2)*Log[2])*Log[5] + (640 + 320*x*Log[
2])*Log[5]^2) + 2^(2*x)*(-32768 - 24576*x - 6144*x^2 - 512*x^3 + (-24576 - 12288*x - 1536*x^2)*Log[5] + (-6144
- 1536*x)*Log[5]^2 - 512*Log[5]^3))/(2^(2*x)*(64*x^3 + 48*x^4 + 12*x^5 + x^6 + (48*x^3 + 24*x^4 + 3*x^5)*Log[
5] + (12*x^3 + 3*x^4)*Log[5]^2 + x^3*Log[5]^3)),x]
[Out]
256/x^2 - (1200*ExpIntegralEi[-2*x*Log[2]])/(4 + Log[5])^4 + (75*2^(12 + Log[25])*ExpIntegralEi[-2*Log[2]*(4 +
x + Log[5])])/(4 + Log[5])^4 - (75*2^(3 - 2*x))/(x*(4 + Log[5])^3) - (1400*ExpIntegralEi[-2*x*Log[2]]*Log[2])
/(4 + Log[5])^3 + (25*2^(11 + Log[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*Log[2])/(4 + Log[5])^3 - (75*
2^(12 + Log[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*Log[2])/(4 + Log[5])^3 + (25*2^(2 - 2*x))/(x^2*(4 +
Log[5])^2) - (25*2^(3 - 2*x)*Log[2])/(x*(4 + Log[5])^2) - (5*2^(10 + Log[5])*ExpIntegralEi[-(Log[2]*(4 + x +
Log[5]))]*Log[2])/(4 + Log[5])^2 - (400*ExpIntegralEi[-2*x*Log[2]]*Log[2]^2)/(4 + Log[5])^2 + (25*2^(12 + Log[
25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*Log[2]^3)/(4 + Log[5]) - (25*2^(2 - 2*x))/((4 + Log[5])^2*(4 +
x + Log[5])^2) - (25*2^(2 - 2*x)*Log[2])/((4 + Log[5])*(4 + x + Log[5])^2) - (75*2^(3 - 2*x))/((4 + Log[5])^3*
(4 + x + Log[5])) - (5*2^(6 - x))/((4 + Log[5])^2*(4 + x + Log[5])) + (25*2^(3 - 2*x)*Log[2]^2)/((4 + Log[5])*
(4 + x + Log[5])) + (600*ExpIntegralEi[-2*x*Log[2]]*(2 + Log[2]*(4 + Log[5])))/(4 + Log[5])^4 - (75*2^(11 + Lo
g[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*(2 + Log[2]*(4 + Log[5])))/(4 + Log[5])^4 + (25*2^(3 - 2*x)*(
2 + Log[2]*(4 + Log[5])))/(x*(4 + Log[5])^3) + (400*ExpIntegralEi[-2*x*Log[2]]*Log[2]*(2 + Log[2]*(4 + Log[5])
))/(4 + Log[5])^3 + (25*2^(13 + Log[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*Log[2]*(2 + Log[2]*(4 + Log
[5])))/(4 + Log[5])^3 - (25*2^(12 + Log[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*Log[2]^2*(2 + Log[2]*(4
+ Log[5])))/(4 + Log[5])^2 + (25*2^(2 - 2*x)*(2 + Log[2]*(4 + Log[5])))/((4 + Log[5])^2*(4 + x + Log[5])^2) +
(25*2^(4 - 2*x)*(2 + Log[2]*(4 + Log[5])))/((4 + Log[5])^3*(4 + x + Log[5])) - (25*2^(3 - 2*x)*Log[2]*(2 + Lo
g[2]*(4 + Log[5])))/((4 + Log[5])^2*(4 + x + Log[5])) - (5*2^(5 - x)*(8 + Log[25]))/(x^2*(4 + Log[5])^2) + (5*
2^(5 - x)*Log[2]*(8 + Log[25]))/(x*(4 + Log[5])^2) + (160*ExpIntegralEi[-(x*Log[2])]*Log[2]^2*(8 + Log[25]))/(
4 + Log[5])^2 - (320*ExpIntegralEi[-(x*Log[2])]*Log[2]*(16 + Log[5]^2 + 4*Log[25]))/(4 + Log[5])^4 + (5*2^(10
+ Log[5])*ExpIntegralEi[-(Log[2]*(4 + x + Log[5]))]*Log[2]*(16 + Log[5]^2 + 4*Log[25]))/(4 + Log[5])^4 + (5*2^
(6 - x)*(4 - Log[2]*Log[5]^2 - Log[16]*Log[25] - Log[65536/5]))/(x*(4 + Log[5])^3) + (320*ExpIntegralEi[-(x*Lo
g[2])]*Log[2]*(4 - Log[2]*Log[5]^2 - Log[16]*Log[25] - Log[65536/5]))/(4 + Log[5])^3
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 2177
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] && !$UseGamma ===
True
Rule 2178
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True
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 \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \left (64+\log ^3(5)\right )} \, dx\\ &=\int \frac {2^{3-2 x} \left (5-2^{3+x} x-2^{5+x} \left (1+\frac {\log (5)}{4}\right )\right ) \left (x^2 \left (2^{3+x}-5 \log (2)\right )+(4+\log (5)) \left (-5+2^{5+x}+2^{3+x} \log (5)\right )+x \left (-10+2^{6+x}+2^{4+x} \log (5)-5 \log (2) (4+\log (5))\right )\right )}{x^3 (4+x+\log (5))^3} \, dx\\ &=\int \left (-\frac {512}{x^3}-\frac {25\ 2^{3-2 x} \log (2)}{x (4+x+\log (5))^3}-\frac {25\ 2^{3-2 x} (4+\log (5))}{x^3 (4+x+\log (5))^3}-\frac {25\ 2^{4-2 x} \left (1+\frac {1}{2} \log (2) (4+\log (5))\right )}{x^2 (4+x+\log (5))^3}+\frac {5\ 2^{6-x} \left (8+x^2 \log (2)+x (3+\log (2) \log (5)+\log (16))+\log (25)\right )}{x^3 (4+x+\log (5))^2}\right ) \, dx\\ &=\frac {256}{x^2}+5 \int \frac {2^{6-x} \left (8+x^2 \log (2)+x (3+\log (2) \log (5)+\log (16))+\log (25)\right )}{x^3 (4+x+\log (5))^2} \, dx-(25 \log (2)) \int \frac {2^{3-2 x}}{x (4+x+\log (5))^3} \, dx-(25 (4+\log (5))) \int \frac {2^{3-2 x}}{x^3 (4+x+\log (5))^3} \, dx-\frac {1}{2} (25 (2+\log (2) (4+\log (5)))) \int \frac {2^{4-2 x}}{x^2 (4+x+\log (5))^3} \, dx\\ &=\frac {256}{x^2}+5 \int \left (\frac {2^{6-x}}{(4+\log (5))^2 (4+x+\log (5))^2}+\frac {2^{6-x} (8+\log (25))}{x^3 (4+\log (5))^2}+\frac {2^{6-x} \left (-4+\log (2) \log ^2(5)+\log (16) \log (25)+\log \left (\frac {65536}{5}\right )\right )}{x^2 (4+\log (5))^3}+\frac {2^{6-x} \left (-\log (2) \log ^2(5)-\log (16) \log (25)-\log (65536)\right )}{x (4+\log (5))^4}+\frac {2^{6-x} \left (\log (2) \log ^2(5)+\log (16) \log (25)+\log (65536)\right )}{(4+\log (5))^4 (4+x+\log (5))}\right ) \, dx-(25 \log (2)) \int \left (\frac {2^{3-2 x}}{x (4+\log (5))^3}-\frac {2^{3-2 x}}{(4+\log (5)) (4+x+\log (5))^3}-\frac {2^{3-2 x}}{(4+\log (5))^2 (4+x+\log (5))^2}-\frac {2^{3-2 x}}{(4+\log (5))^3 (4+x+\log (5))}\right ) \, dx-(25 (4+\log (5))) \int \left (\frac {3\ 2^{4-2 x}}{x (4+\log (5))^5}-\frac {3\ 2^{3-2 x}}{x^2 (4+\log (5))^4}+\frac {2^{3-2 x}}{x^3 (4+\log (5))^3}-\frac {2^{3-2 x}}{(4+\log (5))^3 (4+x+\log (5))^3}-\frac {3\ 2^{3-2 x}}{(4+\log (5))^4 (4+x+\log (5))^2}-\frac {3\ 2^{4-2 x}}{(4+\log (5))^5 (4+x+\log (5))}\right ) \, dx-\frac {1}{2} (25 (2+\log (2) (4+\log (5)))) \int \left (-\frac {3\ 2^{4-2 x}}{x (4+\log (5))^4}+\frac {2^{4-2 x}}{x^2 (4+\log (5))^3}+\frac {2^{4-2 x}}{(4+\log (5))^2 (4+x+\log (5))^3}+\frac {2^{5-2 x}}{(4+\log (5))^3 (4+x+\log (5))^2}+\frac {3\ 2^{4-2 x}}{(4+\log (5))^4 (4+x+\log (5))}\right ) \, dx\\ &=\frac {256}{x^2}-\frac {75 \int \frac {2^{4-2 x}}{x} \, dx}{(4+\log (5))^4}+\frac {75 \int \frac {2^{4-2 x}}{4+x+\log (5)} \, dx}{(4+\log (5))^4}+\frac {75 \int \frac {2^{3-2 x}}{x^2} \, dx}{(4+\log (5))^3}+\frac {75 \int \frac {2^{3-2 x}}{(4+x+\log (5))^2} \, dx}{(4+\log (5))^3}-\frac {(25 \log (2)) \int \frac {2^{3-2 x}}{x} \, dx}{(4+\log (5))^3}+\frac {(25 \log (2)) \int \frac {2^{3-2 x}}{4+x+\log (5)} \, dx}{(4+\log (5))^3}+\frac {5 \int \frac {2^{6-x}}{(4+x+\log (5))^2} \, dx}{(4+\log (5))^2}-\frac {25 \int \frac {2^{3-2 x}}{x^3} \, dx}{(4+\log (5))^2}+\frac {25 \int \frac {2^{3-2 x}}{(4+x+\log (5))^3} \, dx}{(4+\log (5))^2}+\frac {(25 \log (2)) \int \frac {2^{3-2 x}}{(4+x+\log (5))^2} \, dx}{(4+\log (5))^2}+\frac {(25 \log (2)) \int \frac {2^{3-2 x}}{(4+x+\log (5))^3} \, dx}{4+\log (5)}+\frac {(75 (2+\log (2) (4+\log (5)))) \int \frac {2^{4-2 x}}{x} \, dx}{2 (4+\log (5))^4}-\frac {(75 (2+\log (2) (4+\log (5)))) \int \frac {2^{4-2 x}}{4+x+\log (5)} \, dx}{2 (4+\log (5))^4}-\frac {(25 (2+\log (2) (4+\log (5)))) \int \frac {2^{4-2 x}}{x^2} \, dx}{2 (4+\log (5))^3}-\frac {(25 (2+\log (2) (4+\log (5)))) \int \frac {2^{5-2 x}}{(4+x+\log (5))^2} \, dx}{2 (4+\log (5))^3}-\frac {(25 (2+\log (2) (4+\log (5)))) \int \frac {2^{4-2 x}}{(4+x+\log (5))^3} \, dx}{2 (4+\log (5))^2}+\frac {(5 (8+\log (25))) \int \frac {2^{6-x}}{x^3} \, dx}{(4+\log (5))^2}-\frac {\left (5 \log (2) \left (16+\log ^2(5)+4 \log (25)\right )\right ) \int \frac {2^{6-x}}{x} \, dx}{(4+\log (5))^4}+\frac {\left (5 \log (2) \left (16+\log ^2(5)+4 \log (25)\right )\right ) \int \frac {2^{6-x}}{4+x+\log (5)} \, dx}{(4+\log (5))^4}-\frac {\left (5 \left (4-\log (2) \log ^2(5)-\log (16) \log (25)-\log \left (\frac {65536}{5}\right )\right )\right ) \int \frac {2^{6-x}}{x^2} \, dx}{(4+\log (5))^3}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 42, normalized size = 1.75 \begin {gather*} \frac {4^{1-x} \left (-5+2^{5+x}+2^{3+x} x+2^{3+x} \log (5)\right )^2}{x^2 (4+x+\log (5))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-800 - 400*x + (-800*x - 200*x^2)*Log[2] + (-200 - 200*x*Log[2])*Log[5] + 2^x*(10240 + 6400*x + 960
*x^2 + (5120*x + 2560*x^2 + 320*x^3)*Log[2] + (5120 + 1600*x + (2560*x + 640*x^2)*Log[2])*Log[5] + (640 + 320*
x*Log[2])*Log[5]^2) + 2^(2*x)*(-32768 - 24576*x - 6144*x^2 - 512*x^3 + (-24576 - 12288*x - 1536*x^2)*Log[5] +
(-6144 - 1536*x)*Log[5]^2 - 512*Log[5]^3))/(2^(2*x)*(64*x^3 + 48*x^4 + 12*x^5 + x^6 + (48*x^3 + 24*x^4 + 3*x^5
)*Log[5] + (12*x^3 + 3*x^4)*Log[5]^2 + x^3*Log[5]^3)),x]
[Out]
(4^(1 - x)*(-5 + 2^(5 + x) + 2^(3 + x)*x + 2^(3 + x)*Log[5])^2)/(x^2*(4 + x + Log[5])^2)
________________________________________________________________________________________
fricas [B] time = 0.81, size = 84, normalized size = 3.50 \begin {gather*} \frac {4 \, {\left (64 \, {\left (x^{2} + 2 \, {\left (x + 4\right )} \log \relax (5) + \log \relax (5)^{2} + 8 \, x + 16\right )} 2^{2 \, x} - 80 \cdot 2^{x} {\left (x + \log \relax (5) + 4\right )} + 25\right )}}{{\left (x^{4} + x^{2} \log \relax (5)^{2} + 8 \, x^{3} + 16 \, x^{2} + 2 \, {\left (x^{3} + 4 \, x^{2}\right )} \log \relax (5)\right )} 2^{2 \, x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-512*log(5)^3+(-1536*x-6144)*log(5)^2+(-1536*x^2-12288*x-24576)*log(5)-512*x^3-6144*x^2-24576*x-32
768)*exp(x*log(2))^2+((320*x*log(2)+640)*log(5)^2+((640*x^2+2560*x)*log(2)+1600*x+5120)*log(5)+(320*x^3+2560*x
^2+5120*x)*log(2)+960*x^2+6400*x+10240)*exp(x*log(2))+(-200*x*log(2)-200)*log(5)+(-200*x^2-800*x)*log(2)-400*x
-800)/(x^3*log(5)^3+(3*x^4+12*x^3)*log(5)^2+(3*x^5+24*x^4+48*x^3)*log(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*log(2
))^2,x, algorithm="fricas")
[Out]
4*(64*(x^2 + 2*(x + 4)*log(5) + log(5)^2 + 8*x + 16)*2^(2*x) - 80*2^x*(x + log(5) + 4) + 25)/((x^4 + x^2*log(5
)^2 + 8*x^3 + 16*x^2 + 2*(x^3 + 4*x^2)*log(5))*2^(2*x))
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {8 \, {\left (64 \, {\left (x^{3} + 3 \, {\left (x + 4\right )} \log \relax (5)^{2} + \log \relax (5)^{3} + 12 \, x^{2} + 3 \, {\left (x^{2} + 8 \, x + 16\right )} \log \relax (5) + 48 \, x + 64\right )} 2^{2 \, x} - 40 \, {\left ({\left (x \log \relax (2) + 2\right )} \log \relax (5)^{2} + 3 \, x^{2} + {\left (2 \, {\left (x^{2} + 4 \, x\right )} \log \relax (2) + 5 \, x + 16\right )} \log \relax (5) + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \relax (2) + 20 \, x + 32\right )} 2^{x} + 25 \, {\left (x \log \relax (2) + 1\right )} \log \relax (5) + 25 \, {\left (x^{2} + 4 \, x\right )} \log \relax (2) + 50 \, x + 100\right )}}{{\left (x^{6} + x^{3} \log \relax (5)^{3} + 12 \, x^{5} + 48 \, x^{4} + 64 \, x^{3} + 3 \, {\left (x^{4} + 4 \, x^{3}\right )} \log \relax (5)^{2} + 3 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} \log \relax (5)\right )} 2^{2 \, x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-512*log(5)^3+(-1536*x-6144)*log(5)^2+(-1536*x^2-12288*x-24576)*log(5)-512*x^3-6144*x^2-24576*x-32
768)*exp(x*log(2))^2+((320*x*log(2)+640)*log(5)^2+((640*x^2+2560*x)*log(2)+1600*x+5120)*log(5)+(320*x^3+2560*x
^2+5120*x)*log(2)+960*x^2+6400*x+10240)*exp(x*log(2))+(-200*x*log(2)-200)*log(5)+(-200*x^2-800*x)*log(2)-400*x
-800)/(x^3*log(5)^3+(3*x^4+12*x^3)*log(5)^2+(3*x^5+24*x^4+48*x^3)*log(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*log(2
))^2,x, algorithm="giac")
[Out]
integrate(-8*(64*(x^3 + 3*(x + 4)*log(5)^2 + log(5)^3 + 12*x^2 + 3*(x^2 + 8*x + 16)*log(5) + 48*x + 64)*2^(2*x
) - 40*((x*log(2) + 2)*log(5)^2 + 3*x^2 + (2*(x^2 + 4*x)*log(2) + 5*x + 16)*log(5) + (x^3 + 8*x^2 + 16*x)*log(
2) + 20*x + 32)*2^x + 25*(x*log(2) + 1)*log(5) + 25*(x^2 + 4*x)*log(2) + 50*x + 100)/((x^6 + x^3*log(5)^3 + 12
*x^5 + 48*x^4 + 64*x^3 + 3*(x^4 + 4*x^3)*log(5)^2 + 3*(x^5 + 8*x^4 + 16*x^3)*log(5))*2^(2*x)), x)
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maple [A] time = 0.58, size = 41, normalized size = 1.71
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| method |
result |
size |
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| risch |
\(\frac {256}{x^{2}}-\frac {320 \,2^{-x}}{\left (x +\ln \relax (5)+4\right ) x^{2}}+\frac {100 \,2^{-2 x}}{x^{2} \left (x +\ln \relax (5)+4\right )^{2}}\) |
\(41\) |
| norman |
\(\frac {\left (100+\left (-320 \ln \relax (5)-1280\right ) {\mathrm e}^{x \ln \relax (2)}+\left (256 \ln \relax (5)^{2}+2048 \ln \relax (5)+4096\right ) {\mathrm e}^{2 x \ln \relax (2)}+\left (512 \ln \relax (5)+2048\right ) x \,{\mathrm e}^{2 x \ln \relax (2)}+256 \,{\mathrm e}^{2 x \ln \relax (2)} x^{2}-320 \,{\mathrm e}^{x \ln \relax (2)} x \right ) {\mathrm e}^{-2 x \ln \relax (2)}}{x^{2} \left (x +\ln \relax (5)+4\right )^{2}}\) |
\(88\) |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(13774\) |
| default |
\(\text {Expression too large to display}\) |
\(13774\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((-512*ln(5)^3+(-1536*x-6144)*ln(5)^2+(-1536*x^2-12288*x-24576)*ln(5)-512*x^3-6144*x^2-24576*x-32768)*exp(
x*ln(2))^2+((320*x*ln(2)+640)*ln(5)^2+((640*x^2+2560*x)*ln(2)+1600*x+5120)*ln(5)+(320*x^3+2560*x^2+5120*x)*ln(
2)+960*x^2+6400*x+10240)*exp(x*ln(2))+(-200*x*ln(2)-200)*ln(5)+(-200*x^2-800*x)*ln(2)-400*x-800)/(x^3*ln(5)^3+
(3*x^4+12*x^3)*ln(5)^2+(3*x^5+24*x^4+48*x^3)*ln(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*ln(2))^2,x,method=_RETURNVE
RBOSE)
[Out]
256/x^2-320/(x+ln(5)+4)/x^2/(2^x)+100/x^2/(x+ln(5)+4)^2/(2^x)^2
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maxima [B] time = 0.63, size = 1714, normalized size = 71.42 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-512*log(5)^3+(-1536*x-6144)*log(5)^2+(-1536*x^2-12288*x-24576)*log(5)-512*x^3-6144*x^2-24576*x-32
768)*exp(x*log(2))^2+((320*x*log(2)+640)*log(5)^2+((640*x^2+2560*x)*log(2)+1600*x+5120)*log(5)+(320*x^3+2560*x
^2+5120*x)*log(2)+960*x^2+6400*x+10240)*exp(x*log(2))+(-200*x*log(2)-200)*log(5)+(-200*x^2-800*x)*log(2)-400*x
-800)/(x^3*log(5)^3+(3*x^4+12*x^3)*log(5)^2+(3*x^5+24*x^4+48*x^3)*log(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*log(2
))^2,x, algorithm="maxima")
[Out]
-256*((12*x^3 + 18*x^2*(log(5) + 4) - log(5)^3 + 4*(log(5)^2 + 8*log(5) + 16)*x - 12*log(5)^2 - 48*log(5) - 64
)/((log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^4 + 2*(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 +
640*log(5)^2 + 1280*log(5) + 1024)*x^3 + (log(5)^6 + 24*log(5)^5 + 240*log(5)^4 + 1280*log(5)^3 + 3840*log(5)^
2 + 6144*log(5) + 4096)*x^2) - 12*log(x + log(5) + 4)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 +
1280*log(5) + 1024) + 12*log(x)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024))*l
og(5)^3 - 3072*((12*x^3 + 18*x^2*(log(5) + 4) - log(5)^3 + 4*(log(5)^2 + 8*log(5) + 16)*x - 12*log(5)^2 - 48*l
og(5) - 64)/((log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^4 + 2*(log(5)^5 + 20*log(5)^4 + 160*l
og(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024)*x^3 + (log(5)^6 + 24*log(5)^5 + 240*log(5)^4 + 1280*log(5)^3 + 38
40*log(5)^2 + 6144*log(5) + 4096)*x^2) - 12*log(x + log(5) + 4)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*l
og(5)^2 + 1280*log(5) + 1024) + 12*log(x)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5)
+ 1024))*log(5)^2 + 768*((6*x^2 + 9*x*(log(5) + 4) + 2*log(5)^2 + 16*log(5) + 32)/((log(5)^3 + 12*log(5)^2 + 4
8*log(5) + 64)*x^3 + 2*(log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^2 + (log(5)^5 + 20*log(5)^4
+ 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024)*x) - 6*log(x + log(5) + 4)/(log(5)^4 + 16*log(5)^3 + 96*l
og(5)^2 + 256*log(5) + 256) + 6*log(x)/(log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256))*log(5)^2 - 1
2288*((12*x^3 + 18*x^2*(log(5) + 4) - log(5)^3 + 4*(log(5)^2 + 8*log(5) + 16)*x - 12*log(5)^2 - 48*log(5) - 64
)/((log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^4 + 2*(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 +
640*log(5)^2 + 1280*log(5) + 1024)*x^3 + (log(5)^6 + 24*log(5)^5 + 240*log(5)^4 + 1280*log(5)^3 + 3840*log(5)^
2 + 6144*log(5) + 4096)*x^2) - 12*log(x + log(5) + 4)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 +
1280*log(5) + 1024) + 12*log(x)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024))*l
og(5) + 6144*((6*x^2 + 9*x*(log(5) + 4) + 2*log(5)^2 + 16*log(5) + 32)/((log(5)^3 + 12*log(5)^2 + 48*log(5) +
64)*x^3 + 2*(log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^2 + (log(5)^5 + 20*log(5)^4 + 160*log(
5)^3 + 640*log(5)^2 + 1280*log(5) + 1024)*x) - 6*log(x + log(5) + 4)/(log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 2
56*log(5) + 256) + 6*log(x)/(log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256))*log(5) - 768*((2*x + 3*
log(5) + 12)/(log(5)^4 + (log(5)^2 + 8*log(5) + 16)*x^2 + 16*log(5)^3 + 2*(log(5)^3 + 12*log(5)^2 + 48*log(5)
+ 64)*x + 96*log(5)^2 + 256*log(5) + 256) - 2*log(x + log(5) + 4)/(log(5)^3 + 12*log(5)^2 + 48*log(5) + 64) +
2*log(x)/(log(5)^3 + 12*log(5)^2 + 48*log(5) + 64))*log(5) - 16384*(12*x^3 + 18*x^2*(log(5) + 4) - log(5)^3 +
4*(log(5)^2 + 8*log(5) + 16)*x - 12*log(5)^2 - 48*log(5) - 64)/((log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*lo
g(5) + 256)*x^4 + 2*(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024)*x^3 + (log(5)^
6 + 24*log(5)^5 + 240*log(5)^4 + 1280*log(5)^3 + 3840*log(5)^2 + 6144*log(5) + 4096)*x^2) + 12288*(6*x^2 + 9*x
*(log(5) + 4) + 2*log(5)^2 + 16*log(5) + 32)/((log(5)^3 + 12*log(5)^2 + 48*log(5) + 64)*x^3 + 2*(log(5)^4 + 16
*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^2 + (log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280
*log(5) + 1024)*x) - 3072*(2*x + 3*log(5) + 12)/(log(5)^4 + (log(5)^2 + 8*log(5) + 16)*x^2 + 16*log(5)^3 + 2*(
log(5)^3 + 12*log(5)^2 + 48*log(5) + 64)*x + 96*log(5)^2 + 256*log(5) + 256) - 20*(16*(x + log(5) + 4)/2^x - 5
/2^(2*x))/(x^4 + 2*x^3*(log(5) + 4) + (log(5)^2 + 8*log(5) + 16)*x^2) + 196608*log(x + log(5) + 4)/(log(5)^5 +
20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024) - 73728*log(x + log(5) + 4)/(log(5)^4 + 16*lo
g(5)^3 + 96*log(5)^2 + 256*log(5) + 256) + 6144*log(x + log(5) + 4)/(log(5)^3 + 12*log(5)^2 + 48*log(5) + 64)
- 196608*log(x)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024) + 73728*log(x)/(lo
g(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256) - 6144*log(x)/(log(5)^3 + 12*log(5)^2 + 48*log(5) + 64)
+ 256/(x^2 + 2*x*(log(5) + 4) + log(5)^2 + 8*log(5) + 16)
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(exp(-2*x*log(2))*(400*x + log(2)*(800*x + 200*x^2) + log(5)*(200*x*log(2) + 200) + exp(2*x*log(2))*(2457
6*x + log(5)*(12288*x + 1536*x^2 + 24576) + log(5)^2*(1536*x + 6144) + 512*log(5)^3 + 6144*x^2 + 512*x^3 + 327
68) - exp(x*log(2))*(6400*x + log(5)*(1600*x + log(2)*(2560*x + 640*x^2) + 5120) + log(2)*(5120*x + 2560*x^2 +
320*x^3) + log(5)^2*(320*x*log(2) + 640) + 960*x^2 + 10240) + 800))/(x^3*log(5)^3 + log(5)*(48*x^3 + 24*x^4 +
3*x^5) + 64*x^3 + 48*x^4 + 12*x^5 + x^6 + log(5)^2*(12*x^3 + 3*x^4)),x)
[Out]
\text{Hanged}
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sympy [B] time = 0.50, size = 155, normalized size = 6.46 \begin {gather*} \frac {\left (100 x^{3} + 100 x^{2} \log {\relax (5 )} + 400 x^{2}\right ) e^{- 2 x \log {\relax (2 )}} + \left (- 320 x^{4} - 2560 x^{3} - 640 x^{3} \log {\relax (5 )} - 5120 x^{2} - 2560 x^{2} \log {\relax (5 )} - 320 x^{2} \log {\relax (5 )}^{2}\right ) e^{- x \log {\relax (2 )}}}{x^{7} + 3 x^{6} \log {\relax (5 )} + 12 x^{6} + 3 x^{5} \log {\relax (5 )}^{2} + 24 x^{5} \log {\relax (5 )} + 48 x^{5} + x^{4} \log {\relax (5 )}^{3} + 12 x^{4} \log {\relax (5 )}^{2} + 64 x^{4} + 48 x^{4} \log {\relax (5 )}} + \frac {256}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-512*ln(5)**3+(-1536*x-6144)*ln(5)**2+(-1536*x**2-12288*x-24576)*ln(5)-512*x**3-6144*x**2-24576*x-
32768)*exp(x*ln(2))**2+((320*x*ln(2)+640)*ln(5)**2+((640*x**2+2560*x)*ln(2)+1600*x+5120)*ln(5)+(320*x**3+2560*
x**2+5120*x)*ln(2)+960*x**2+6400*x+10240)*exp(x*ln(2))+(-200*x*ln(2)-200)*ln(5)+(-200*x**2-800*x)*ln(2)-400*x-
800)/(x**3*ln(5)**3+(3*x**4+12*x**3)*ln(5)**2+(3*x**5+24*x**4+48*x**3)*ln(5)+x**6+12*x**5+48*x**4+64*x**3)/exp
(x*ln(2))**2,x)
[Out]
((100*x**3 + 100*x**2*log(5) + 400*x**2)*exp(-2*x*log(2)) + (-320*x**4 - 2560*x**3 - 640*x**3*log(5) - 5120*x*
*2 - 2560*x**2*log(5) - 320*x**2*log(5)**2)*exp(-x*log(2)))/(x**7 + 3*x**6*log(5) + 12*x**6 + 3*x**5*log(5)**2
+ 24*x**5*log(5) + 48*x**5 + x**4*log(5)**3 + 12*x**4*log(5)**2 + 64*x**4 + 48*x**4*log(5)) + 256/x**2
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