3.27.30 \(\int \frac {27 x^3+27 x^4+27 x^5+9 x^6+(90 x+270 x^2+135 x^3+270 x^4+135 x^5) (i \pi +\log (4))^2+(450+1350 x+675 x^3+675 x^4) (i \pi +\log (4))^4+(2250+1125 x^3) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+(15 x^3+30 x^4+15 x^5) (i \pi +\log (4))^2+(75 x^3+75 x^4) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx\)

Optimal. Leaf size=29 \[ 9 \left (3+x-\frac {1}{\left (x+\frac {x}{x+5 (i \pi +\log (4))^2}\right )^2}\right ) \]

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Rubi [B]  time = 0.49, antiderivative size = 182, normalized size of antiderivative = 6.28, number of steps used = 3, number of rules used = 2, integrand size = 184, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6, 2074} \begin {gather*} -\frac {225 (\pi -i \log (4))^4}{x^2 \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^2}+9 x-\frac {90 (\pi -i \log (4))^2}{\left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^3 \left (x-5 \pi ^2+1+5 \log ^2(4)+10 i \pi \log (4)\right )}-\frac {9}{\left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^2 \left (x-5 \pi ^2+1+5 \log ^2(4)+10 i \pi \log (4)\right )^2}+\frac {90 (\pi -i \log (4))^2}{x \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(27*x^3 + 27*x^4 + 27*x^5 + 9*x^6 + (90*x + 270*x^2 + 135*x^3 + 270*x^4 + 135*x^5)*(I*Pi + Log[4])^2 + (45
0 + 1350*x + 675*x^3 + 675*x^4)*(I*Pi + Log[4])^4 + (2250 + 1125*x^3)*(I*Pi + Log[4])^6)/(x^3 + 3*x^4 + 3*x^5
+ x^6 + (15*x^3 + 30*x^4 + 15*x^5)*(I*Pi + Log[4])^2 + (75*x^3 + 75*x^4)*(I*Pi + Log[4])^4 + 125*x^3*(I*Pi + L
og[4])^6),x]

[Out]

9*x + (90*(Pi - I*Log[4])^2)/(x*(1 - 5*Pi^2 + (10*I)*Pi*Log[4] + 5*Log[4]^2)^3) - (225*(Pi - I*Log[4])^4)/(x^2
*(1 - 5*Pi^2 + (10*I)*Pi*Log[4] + 5*Log[4]^2)^2) - 9/((1 - 5*Pi^2 + (10*I)*Pi*Log[4] + 5*Log[4]^2)^2*(1 - 5*Pi
^2 + x + (10*I)*Pi*Log[4] + 5*Log[4]^2)^2) - (90*(Pi - I*Log[4])^2)/((1 - 5*Pi^2 + (10*I)*Pi*Log[4] + 5*Log[4]
^2)^3*(1 - 5*Pi^2 + x + (10*I)*Pi*Log[4] + 5*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 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 {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+x^3 \left (1+125 (i \pi +\log (4))^6\right )} \, dx\\ &=\int \left (9+\frac {90 (\pi -i \log (4))^2}{x^2 \left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^3}+\frac {450 (\pi -i \log (4))^4}{x^3 \left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^2}-\frac {18}{\left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^2 \left (-1+5 \pi ^2-x-10 i \pi \log (4)-5 \log ^2(4)\right )^3}-\frac {90 (\pi -i \log (4))^2}{\left (-1+5 \pi ^2-10 i \pi \log (4)-5 \log ^2(4)\right )^3 \left (-1+5 \pi ^2-x-10 i \pi \log (4)-5 \log ^2(4)\right )^2}\right ) \, dx\\ &=9 x+\frac {90 (\pi -i \log (4))^2}{x \left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^3}-\frac {225 (\pi -i \log (4))^4}{x^2 \left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^2}-\frac {9}{\left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^2 \left (1-5 \pi ^2+x+10 i \pi \log (4)+5 \log ^2(4)\right )^2}-\frac {90 (\pi -i \log (4))^2}{\left (1-5 \pi ^2+10 i \pi \log (4)+5 \log ^2(4)\right )^3 \left (1-5 \pi ^2+x+10 i \pi \log (4)+5 \log ^2(4)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.20, size = 163, normalized size = 5.62 \begin {gather*} \frac {9 \left (-x^2+x^5+25 \pi ^4 \left (-1+x^3\right )-100 i \pi ^3 \left (-1+x^3\right ) \log (4)-10 x \log ^2(4)-25 \log ^4(4)+2 x^4 \left (1+5 \log ^2(4)\right )+x^3 \left (1+5 \log ^2(4)\right )^2+20 i \pi \log (4) \left (-x+x^4-5 \log ^2(4)+x^3 \left (1+5 \log ^2(4)\right )\right )-10 \pi ^2 \left (-x+x^4-15 \log ^2(4)+x^3 \left (1+15 \log ^2(4)\right )\right )\right )}{x^2 \left (1-5 \pi ^2+x+10 i \pi \log (4)+5 \log ^2(4)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(27*x^3 + 27*x^4 + 27*x^5 + 9*x^6 + (90*x + 270*x^2 + 135*x^3 + 270*x^4 + 135*x^5)*(I*Pi + Log[4])^2
+ (450 + 1350*x + 675*x^3 + 675*x^4)*(I*Pi + Log[4])^4 + (2250 + 1125*x^3)*(I*Pi + Log[4])^6)/(x^3 + 3*x^4 +
3*x^5 + x^6 + (15*x^3 + 30*x^4 + 15*x^5)*(I*Pi + Log[4])^2 + (75*x^3 + 75*x^4)*(I*Pi + Log[4])^4 + 125*x^3*(I*
Pi + Log[4])^6),x]

[Out]

(9*(-x^2 + x^5 + 25*Pi^4*(-1 + x^3) - (100*I)*Pi^3*(-1 + x^3)*Log[4] - 10*x*Log[4]^2 - 25*Log[4]^4 + 2*x^4*(1
+ 5*Log[4]^2) + x^3*(1 + 5*Log[4]^2)^2 + (20*I)*Pi*Log[4]*(-x + x^4 - 5*Log[4]^2 + x^3*(1 + 5*Log[4]^2)) - 10*
Pi^2*(-x + x^4 - 15*Log[4]^2 + x^3*(1 + 15*Log[4]^2))))/(x^2*(1 - 5*Pi^2 + x + (10*I)*Pi*Log[4] + 5*Log[4]^2)^
2)

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fricas [B]  time = 0.55, size = 240, normalized size = 8.28 \begin {gather*} -\frac {9 \, {\left (2 \, {\left (5 \, \pi ^{2} - 1\right )} x^{4} - x^{5} - 400 \, {\left (x^{3} - 1\right )} \log \relax (2)^{4} + 25 \, \pi ^{4} - {\left (25 \, \pi ^{4} - 10 \, \pi ^{2} + 1\right )} x^{3} + 800 \, {\left (i \, \pi - i \, \pi x^{3}\right )} \log \relax (2)^{3} - 10 \, \pi ^{2} x + 40 \, {\left ({\left (15 \, \pi ^{2} - 1\right )} x^{3} - x^{4} - 15 \, \pi ^{2} + x\right )} \log \relax (2)^{2} + x^{2} + 40 \, {\left (-i \, \pi x^{4} + {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} x^{3} - 5 i \, \pi ^{3} + i \, \pi x\right )} \log \relax (2)\right )}}{800 i \, \pi x^{2} \log \relax (2)^{3} + 400 \, x^{2} \log \relax (2)^{4} - 2 \, {\left (5 \, \pi ^{2} - 1\right )} x^{3} + x^{4} + {\left (25 \, \pi ^{4} - 10 \, \pi ^{2} + 1\right )} x^{2} - 40 \, {\left ({\left (15 \, \pi ^{2} - 1\right )} x^{2} - x^{3}\right )} \log \relax (2)^{2} - 40 \, {\left (-i \, \pi x^{3} + {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} x^{2}\right )} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1125*x^3+2250)*(2*log(2)+I*pi)^6+(675*x^4+675*x^3+1350*x+450)*(2*log(2)+I*pi)^4+(135*x^5+270*x^4+1
35*x^3+270*x^2+90*x)*(2*log(2)+I*pi)^2+9*x^6+27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*pi)^6+(75*x^4+75*x^3)*
(2*log(2)+I*pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*pi)^2+x^6+3*x^5+3*x^4+x^3),x, algorithm="fricas")

[Out]

-9*(2*(5*pi^2 - 1)*x^4 - x^5 - 400*(x^3 - 1)*log(2)^4 + 25*pi^4 - (25*pi^4 - 10*pi^2 + 1)*x^3 + 800*(I*pi - I*
pi*x^3)*log(2)^3 - 10*pi^2*x + 40*((15*pi^2 - 1)*x^3 - x^4 - 15*pi^2 + x)*log(2)^2 + x^2 + 40*(-I*pi*x^4 + (-I
*pi + 5*I*pi^3)*x^3 - 5*I*pi^3 + I*pi*x)*log(2))/(800*I*pi*x^2*log(2)^3 + 400*x^2*log(2)^4 - 2*(5*pi^2 - 1)*x^
3 + x^4 + (25*pi^4 - 10*pi^2 + 1)*x^2 - 40*((15*pi^2 - 1)*x^2 - x^3)*log(2)^2 - 40*(-I*pi*x^3 + (-I*pi + 5*I*p
i^3)*x^2)*log(2))

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giac [B]  time = 0.21, size = 93, normalized size = 3.21 \begin {gather*} 9 \, x - \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \relax (2) - 600 \, \pi ^{2} \log \relax (2)^{2} + 800 i \, \pi \log \relax (2)^{3} + 400 \, \log \relax (2)^{4} - 10 \, \pi ^{2} x + 40 i \, \pi x \log \relax (2) + 40 \, x \log \relax (2)^{2} + x^{2}\right )}}{{\left (5 \, \pi ^{2} x - 20 i \, \pi x \log \relax (2) - 20 \, x \log \relax (2)^{2} - x^{2} - x\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1125*x^3+2250)*(2*log(2)+I*pi)^6+(675*x^4+675*x^3+1350*x+450)*(2*log(2)+I*pi)^4+(135*x^5+270*x^4+1
35*x^3+270*x^2+90*x)*(2*log(2)+I*pi)^2+9*x^6+27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*pi)^6+(75*x^4+75*x^3)*
(2*log(2)+I*pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*pi)^2+x^6+3*x^5+3*x^4+x^3),x, algorithm="giac")

[Out]

9*x - 9*(25*pi^4 - 200*I*pi^3*log(2) - 600*pi^2*log(2)^2 + 800*I*pi*log(2)^3 + 400*log(2)^4 - 10*pi^2*x + 40*I
*pi*x*log(2) + 40*x*log(2)^2 + x^2)/(5*pi^2*x - 20*I*pi*x*log(2) - 20*x*log(2)^2 - x^2 - x)^2

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maple [B]  time = 2.32, size = 155, normalized size = 5.34

method	result	size
risch	\(9 x +\frac {-\frac {9 x^{2}}{25}-\left (\frac {72 i \pi \ln \relax (2)}{5}-\frac {18 \pi ^{2}}{5}+\frac {72 \ln \relax (2)^{2}}{5}\right ) x +72 i \pi ^{3} \ln \relax (2)-288 i \pi \ln \relax (2)^{3}-9 \pi ^{4}+216 \pi ^{2} \ln \relax (2)^{2}-144 \ln \relax (2)^{4}}{x^{2} \left (-8 i \pi ^{3} \ln \relax (2)+32 i \pi \ln \relax (2)^{3}+\pi ^{4}-24 \pi ^{2} \ln \relax (2)^{2}+\frac {8 i \pi \ln \relax (2) x}{5}+16 \ln \relax (2)^{4}-\frac {2 x \,\pi ^{2}}{5}+\frac {8 i \pi \ln \relax (2)}{5}+\frac {8 x \ln \relax (2)^{2}}{5}-\frac {2 \pi ^{2}}{5}+\frac {8 \ln \relax (2)^{2}}{5}+\frac {x^{2}}{25}+\frac {2 x}{25}+\frac {1}{25}\right )}\)	\(155\)
default	\(9 x -\frac {9 \left (6000 i \pi ^{5} \ln \relax (2)+96000 i \pi \ln \relax (2)^{5}-2240000 i \pi ^{3} \ln \relax (2)^{5}-80000 i \pi ^{3} \ln \relax (2)^{3}+1250 \pi ^{8}-140000 \pi ^{6} \ln \relax (2)^{2}+1400000 \pi ^{4} \ln \relax (2)^{4}-2240000 \pi ^{2} \ln \relax (2)^{6}+320000 \ln \relax (2)^{8}+560000 i \pi ^{5} \ln \relax (2)^{3}-20000 i \pi ^{7} \ln \relax (2)+1280000 i \pi \ln \relax (2)^{7}-500 \pi ^{6}+30000 \pi ^{4} \ln \relax (2)^{2}-120000 \pi ^{2} \ln \relax (2)^{4}+32000 \ln \relax (2)^{6}-400 i \pi ^{3} \ln \relax (2)+1600 i \pi \ln \relax (2)^{3}+50 \pi ^{4}-1200 \pi ^{2} \ln \relax (2)^{2}+800 \ln \relax (2)^{4}\right )}{2 \left (-20 i \pi \ln \relax (2)+5 \pi ^{2}-20 \ln \relax (2)^{2}-1\right )^{4} x^{2}}-\frac {9 \left (-400 i \pi ^{3} \ln \relax (2)+1600 i \pi \ln \relax (2)^{3}+50 \pi ^{4}-1200 \pi ^{2} \ln \relax (2)^{2}+800 \ln \relax (2)^{4}+40 i \pi \ln \relax (2)-10 \pi ^{2}+40 \ln \relax (2)^{2}\right )}{\left (-20 i \pi \ln \relax (2)+5 \pi ^{2}-20 \ln \relax (2)^{2}-1\right )^{4} x}-\frac {9 \left (-400 i \pi ^{3} \ln \relax (2)+1600 i \pi \ln \relax (2)^{3}+50 \pi ^{4}-1200 \pi ^{2} \ln \relax (2)^{2}+800 \ln \relax (2)^{4}+80 i \pi \ln \relax (2)-20 \pi ^{2}+80 \ln \relax (2)^{2}+2\right )}{2 \left (-20 i \pi \ln \relax (2)+5 \pi ^{2}-20 \ln \relax (2)^{2}-1\right )^{4} \left (-5 \pi ^{2}+20 i \pi \ln \relax (2)+20 \ln \relax (2)^{2}+x +1\right )^{2}}-\frac {9 \left (400 i \pi ^{3} \ln \relax (2)-1600 i \pi \ln \relax (2)^{3}-50 \pi ^{4}+1200 \pi ^{2} \ln \relax (2)^{2}-800 \ln \relax (2)^{4}-40 i \pi \ln \relax (2)+10 \pi ^{2}-40 \ln \relax (2)^{2}\right )}{\left (-20 i \pi \ln \relax (2)+5 \pi ^{2}-20 \ln \relax (2)^{2}-1\right )^{4} \left (-5 \pi ^{2}+20 i \pi \ln \relax (2)+20 \ln \relax (2)^{2}+x +1\right )}\)	\(476\)
norman	\(\frac {\left (-2250 \pi ^{4}+25200 \pi ^{2} \ln \relax (2)^{2}-36000 \ln \relax (2)^{4}+900 \pi ^{2}-3600 \ln \relax (2)^{2}-90\right ) x^{5}+\left (4500 \pi ^{6}+18000 \pi ^{4} \ln \relax (2)^{2}-72000 \pi ^{2} \ln \relax (2)^{4}-288000 \ln \relax (2)^{6}+3600 i \pi ^{3} \ln \relax (2)-14400 i \pi \ln \relax (2)^{3}-1350 \pi ^{4}+3600 \pi ^{2} \ln \relax (2)^{2}-21600 \ln \relax (2)^{4}-360 i \pi \ln \relax (2)+90 \pi ^{2}-360 \ln \relax (2)^{2}\right ) x +\left (22500 \pi ^{6}-198000 \pi ^{4} \ln \relax (2)^{2}+792000 \pi ^{2} \ln \relax (2)^{4}-1440000 \ln \relax (2)^{6}-13500 \pi ^{4}+93600 \pi ^{2} \ln \relax (2)^{2}-216000 \ln \relax (2)^{4}+2700 \pi ^{2}-10800 \ln \relax (2)^{2}-189\right ) x^{4}+\left (-84375 \pi ^{8}+90000 \pi ^{6} \ln \relax (2)^{2}+3420000 \pi ^{4} \ln \relax (2)^{4}+1440000 \pi ^{2} \ln \relax (2)^{6}-21600000 \ln \relax (2)^{8}+67500 \pi ^{6}-306000 \pi ^{4} \ln \relax (2)^{2}+1224000 \pi ^{2} \ln \relax (2)^{4}-4320000 \ln \relax (2)^{6}-20250 \pi ^{4}+111600 \pi ^{2} \ln \relax (2)^{2}-324000 \ln \relax (2)^{4}+2880 \pi ^{2}-11520 \ln \relax (2)^{2}-153\right ) x^{3}+\left (112500 \pi ^{10}+1350000 \pi ^{8} \ln \relax (2)^{2}+3600000 \pi ^{6} \ln \relax (2)^{4}-14400000 \pi ^{4} \ln \relax (2)^{6}-86400000 \pi ^{2} \ln \relax (2)^{8}-115200000 \ln \relax (2)^{10}-112500 \pi ^{8}-360000 \pi ^{6} \ln \relax (2)^{2}+720000 \pi ^{4} \ln \relax (2)^{4}-5760000 \pi ^{2} \ln \relax (2)^{6}-28800000 \ln \relax (2)^{8}+45000 \pi ^{6}-108000 \pi ^{4} \ln \relax (2)^{2}+432000 \pi ^{2} \ln \relax (2)^{4}-2880000 \ln \relax (2)^{6}-10350 \pi ^{4}+46800 \pi ^{2} \ln \relax (2)^{2}-165600 \ln \relax (2)^{4}-360 i \pi \ln \relax (2)+1170 \pi ^{2}-4680 \ln \relax (2)^{2}-45\right ) x^{2}+9 x^{7}-5625 \pi ^{8}-90000 \pi ^{6} \ln \relax (2)^{2}-540000 \pi ^{4} \ln \relax (2)^{4}-1440000 \pi ^{2} \ln \relax (2)^{6}-1440000 \ln \relax (2)^{8}-9000 i \pi ^{5} \ln \relax (2)-144000 i \pi \ln \relax (2)^{5}-72000 i \pi ^{3} \ln \relax (2)^{3}+2250 \pi ^{6}+9000 \pi ^{4} \ln \relax (2)^{2}-36000 \pi ^{2} \ln \relax (2)^{4}-144000 \ln \relax (2)^{6}+1800 i \pi ^{3} \ln \relax (2)-7200 i \pi \ln \relax (2)^{3}-225 \pi ^{4}+5400 \pi ^{2} \ln \relax (2)^{2}-3600 \ln \relax (2)^{4}}{x^{2} \left (25 \pi ^{4}+200 \pi ^{2} \ln \relax (2)^{2}+400 \ln \relax (2)^{4}-10 x \,\pi ^{2}+40 x \ln \relax (2)^{2}-10 \pi ^{2}+40 \ln \relax (2)^{2}+x^{2}+2 x +1\right )^{2}}\)	\(643\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1125*x^3+2250)*(2*ln(2)+I*Pi)^6+(675*x^4+675*x^3+1350*x+450)*(2*ln(2)+I*Pi)^4+(135*x^5+270*x^4+135*x^3+2
70*x^2+90*x)*(2*ln(2)+I*Pi)^2+9*x^6+27*x^5+27*x^4+27*x^3)/(125*x^3*(2*ln(2)+I*Pi)^6+(75*x^4+75*x^3)*(2*ln(2)+I
*Pi)^4+(15*x^5+30*x^4+15*x^3)*(2*ln(2)+I*Pi)^2+x^6+3*x^5+3*x^4+x^3),x,method=_RETURNVERBOSE)

[Out]

9*x+(-9/25*x^2-(72/5*I*Pi*ln(2)-18/5*Pi^2+72/5*ln(2)^2)*x+72*I*Pi^3*ln(2)-288*I*Pi*ln(2)^3-9*Pi^4+216*Pi^2*ln(
2)^2-144*ln(2)^4)/x^2/(-8*I*Pi^3*ln(2)+32*I*Pi*ln(2)^3+Pi^4-24*Pi^2*ln(2)^2+8/5*I*Pi*ln(2)*x+16*ln(2)^4-2/5*x*
Pi^2+8/5*I*Pi*ln(2)+8/5*x*ln(2)^2-2/5*Pi^2+8/5*ln(2)^2+1/25*x^2+2/25*x+1/25)

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maxima [B]  time = 0.44, size = 149, normalized size = 5.14 \begin {gather*} 9 \, x + \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \relax (2) - 600 \, \pi ^{2} \log \relax (2)^{2} + 800 i \, \pi \log \relax (2)^{3} + 400 \, \log \relax (2)^{4} - 10 \, {\left (\pi ^{2} - 4 i \, \pi \log \relax (2) - 4 \, \log \relax (2)^{2}\right )} x + x^{2}\right )}}{2 \, {\left (5 \, \pi ^{2} - 20 i \, \pi \log \relax (2) - 20 \, \log \relax (2)^{2} - 1\right )} x^{3} - x^{4} - {\left (25 \, \pi ^{4} + 800 i \, \pi \log \relax (2)^{3} + 400 \, \log \relax (2)^{4} - 40 \, {\left (15 \, \pi ^{2} - 1\right )} \log \relax (2)^{2} - 10 \, \pi ^{2} - 40 \, {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} \log \relax (2) + 1\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1125*x^3+2250)*(2*log(2)+I*pi)^6+(675*x^4+675*x^3+1350*x+450)*(2*log(2)+I*pi)^4+(135*x^5+270*x^4+1
35*x^3+270*x^2+90*x)*(2*log(2)+I*pi)^2+9*x^6+27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*pi)^6+(75*x^4+75*x^3)*
(2*log(2)+I*pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*pi)^2+x^6+3*x^5+3*x^4+x^3),x, algorithm="maxima")

[Out]

9*x + 9*(25*pi^4 - 200*I*pi^3*log(2) - 600*pi^2*log(2)^2 + 800*I*pi*log(2)^3 + 400*log(2)^4 - 10*(pi^2 - 4*I*p
i*log(2) - 4*log(2)^2)*x + x^2)/(2*(5*pi^2 - 20*I*pi*log(2) - 20*log(2)^2 - 1)*x^3 - x^4 - (25*pi^4 + 800*I*pi
*log(2)^3 + 400*log(2)^4 - 40*(15*pi^2 - 1)*log(2)^2 - 10*pi^2 - 40*(-I*pi + 5*I*pi^3)*log(2) + 1)*x^2)

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mupad [B]  time = 20.40, size = 640, normalized size = 22.07 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1125*x^3 + 2250)*(Pi*1i + 2*log(2))^6 + (Pi*1i + 2*log(2))^4*(1350*x + 675*x^3 + 675*x^4 + 450) + (Pi*1i
+ 2*log(2))^2*(90*x + 270*x^2 + 135*x^3 + 270*x^4 + 135*x^5) + 27*x^3 + 27*x^4 + 27*x^5 + 9*x^6)/((75*x^3 + 7
5*x^4)*(Pi*1i + 2*log(2))^4 + 125*x^3*(Pi*1i + 2*log(2))^6 + (Pi*1i + 2*log(2))^2*(15*x^3 + 30*x^4 + 15*x^5) +
x^3 + 3*x^4 + 3*x^5 + x^6),x)

[Out]

9*x + (900*Pi*log(4)^3 - 900*Pi^3*log(4) - Pi^4*225i - log(4)^4*225i + Pi^2*log(4)^2*1350i - (90*x*(Pi*log(4)*
2i - Pi^3*log(2)*60i - Pi*log(4)^3*40i + Pi^3*log(4)*10i - Pi^2 + 5*Pi^4 + log(4)^2 - 10*log(4)^4 - 60*Pi^2*lo
g(2)^2 + 45*Pi^2*log(4)^2 + 60*log(2)^2*log(4)^2 + Pi*log(2)*log(4)^2*60i + Pi*log(2)^2*log(4)*120i - 120*Pi^2
*log(2)*log(4)))/(20*Pi*log(2) + Pi^2*5i - log(2)^2*20i - 1i) + (9*x^2*(Pi*log(2)*40i + Pi*log(2)^3*800i - Pi^
3*log(2)*2000i - Pi*log(4)^3*1800i + Pi^3*log(4)*900i - 10*Pi^2 + 25*Pi^4 + 40*log(2)^2 + 400*log(2)^4 - 450*l
og(4)^4 - 2400*Pi^2*log(2)^2 + 2250*Pi^2*log(4)^2 + 1800*log(2)^2*log(4)^2 + Pi*log(2)*log(4)^2*1800i + Pi*log
(2)^2*log(4)*3600i - 3600*Pi^2*log(2)*log(4) + 1))/(200*Pi^3*log(2) - 800*Pi*log(2)^3 - 40*Pi*log(2) - Pi^2*10
i + Pi^4*25i + log(2)^2*40i + log(2)^4*400i - Pi^2*log(2)^2*600i + 1i) + (2700*x^3*(2*Pi^3*log(4) - 4*Pi*log(4
)^3 - 4*Pi^3*log(2) + log(4)^4*1i + Pi^2*log(2)^2*4i - Pi^2*log(4)^2*5i - log(2)^2*log(4)^2*4i + 4*Pi*log(2)*l
og(4)^2 + 8*Pi*log(2)^2*log(4) + Pi^2*log(2)*log(4)*8i))/((20*Pi*log(2) + Pi^2*5i - log(2)^2*20i - 1i)*(200*Pi
^3*log(2) - 800*Pi*log(2)^3 - 40*Pi*log(2) - Pi^2*10i + Pi^4*25i + log(2)^2*40i + log(2)^4*400i - Pi^2*log(2)^
2*600i + 1i)))/(x^4*1i - x^3*(40*Pi*log(2) + Pi^2*10i - log(2)^2*40i - 2i) + x^2*(200*Pi^3*log(2) - 800*Pi*log
(2)^3 - 40*Pi*log(2) - Pi^2*10i + Pi^4*25i + log(2)^2*40i + log(2)^4*400i - Pi^2*log(2)^2*600i + 1i))

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sympy [B]  time = 36.28, size = 168, normalized size = 5.79 \begin {gather*} 9 x + \frac {9 x^{2} + x \left (- 90 \pi ^{2} + 360 \log {\relax (2 )}^{2} + 360 i \pi \log {\relax (2 )}\right ) - 5400 \pi ^{2} \log {\relax (2 )}^{2} + 3600 \log {\relax (2 )}^{4} + 225 \pi ^{4} - 1800 i \pi ^{3} \log {\relax (2 )} + 7200 i \pi \log {\relax (2 )}^{3}}{- x^{4} + x^{3} \left (- 40 \log {\relax (2 )}^{2} - 2 + 10 \pi ^{2} - 40 i \pi \log {\relax (2 )}\right ) + x^{2} \left (- 25 \pi ^{4} - 400 \log {\relax (2 )}^{4} - 40 \log {\relax (2 )}^{2} - 1 + 10 \pi ^{2} + 600 \pi ^{2} \log {\relax (2 )}^{2} - 800 i \pi \log {\relax (2 )}^{3} - 40 i \pi \log {\relax (2 )} + 200 i \pi ^{3} \log {\relax (2 )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1125*x**3+2250)*(2*ln(2)+I*pi)**6+(675*x**4+675*x**3+1350*x+450)*(2*ln(2)+I*pi)**4+(135*x**5+270*x
**4+135*x**3+270*x**2+90*x)*(2*ln(2)+I*pi)**2+9*x**6+27*x**5+27*x**4+27*x**3)/(125*x**3*(2*ln(2)+I*pi)**6+(75*
x**4+75*x**3)*(2*ln(2)+I*pi)**4+(15*x**5+30*x**4+15*x**3)*(2*ln(2)+I*pi)**2+x**6+3*x**5+3*x**4+x**3),x)

[Out]

9*x + (9*x**2 + x*(-90*pi**2 + 360*log(2)**2 + 360*I*pi*log(2)) - 5400*pi**2*log(2)**2 + 3600*log(2)**4 + 225*
pi**4 - 1800*I*pi**3*log(2) + 7200*I*pi*log(2)**3)/(-x**4 + x**3*(-40*log(2)**2 - 2 + 10*pi**2 - 40*I*pi*log(2
)) + x**2*(-25*pi**4 - 400*log(2)**4 - 40*log(2)**2 - 1 + 10*pi**2 + 600*pi**2*log(2)**2 - 800*I*pi*log(2)**3
- 40*I*pi*log(2) + 200*I*pi**3*log(2)))

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