3.27.30 \(\int \frac {27 x^3+27 x^4+27 x^5+9 x^6+(90 x+270 x^2+135 x^3+270 x^4+135 x^5) (i \pi +\log (4))^2+(450+1350 x+675 x^3+675 x^4) (i \pi +\log (4))^4+(2250+1125 x^3) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+(15 x^3+30 x^4+15 x^5) (i \pi +\log (4))^2+(75 x^3+75 x^4) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx\) [2630]

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

[Out]

9*x+27-9/(x+x/(x+5*(2*ln(2)+I*Pi)^2))^2

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Rubi [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(182\) vs. \(2(29)=58\).
time = 0.33, 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, 2099} \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 2099

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] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(29)=58\).
time = 0.14, 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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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (30 ) = 60\).
time = 2.44, size = 476, normalized size = 16.41 Too large to display

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/(20*I*ln(2)*Pi+20*ln(2)^2-5*Pi^2+1)^4*(-1600*I*ln(2)^3*Pi+400*I*ln(2)*Pi^3-800*ln(2)^4+1200*ln(2)^2*Pi^2
-50*Pi^4-40*I*ln(2)*Pi-40*ln(2)^2+10*Pi^2)/(-5*Pi^2+20*I*ln(2)*Pi+20*ln(2)^2+x+1)-9/2/(20*I*ln(2)*Pi+20*ln(2)^
2-5*Pi^2+1)^4*(-1200*ln(2)^2*Pi^2+1600*I*ln(2)^3*Pi-400*I*ln(2)*Pi^3+800*ln(2)^4+50*Pi^4+80*I*ln(2)*Pi+80*ln(2
)^2-20*Pi^2+2)/(-5*Pi^2+20*I*ln(2)*Pi+20*ln(2)^2+x+1)^2-9/2/(20*I*ln(2)*Pi+20*ln(2)^2-5*Pi^2+1)^4*(-80000*I*ln
(2)^3*Pi^3-2240000*I*ln(2)^5*Pi^3+1280000*I*ln(2)^7*Pi+1600*I*ln(2)^3*Pi+320000*ln(2)^8-2240000*ln(2)^6*Pi^2+1
400000*ln(2)^4*Pi^4-140000*ln(2)^2*Pi^6+1250*Pi^8-400*I*ln(2)*Pi^3+6000*I*ln(2)*Pi^5+560000*I*ln(2)^3*Pi^5+320
00*ln(2)^6-120000*ln(2)^4*Pi^2+30000*ln(2)^2*Pi^4-500*Pi^6-20000*I*ln(2)*Pi^7+96000*I*ln(2)^5*Pi+800*ln(2)^4-1
200*ln(2)^2*Pi^2+50*Pi^4)/x^2-9/(20*I*ln(2)*Pi+20*ln(2)^2-5*Pi^2+1)^4*(1600*I*ln(2)^3*Pi-400*I*ln(2)*Pi^3+800*
ln(2)^4-1200*ln(2)^2*Pi^2+50*Pi^4+40*I*ln(2)*Pi+40*ln(2)^2-10*Pi^2)/x

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (29) = 58\).
time = 0.31, size = 149, normalized size = 5.14 \begin {gather*} 9 \, x + \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \left (2\right ) - 600 \, \pi ^{2} \log \left (2\right )^{2} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 10 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (2\right ) - 4 \, \log \left (2\right )^{2}\right )} x + x^{2}\right )}}{2 \, {\left (5 \, \pi ^{2} - 20 i \, \pi \log \left (2\right ) - 20 \, \log \left (2\right )^{2} - 1\right )} x^{3} - x^{4} - {\left (25 \, \pi ^{4} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 40 \, {\left (15 \, \pi ^{2} - 1\right )} \log \left (2\right )^{2} - 10 \, \pi ^{2} - 40 \, {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} \log \left (2\right ) + 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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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (29) = 58\).
time = 0.36, 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 \left (2\right )^{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 \left (2\right )^{3} - 10 \, \pi ^{2} x + 40 \, {\left ({\left (15 \, \pi ^{2} - 1\right )} x^{3} - x^{4} - 15 \, \pi ^{2} + x\right )} \log \left (2\right )^{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 \left (2\right )\right )}}{800 i \, \pi x^{2} \log \left (2\right )^{3} + 400 \, x^{2} \log \left (2\right )^{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 \left (2\right )^{2} - 40 \, {\left (-i \, \pi x^{3} + {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} x^{2}\right )} \log \left (2\right )} \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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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (24) = 48\).
time = 18.39, size = 168, normalized size = 5.79 \begin {gather*} 9 x + \frac {- 9 x^{2} + x \left (- 360 \log {\left (2 \right )}^{2} + 90 \pi ^{2} - 360 i \pi \log {\left (2 \right )}\right ) - 225 \pi ^{4} - 3600 \log {\left (2 \right )}^{4} + 5400 \pi ^{2} \log {\left (2 \right )}^{2} - 7200 i \pi \log {\left (2 \right )}^{3} + 1800 i \pi ^{3} \log {\left (2 \right )}}{x^{4} + x^{3} \left (- 10 \pi ^{2} + 2 + 40 \log {\left (2 \right )}^{2} + 40 i \pi \log {\left (2 \right )}\right ) + x^{2} \left (- 600 \pi ^{2} \log {\left (2 \right )}^{2} - 10 \pi ^{2} + 1 + 40 \log {\left (2 \right )}^{2} + 400 \log {\left (2 \right )}^{4} + 25 \pi ^{4} - 200 i \pi ^{3} \log {\left (2 \right )} + 40 i \pi \log {\left (2 \right )} + 800 i \pi \log {\left (2 \right )}^{3}\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*(-360*log(2)**2 + 90*pi**2 - 360*I*pi*log(2)) - 225*pi**4 - 3600*log(2)**4 + 5400*pi**2*log
(2)**2 - 7200*I*pi*log(2)**3 + 1800*I*pi**3*log(2))/(x**4 + x**3*(-10*pi**2 + 2 + 40*log(2)**2 + 40*I*pi*log(2
)) + x**2*(-600*pi**2*log(2)**2 - 10*pi**2 + 1 + 40*log(2)**2 + 400*log(2)**4 + 25*pi**4 - 200*I*pi**3*log(2)
+ 40*I*pi*log(2) + 800*I*pi*log(2)**3))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (29) = 58\).
time = 0.42, size = 93, normalized size = 3.21 \begin {gather*} 9 \, x - \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \left (2\right ) - 600 \, \pi ^{2} \log \left (2\right )^{2} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 10 \, \pi ^{2} x + 40 i \, \pi x \log \left (2\right ) + 40 \, x \log \left (2\right )^{2} + x^{2}\right )}}{{\left (5 \, \pi ^{2} x - 20 i \, \pi x \log \left (2\right ) - 20 \, x \log \left (2\right )^{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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Mupad [B]
time = 20.40, size = 640, normalized size = 22.07 \begin {gather*} 9\,x+\frac {900\,\Pi \,{\ln \left (4\right )}^3-900\,\Pi ^3\,\ln \left (4\right )-\Pi ^4\,225{}\mathrm {i}-{\ln \left (4\right )}^4\,225{}\mathrm {i}+\Pi ^2\,{\ln \left (4\right )}^2\,1350{}\mathrm {i}-\frac {90\,x\,\left (5\,\Pi ^4-\Pi ^2+{\ln \left (4\right )}^2-10\,{\ln \left (4\right )}^4-60\,\Pi ^2\,{\ln \left (2\right )}^2+45\,\Pi ^2\,{\ln \left (4\right )}^2+60\,{\ln \left (2\right )}^2\,{\ln \left (4\right )}^2-120\,\Pi ^2\,\ln \left (2\right )\,\ln \left (4\right )+\Pi \,\ln \left (4\right )\,2{}\mathrm {i}-\Pi ^3\,\ln \left (2\right )\,60{}\mathrm {i}-\Pi \,{\ln \left (4\right )}^3\,40{}\mathrm {i}+\Pi ^3\,\ln \left (4\right )\,10{}\mathrm {i}+\Pi \,\ln \left (2\right )\,{\ln \left (4\right )}^2\,60{}\mathrm {i}+\Pi \,{\ln \left (2\right )}^2\,\ln \left (4\right )\,120{}\mathrm {i}\right )}{\Pi ^2\,5{}\mathrm {i}+20\,\ln \left (2\right )\,\Pi -{\ln \left (2\right )}^2\,20{}\mathrm {i}-\mathrm {i}}+\frac {9\,x^2\,\left (25\,\Pi ^4-10\,\Pi ^2+40\,{\ln \left (2\right )}^2+400\,{\ln \left (2\right )}^4-450\,{\ln \left (4\right )}^4-2400\,\Pi ^2\,{\ln \left (2\right )}^2+2250\,\Pi ^2\,{\ln \left (4\right )}^2+1800\,{\ln \left (2\right )}^2\,{\ln \left (4\right )}^2-3600\,\Pi ^2\,\ln \left (2\right )\,\ln \left (4\right )+1+\Pi \,\ln \left (2\right )\,40{}\mathrm {i}+\Pi \,{\ln \left (2\right )}^3\,800{}\mathrm {i}-\Pi ^3\,\ln \left (2\right )\,2000{}\mathrm {i}-\Pi \,{\ln \left (4\right )}^3\,1800{}\mathrm {i}+\Pi ^3\,\ln \left (4\right )\,900{}\mathrm {i}+\Pi \,\ln \left (2\right )\,{\ln \left (4\right )}^2\,1800{}\mathrm {i}+\Pi \,{\ln \left (2\right )}^2\,\ln \left (4\right )\,3600{}\mathrm {i}\right )}{200\,\Pi ^3\,\ln \left (2\right )-800\,\Pi \,{\ln \left (2\right )}^3-40\,\Pi \,\ln \left (2\right )-\Pi ^2\,10{}\mathrm {i}+\Pi ^4\,25{}\mathrm {i}+{\ln \left (2\right )}^2\,40{}\mathrm {i}+{\ln \left (2\right )}^4\,400{}\mathrm {i}-\Pi ^2\,{\ln \left (2\right )}^2\,600{}\mathrm {i}+1{}\mathrm {i}}+\frac {2700\,x^3\,\left (2\,\Pi ^3\,\ln \left (4\right )-4\,\Pi \,{\ln \left (4\right )}^3-4\,\Pi ^3\,\ln \left (2\right )+{\ln \left (4\right )}^4\,1{}\mathrm {i}+\Pi ^2\,{\ln \left (2\right )}^2\,4{}\mathrm {i}-\Pi ^2\,{\ln \left (4\right )}^2\,5{}\mathrm {i}-{\ln \left (2\right )}^2\,{\ln \left (4\right )}^2\,4{}\mathrm {i}+4\,\Pi \,\ln \left (2\right )\,{\ln \left (4\right )}^2+8\,\Pi \,{\ln \left (2\right )}^2\,\ln \left (4\right )+\Pi ^2\,\ln \left (2\right )\,\ln \left (4\right )\,8{}\mathrm {i}\right )}{\left (\Pi ^2\,5{}\mathrm {i}+20\,\ln \left (2\right )\,\Pi -{\ln \left (2\right )}^2\,20{}\mathrm {i}-\mathrm {i}\right )\,\left (200\,\Pi ^3\,\ln \left (2\right )-800\,\Pi \,{\ln \left (2\right )}^3-40\,\Pi \,\ln \left (2\right )-\Pi ^2\,10{}\mathrm {i}+\Pi ^4\,25{}\mathrm {i}+{\ln \left (2\right )}^2\,40{}\mathrm {i}+{\ln \left (2\right )}^4\,400{}\mathrm {i}-\Pi ^2\,{\ln \left (2\right )}^2\,600{}\mathrm {i}+1{}\mathrm {i}\right )}}{x^4\,1{}\mathrm {i}+\left (-\Pi ^2\,10{}\mathrm {i}-40\,\ln \left (2\right )\,\Pi +{\ln \left (2\right )}^2\,40{}\mathrm {i}+2{}\mathrm {i}\right )\,x^3+\left (200\,\Pi ^3\,\ln \left (2\right )-800\,\Pi \,{\ln \left (2\right )}^3-40\,\Pi \,\ln \left (2\right )-\Pi ^2\,10{}\mathrm {i}+\Pi ^4\,25{}\mathrm {i}+{\ln \left (2\right )}^2\,40{}\mathrm {i}+{\ln \left (2\right )}^4\,400{}\mathrm {i}-\Pi ^2\,{\ln \left (2\right )}^2\,600{}\mathrm {i}+1{}\mathrm {i}\right )\,x^2} \end {gather*}

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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