3.41.10 \(\int \frac {2 x^2+(-4 x+2 x^2) (i \pi +\log (\frac {5}{4}))+(2-4 x+x^2) (i \pi +\log (\frac {5}{4}))^2+(x^3+(-2 x^2+x^3) (i \pi +\log (\frac {5}{4}))+(x-x^2) (i \pi +\log (\frac {5}{4}))^2) \log (\frac {x^2+(-x+x^2) (i \pi +\log (\frac {5}{4}))}{i \pi -x+\log (\frac {5}{4})})}{-x^3+(2 x^2-x^3) (i \pi +\log (\frac {5}{4}))+(-x+x^2) (i \pi +\log (\frac {5}{4}))^2+(-2 x^3+(4 x^2-2 x^3) (i \pi +\log (\frac {5}{4}))+(-2 x+2 x^2) (i \pi +\log (\frac {5}{4}))^2) \log (\frac {x^2+(-x+x^2) (i \pi +\log (\frac {5}{4}))}{i \pi -x+\log (\frac {5}{4})})+(-x^3+(2 x^2-x^3) (i \pi +\log (\frac {5}{4}))+(-x+x^2) (i \pi +\log (\frac {5}{4}))^2) \log ^2(\frac {x^2+(-x+x^2) (i \pi +\log (\frac {5}{4}))}{i \pi -x+\log (\frac {5}{4})})} \, dx\)

Optimal. Leaf size=38 \[ \frac {2-x}{1+\log \left (-x+\frac {x^2}{1-\frac {x}{i \pi +\log \left (\frac {5}{4}\right )}}\right )} \]

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Rubi [F]  time = 13.69, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2 x^2+\left (-4 x+2 x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+\left (2-4 x+x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )^2+\left (x^3+\left (-2 x^2+x^3\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+\left (x-x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )^2\right ) \log \left (\frac {x^2+\left (-x+x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )}{-x^3+\left (2 x^2-x^3\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+\left (-x+x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )^2+\left (-2 x^3+\left (4 x^2-2 x^3\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+\left (-2 x+2 x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )^2\right ) \log \left (\frac {x^2+\left (-x+x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )+\left (-x^3+\left (2 x^2-x^3\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+\left (-x+x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )^2\right ) \log ^2\left (\frac {x^2+\left (-x+x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^2 + (-4*x + 2*x^2)*(I*Pi + Log[5/4]) + (2 - 4*x + x^2)*(I*Pi + Log[5/4])^2 + (x^3 + (-2*x^2 + x^3)*(I
*Pi + Log[5/4]) + (x - x^2)*(I*Pi + Log[5/4])^2)*Log[(x^2 + (-x + x^2)*(I*Pi + Log[5/4]))/(I*Pi - x + Log[5/4]
)])/(-x^3 + (2*x^2 - x^3)*(I*Pi + Log[5/4]) + (-x + x^2)*(I*Pi + Log[5/4])^2 + (-2*x^3 + (4*x^2 - 2*x^3)*(I*Pi
+ Log[5/4]) + (-2*x + 2*x^2)*(I*Pi + Log[5/4])^2)*Log[(x^2 + (-x + x^2)*(I*Pi + Log[5/4]))/(I*Pi - x + Log[5/
4])] + (-x^3 + (2*x^2 - x^3)*(I*Pi + Log[5/4]) + (-x + x^2)*(I*Pi + Log[5/4])^2)*Log[(x^2 + (-x + x^2)*(I*Pi +
Log[5/4]))/(I*Pi - x + Log[5/4])]^2),x]

[Out]

Defer[Int][(-1 - Log[(x*(x + (-1 + x)*(I*Pi + Log[5/4])))/(I*Pi - x + Log[5/4])])^(-1), x] + Defer[Int][(1 + L
og[(x*(x + (-1 + x)*(I*Pi + Log[5/4])))/(I*Pi - x + Log[5/4])])^(-2), x] - 2*Defer[Int][1/(x*(1 + Log[(x*(x +
(-1 + x)*(I*Pi + Log[5/4])))/(I*Pi - x + Log[5/4])])^2), x] - (2*I)*Defer[Int][1/((-Pi - I*x + I*Log[5/4])*(1
+ Log[(x*(x + (-1 + x)*(I*Pi + Log[5/4])))/(I*Pi - x + Log[5/4])])^2), x] - I*(Pi - I*Log[5/4])^2*(1 + ((Pi -
I*Log[5/4])*(2 + I*Pi + Log[5/4]))/Sqrt[-(Pi - I*Log[5/4])^4])*Defer[Int][1/((-Sqrt[-(Pi - I*Log[5/4])^4] + (2
*I - Pi + I*Log[5/4])*(I*Pi + Log[5/4]) + 2*x*(Pi - I*(1 + Log[5/4])))*(1 + Log[(x*(x + (-1 + x)*(I*Pi + Log[5
/4])))/(I*Pi - x + Log[5/4])])^2), x] - I*(Pi - I*Log[5/4])^2*(1 - ((Pi - I*Log[5/4])*(2 + I*Pi + Log[5/4]))/S
qrt[-(Pi - I*Log[5/4])^4])*Defer[Int][1/((Sqrt[-(Pi - I*Log[5/4])^4] + (2*I - Pi + I*Log[5/4])*(I*Pi + Log[5/4
]) + 2*x*(Pi - I*(1 + Log[5/4])))*(1 + Log[(x*(x + (-1 + x)*(I*Pi + Log[5/4])))/(I*Pi - x + Log[5/4])])^2), x]
- 4*(Pi - I*(1 + Log[5/4]))*Defer[Int][1/((-2*Pi + 2*x*(Pi - I*(1 + Log[5/4])) + I*Log[25/16])*(1 + Log[(x*(x
+ (-1 + x)*(I*Pi + Log[5/4])))/(I*Pi - x + Log[5/4])])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x^2+2 (-2+x) x \left (i \pi +\log \left (\frac {5}{4}\right )\right )+\left (2-4 x+x^2\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )^2+x \left (x^2+(-1+x) \left (\pi -i \log \left (\frac {5}{4}\right )\right )^2+(-2+x) x \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right ) \log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )}{x \left (\pi ^2+x \left (\pi -i \log \left (\frac {5}{4}\right )\right ) \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right )-\log ^2\left (\frac {5}{4}\right )-x^2 \left (1+i \pi +\log \left (\frac {5}{4}\right )\right )-i \pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\\ &=\int \left (\frac {1}{-1-\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )}+\frac {(2-x) \left (-i \pi ^2+i \log ^2\left (\frac {5}{4}\right )-x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )-\pi \log \left (\frac {25}{16}\right )+2 x \left (i \pi ^2-i \log \left (\frac {5}{4}\right ) \left (1+\log \left (\frac {5}{4}\right )\right )+\pi \left (1+\log \left (\frac {25}{16}\right )\right )\right )\right )}{x \left (i \pi ^2-i \log ^2\left (\frac {5}{4}\right )+x \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )+\pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2}\right ) \, dx\\ &=\int \frac {1}{-1-\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )} \, dx+\int \frac {(2-x) \left (-i \pi ^2+i \log ^2\left (\frac {5}{4}\right )-x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )-\pi \log \left (\frac {25}{16}\right )+2 x \left (i \pi ^2-i \log \left (\frac {5}{4}\right ) \left (1+\log \left (\frac {5}{4}\right )\right )+\pi \left (1+\log \left (\frac {25}{16}\right )\right )\right )\right )}{x \left (i \pi ^2-i \log ^2\left (\frac {5}{4}\right )+x \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )+\pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\\ &=\int \frac {1}{-1-\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )} \, dx+\int \left (\frac {1}{\left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2}-\frac {2}{x \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2}-\frac {i (-2+x) \left (\pi -i \log \left (\frac {5}{4}\right )\right )^2}{\left (i \pi ^2-i \log ^2\left (\frac {5}{4}\right )+x \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )+\pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\right )-\left (i \left (\pi -i \log \left (\frac {5}{4}\right )\right )^2\right ) \int \frac {-2+x}{\left (i \pi ^2-i \log ^2\left (\frac {5}{4}\right )+x \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )+\pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx+\int \frac {1}{-1-\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )} \, dx+\int \frac {1}{\left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {1}{x \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\right )-\left (i \left (\pi -i \log \left (\frac {5}{4}\right )\right )^2\right ) \int \left (-\frac {2 i}{\left (-\pi ^2-x \left (\pi -i \log \left (\frac {5}{4}\right )\right ) \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right )+\log ^2\left (\frac {5}{4}\right )+x^2 \left (1+i \pi +\log \left (\frac {5}{4}\right )\right )+i \pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2}+\frac {x}{\left (i \pi ^2-i \log ^2\left (\frac {5}{4}\right )+x \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )+\pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2}\right ) \, dx+\int \frac {1}{-1-\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )} \, dx+\int \frac {1}{\left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {1}{x \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\right )-\left (i \left (\pi -i \log \left (\frac {5}{4}\right )\right )^2\right ) \int \frac {x}{\left (i \pi ^2-i \log ^2\left (\frac {5}{4}\right )+x \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right ) \left (i \pi +\log \left (\frac {5}{4}\right )\right )+x^2 \left (\pi -i \left (1+\log \left (\frac {5}{4}\right )\right )\right )+\pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx-\left (2 \left (\pi -i \log \left (\frac {5}{4}\right )\right )^2\right ) \int \frac {1}{\left (-\pi ^2-x \left (\pi -i \log \left (\frac {5}{4}\right )\right ) \left (2 i-\pi +i \log \left (\frac {5}{4}\right )\right )+\log ^2\left (\frac {5}{4}\right )+x^2 \left (1+i \pi +\log \left (\frac {5}{4}\right )\right )+i \pi \log \left (\frac {25}{16}\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx+\int \frac {1}{-1-\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )} \, dx+\int \frac {1}{\left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.47, size = 143, normalized size = 3.76 \begin {gather*} -\frac {i (-2+x) \left (-i \pi +x-\log \left (\frac {5}{4}\right )\right ) \left (\pi (-1+x)-i \left (x-\log \left (\frac {5}{4}\right )+x \log \left (\frac {5}{4}\right )\right )\right )}{\left (\pi ^2 (-1+x)+\log ^2\left (\frac {5}{4}\right )+x^2 \left (1+\log \left (\frac {5}{4}\right )\right )-x \log \left (\frac {5}{4}\right ) \left (2+\log \left (\frac {5}{4}\right )\right )+i \pi \left (x^2-2 x \left (1+\log \left (\frac {5}{4}\right )\right )+\log \left (\frac {25}{16}\right )\right )\right ) \left (1+\log \left (\frac {x \left (x+(-1+x) \left (i \pi +\log \left (\frac {5}{4}\right )\right )\right )}{i \pi -x+\log \left (\frac {5}{4}\right )}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 + (-4*x + 2*x^2)*(I*Pi + Log[5/4]) + (2 - 4*x + x^2)*(I*Pi + Log[5/4])^2 + (x^3 + (-2*x^2 + x
^3)*(I*Pi + Log[5/4]) + (x - x^2)*(I*Pi + Log[5/4])^2)*Log[(x^2 + (-x + x^2)*(I*Pi + Log[5/4]))/(I*Pi - x + Lo
g[5/4])])/(-x^3 + (2*x^2 - x^3)*(I*Pi + Log[5/4]) + (-x + x^2)*(I*Pi + Log[5/4])^2 + (-2*x^3 + (4*x^2 - 2*x^3)
*(I*Pi + Log[5/4]) + (-2*x + 2*x^2)*(I*Pi + Log[5/4])^2)*Log[(x^2 + (-x + x^2)*(I*Pi + Log[5/4]))/(I*Pi - x +
Log[5/4])] + (-x^3 + (2*x^2 - x^3)*(I*Pi + Log[5/4]) + (-x + x^2)*(I*Pi + Log[5/4])^2)*Log[(x^2 + (-x + x^2)*(
I*Pi + Log[5/4]))/(I*Pi - x + Log[5/4])]^2),x]

[Out]

((-I)*(-2 + x)*((-I)*Pi + x - Log[5/4])*(Pi*(-1 + x) - I*(x - Log[5/4] + x*Log[5/4])))/((Pi^2*(-1 + x) + Log[5
/4]^2 + x^2*(1 + Log[5/4]) - x*Log[5/4]*(2 + Log[5/4]) + I*Pi*(x^2 - 2*x*(1 + Log[5/4]) + Log[25/16]))*(1 + Lo
g[(x*(x + (-1 + x)*(I*Pi + Log[5/4])))/(I*Pi - x + Log[5/4])]))

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fricas [A]  time = 1.08, size = 46, normalized size = 1.21 \begin {gather*} -\frac {x - 2}{\log \left (\frac {{\left (i \, \pi + 1\right )} x^{2} - i \, \pi x + {\left (x^{2} - x\right )} \log \left (\frac {5}{4}\right )}{i \, \pi - x + \log \left (\frac {5}{4}\right )}\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*(log(5/4)+I*pi)^2+(x^3-2*x^2)*(log(5/4)+I*pi)+x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log
(5/4)+I*pi-x))+(x^2-4*x+2)*(log(5/4)+I*pi)^2+(2*x^2-4*x)*(log(5/4)+I*pi)+2*x^2)/(((x^2-x)*(log(5/4)+I*pi)^2+(-
x^3+2*x^2)*(log(5/4)+I*pi)-x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log(5/4)+I*pi-x))^2+((2*x^2-2*x)*(log(5/4)+
I*pi)^2+(-2*x^3+4*x^2)*(log(5/4)+I*pi)-2*x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log(5/4)+I*pi-x))+(x^2-x)*(lo
g(5/4)+I*pi)^2+(-x^3+2*x^2)*(log(5/4)+I*pi)-x^3),x, algorithm="fricas")

[Out]

-(x - 2)/(log(((I*pi + 1)*x^2 - I*pi*x + (x^2 - x)*log(5/4))/(I*pi - x + log(5/4))) + 1)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*(log(5/4)+I*pi)^2+(x^3-2*x^2)*(log(5/4)+I*pi)+x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log
(5/4)+I*pi-x))+(x^2-4*x+2)*(log(5/4)+I*pi)^2+(2*x^2-4*x)*(log(5/4)+I*pi)+2*x^2)/(((x^2-x)*(log(5/4)+I*pi)^2+(-
x^3+2*x^2)*(log(5/4)+I*pi)-x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log(5/4)+I*pi-x))^2+((2*x^2-2*x)*(log(5/4)+
I*pi)^2+(-2*x^3+4*x^2)*(log(5/4)+I*pi)-2*x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log(5/4)+I*pi-x))+(x^2-x)*(lo
g(5/4)+I*pi)^2+(-x^3+2*x^2)*(log(5/4)+I*pi)-x^3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 110.66, size = 51, normalized size = 1.34

method	result	size
risch	\(-\frac {x -2}{\ln \left (\frac {\left (x^{2}-x \right ) \left (\ln \relax (5)-2 \ln \relax (2)+i \pi \right )+x^{2}}{\ln \relax (5)-2 \ln \relax (2)+i \pi -x}\right )+1}\)	\(51\)
norman	\(\frac {\frac {4 i \pi \ln \relax (5)+2 \pi ^{2}+2 i \pi \ln \relax (5)^{2}-8 i \ln \relax (2) \pi \ln \relax (5)-8 i \pi \ln \relax (2)-4 \ln \relax (2) \pi ^{2}+2 \pi ^{2} \ln \relax (5)+2 i \pi ^{3}+8 i \ln \relax (2)^{2} \pi -16 \ln \relax (2)^{3}+24 \ln \relax (2)^{2} \ln \relax (5)-12 \ln \relax (2) \ln \relax (5)^{2}+2 \ln \relax (5)^{3}+2 i \pi +24 \ln \relax (2)^{2}-24 \ln \relax (2) \ln \relax (5)+6 \ln \relax (5)^{2}-12 \ln \relax (2)+6 \ln \relax (5)+2}{2 i \pi \ln \relax (5)+i \pi \ln \relax (5)^{2}-4 i \ln \relax (2) \pi \ln \relax (5)-4 i \pi \ln \relax (2)-2 \ln \relax (2) \pi ^{2}+\pi ^{2} \ln \relax (5)+i \pi ^{3}+4 i \ln \relax (2)^{2} \pi -8 \ln \relax (2)^{3}+12 \ln \relax (2)^{2} \ln \relax (5)-6 \ln \relax (2) \ln \relax (5)^{2}+\ln \relax (5)^{3}+\pi ^{2}+i \pi +12 \ln \relax (2)^{2}-12 \ln \relax (2) \ln \relax (5)+3 \ln \relax (5)^{2}-6 \ln \relax (2)+3 \ln \relax (5)+1}-x}{\ln \left (\frac {\left (x^{2}-x \right ) \left (\ln \left (\frac {5}{4}\right )+i \pi \right )+x^{2}}{\ln \left (\frac {5}{4}\right )+i \pi -x}\right )+1}\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+x)*(ln(5/4)+I*Pi)^2+(x^3-2*x^2)*(ln(5/4)+I*Pi)+x^3)*ln(((x^2-x)*(ln(5/4)+I*Pi)+x^2)/(ln(5/4)+I*Pi-
x))+(x^2-4*x+2)*(ln(5/4)+I*Pi)^2+(2*x^2-4*x)*(ln(5/4)+I*Pi)+2*x^2)/(((x^2-x)*(ln(5/4)+I*Pi)^2+(-x^3+2*x^2)*(ln
(5/4)+I*Pi)-x^3)*ln(((x^2-x)*(ln(5/4)+I*Pi)+x^2)/(ln(5/4)+I*Pi-x))^2+((2*x^2-2*x)*(ln(5/4)+I*Pi)^2+(-2*x^3+4*x
^2)*(ln(5/4)+I*Pi)-2*x^3)*ln(((x^2-x)*(ln(5/4)+I*Pi)+x^2)/(ln(5/4)+I*Pi-x))+(x^2-x)*(ln(5/4)+I*Pi)^2+(-x^3+2*x
^2)*(ln(5/4)+I*Pi)-x^3),x,method=_RETURNVERBOSE)

[Out]

-(x-2)/(ln(((x^2-x)*(ln(5)-2*ln(2)+I*Pi)+x^2)/(ln(5)-2*ln(2)+I*Pi-x))+1)

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maxima [A]  time = 18.75, size = 54, normalized size = 1.42 \begin {gather*} \frac {-i \, x + 2 i}{\log \left (i \, \pi + {\left (-i \, \pi - \log \relax (5) + 2 \, \log \relax (2) - 1\right )} x + \log \relax (5) - 2 \, \log \relax (2)\right ) - \log \left (-i \, \pi + x - \log \relax (5) + 2 \, \log \relax (2)\right ) + \log \relax (x) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+x)*(log(5/4)+I*pi)^2+(x^3-2*x^2)*(log(5/4)+I*pi)+x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log
(5/4)+I*pi-x))+(x^2-4*x+2)*(log(5/4)+I*pi)^2+(2*x^2-4*x)*(log(5/4)+I*pi)+2*x^2)/(((x^2-x)*(log(5/4)+I*pi)^2+(-
x^3+2*x^2)*(log(5/4)+I*pi)-x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log(5/4)+I*pi-x))^2+((2*x^2-2*x)*(log(5/4)+
I*pi)^2+(-2*x^3+4*x^2)*(log(5/4)+I*pi)-2*x^3)*log(((x^2-x)*(log(5/4)+I*pi)+x^2)/(log(5/4)+I*pi-x))+(x^2-x)*(lo
g(5/4)+I*pi)^2+(-x^3+2*x^2)*(log(5/4)+I*pi)-x^3),x, algorithm="maxima")

[Out]

(-I*x + 2*I)/(log(I*pi + (-I*pi - log(5) + 2*log(2) - 1)*x + log(5) - 2*log(2)) - log(-I*pi + x - log(5) + 2*l
og(2)) + log(x) + 1)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((x^2 - (x - x^2)*(Pi*1i + log(5/4)))/(Pi*1i - x + log(5/4)))*((x - x^2)*(Pi*1i + log(5/4))^2 - (Pi*1
i + log(5/4))*(2*x^2 - x^3) + x^3) - (4*x - 2*x^2)*(Pi*1i + log(5/4)) + (Pi*1i + log(5/4))^2*(x^2 - 4*x + 2) +
2*x^2)/((x - x^2)*(Pi*1i + log(5/4))^2 + log((x^2 - (x - x^2)*(Pi*1i + log(5/4)))/(Pi*1i - x + log(5/4)))^2*(
(x - x^2)*(Pi*1i + log(5/4))^2 - (Pi*1i + log(5/4))*(2*x^2 - x^3) + x^3) - (Pi*1i + log(5/4))*(2*x^2 - x^3) +
x^3 + log((x^2 - (x - x^2)*(Pi*1i + log(5/4)))/(Pi*1i - x + log(5/4)))*((2*x - 2*x^2)*(Pi*1i + log(5/4))^2 - (
Pi*1i + log(5/4))*(4*x^2 - 2*x^3) + 2*x^3)),x)

[Out]

\text{Hanged}

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+x)*(ln(5/4)+I*pi)**2+(x**3-2*x**2)*(ln(5/4)+I*pi)+x**3)*ln(((x**2-x)*(ln(5/4)+I*pi)+x**2)/(
ln(5/4)+I*pi-x))+(x**2-4*x+2)*(ln(5/4)+I*pi)**2+(2*x**2-4*x)*(ln(5/4)+I*pi)+2*x**2)/(((x**2-x)*(ln(5/4)+I*pi)*
*2+(-x**3+2*x**2)*(ln(5/4)+I*pi)-x**3)*ln(((x**2-x)*(ln(5/4)+I*pi)+x**2)/(ln(5/4)+I*pi-x))**2+((2*x**2-2*x)*(l
n(5/4)+I*pi)**2+(-2*x**3+4*x**2)*(ln(5/4)+I*pi)-2*x**3)*ln(((x**2-x)*(ln(5/4)+I*pi)+x**2)/(ln(5/4)+I*pi-x))+(x
**2-x)*(ln(5/4)+I*pi)**2+(-x**3+2*x**2)*(ln(5/4)+I*pi)-x**3),x)

[Out]

Timed out

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