3.86.1 \(\int \frac {-64 x-60 x \log (x)+e^x (30 x-32 x^2) \log ^2(x)+(64 x+128 x \log (x)+e^x (-64 x+32 x^2) \log ^2(x)) \log (\frac {2-e^x \log (x)}{\log (x)})+(-64 x \log (x)+32 e^x x \log ^2(x)) \log ^2(\frac {2-e^x \log (x)}{\log (x)})}{-2 \log (x)+e^x \log ^2(x)} \, dx\)

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

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Rubi [F]  time = 3.93, 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 {-64 x-60 x \log (x)+e^x \left (30 x-32 x^2\right ) \log ^2(x)+\left (64 x+128 x \log (x)+e^x \left (-64 x+32 x^2\right ) \log ^2(x)\right ) \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\left (-64 x \log (x)+32 e^x x \log ^2(x)\right ) \log ^2\left (\frac {2-e^x \log (x)}{\log (x)}\right )}{-2 \log (x)+e^x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-64*x - 60*x*Log[x] + E^x*(30*x - 32*x^2)*Log[x]^2 + (64*x + 128*x*Log[x] + E^x*(-64*x + 32*x^2)*Log[x]^2
)*Log[(2 - E^x*Log[x])/Log[x]] + (-64*x*Log[x] + 32*E^x*x*Log[x]^2)*Log[(2 - E^x*Log[x])/Log[x]]^2)/(-2*Log[x]
+ E^x*Log[x]^2),x]

[Out]

15*x^2 - (8*x^4)/3 - 32*x^2*Log[(2 - E^x*Log[x])/Log[x]] + (32*x^3*Log[(2 - E^x*Log[x])/Log[x]])/3 - (64*Defer
[Int][x^3/(-2 + E^x*Log[x]), x])/3 - (64*Defer[Int][x^2/(Log[x]*(-2 + E^x*Log[x])), x])/3 + 64*Defer[Int][(x^2
*Log[-((-2 + E^x*Log[x])/Log[x])])/(-2 + E^x*Log[x]), x] + 64*Defer[Int][(x*Log[-((-2 + E^x*Log[x])/Log[x])])/
(Log[x]*(-2 + E^x*Log[x])), x] + 32*Defer[Int][x*Log[-((-2 + E^x*Log[x])/Log[x])]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {64 x+60 x \log (x)-e^x \left (30 x-32 x^2\right ) \log ^2(x)-\left (64 x+128 x \log (x)+e^x \left (-64 x+32 x^2\right ) \log ^2(x)\right ) \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\left (-64 x \log (x)+32 e^x x \log ^2(x)\right ) \log ^2\left (\frac {2-e^x \log (x)}{\log (x)}\right )}{\log (x) \left (2-e^x \log (x)\right )} \, dx\\ &=\int \left (\frac {64 x (1+x \log (x)) \left (-1+\log \left (-e^x+\frac {2}{\log (x)}\right )\right )}{\log (x) \left (-2+e^x \log (x)\right )}+2 x \left (15-16 x-32 \log \left (-e^x+\frac {2}{\log (x)}\right )+16 x \log \left (-e^x+\frac {2}{\log (x)}\right )+16 \log ^2\left (-e^x+\frac {2}{\log (x)}\right )\right )\right ) \, dx\\ &=2 \int x \left (15-16 x-32 \log \left (-e^x+\frac {2}{\log (x)}\right )+16 x \log \left (-e^x+\frac {2}{\log (x)}\right )+16 \log ^2\left (-e^x+\frac {2}{\log (x)}\right )\right ) \, dx+64 \int \frac {x (1+x \log (x)) \left (-1+\log \left (-e^x+\frac {2}{\log (x)}\right )\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=2 \int \left (-x (-15+16 x)+16 (-2+x) x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )+16 x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right )\right ) \, dx+64 \int \left (-\frac {x^2}{-2+e^x \log (x)}-\frac {x}{\log (x) \left (-2+e^x \log (x)\right )}+\frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)}+\frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )}\right ) \, dx\\ &=-(2 \int x (-15+16 x) \, dx)+32 \int (-2+x) x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-2 \int \left (-15 x+16 x^2\right ) \, dx-32 \int \frac {(3-x) x \left (2+e^x x \log ^2(x)\right )}{3 \log (x) \left (2-e^x \log (x)\right )} \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=15 x^2-\frac {32 x^3}{3}-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\frac {32}{3} \int \frac {(3-x) x \left (2+e^x x \log ^2(x)\right )}{\log (x) \left (2-e^x \log (x)\right )} \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=15 x^2-\frac {32 x^3}{3}-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\frac {32}{3} \int \left ((-3+x) x^2+\frac {2 (-3+x) x (1+x \log (x))}{\log (x) \left (-2+e^x \log (x)\right )}\right ) \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=15 x^2-\frac {32 x^3}{3}-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\frac {32}{3} \int (-3+x) x^2 \, dx-\frac {64}{3} \int \frac {(-3+x) x (1+x \log (x))}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=15 x^2-\frac {32 x^3}{3}-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\frac {32}{3} \int \left (-3 x^2+x^3\right ) \, dx-\frac {64}{3} \int \left (-\frac {3 x (1+x \log (x))}{\log (x) \left (-2+e^x \log (x)\right )}+\frac {x^2 (1+x \log (x))}{\log (x) \left (-2+e^x \log (x)\right )}\right ) \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=15 x^2-\frac {8 x^4}{3}-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\frac {64}{3} \int \frac {x^2 (1+x \log (x))}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x (1+x \log (x))}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=15 x^2-\frac {8 x^4}{3}-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\frac {64}{3} \int \left (\frac {x^3}{-2+e^x \log (x)}+\frac {x^2}{\log (x) \left (-2+e^x \log (x)\right )}\right ) \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx-64 \int \frac {x^2}{-2+e^x \log (x)} \, dx-64 \int \frac {x}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+64 \int \left (\frac {x^2}{-2+e^x \log (x)}+\frac {x}{\log (x) \left (-2+e^x \log (x)\right )}\right ) \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ &=15 x^2-\frac {8 x^4}{3}-32 x^2 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )+\frac {32}{3} x^3 \log \left (\frac {2-e^x \log (x)}{\log (x)}\right )-\frac {64}{3} \int \frac {x^3}{-2+e^x \log (x)} \, dx-\frac {64}{3} \int \frac {x^2}{\log (x) \left (-2+e^x \log (x)\right )} \, dx+32 \int x \log ^2\left (-\frac {-2+e^x \log (x)}{\log (x)}\right ) \, dx+64 \int \frac {x^2 \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{-2+e^x \log (x)} \, dx+64 \int \frac {x \log \left (-\frac {-2+e^x \log (x)}{\log (x)}\right )}{\log (x) \left (-2+e^x \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 48, normalized size = 1.71 \begin {gather*} 2 \left (\frac {15 x^2}{2}-16 x^2 \log \left (-e^x+\frac {2}{\log (x)}\right )+8 x^2 \log ^2\left (-e^x+\frac {2}{\log (x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-64*x - 60*x*Log[x] + E^x*(30*x - 32*x^2)*Log[x]^2 + (64*x + 128*x*Log[x] + E^x*(-64*x + 32*x^2)*Lo
g[x]^2)*Log[(2 - E^x*Log[x])/Log[x]] + (-64*x*Log[x] + 32*E^x*x*Log[x]^2)*Log[(2 - E^x*Log[x])/Log[x]]^2)/(-2*
Log[x] + E^x*Log[x]^2),x]

[Out]

2*((15*x^2)/2 - 16*x^2*Log[-E^x + 2/Log[x]] + 8*x^2*Log[-E^x + 2/Log[x]]^2)

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fricas [A]  time = 0.58, size = 46, normalized size = 1.64 \begin {gather*} 16 \, x^{2} \log \left (-\frac {e^{x} \log \relax (x) - 2}{\log \relax (x)}\right )^{2} - 32 \, x^{2} \log \left (-\frac {e^{x} \log \relax (x) - 2}{\log \relax (x)}\right ) + 15 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)*log(x)^2-64*x*log(x))*log((-exp(x)*log(x)+2)/log(x))^2+((32*x^2-64*x)*exp(x)*log(x)^2+
128*x*log(x)+64*x)*log((-exp(x)*log(x)+2)/log(x))+(-32*x^2+30*x)*exp(x)*log(x)^2-60*x*log(x)-64*x)/(exp(x)*log
(x)^2-2*log(x)),x, algorithm="fricas")

[Out]

16*x^2*log(-(e^x*log(x) - 2)/log(x))^2 - 32*x^2*log(-(e^x*log(x) - 2)/log(x)) + 15*x^2

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giac [B]  time = 0.33, size = 71, normalized size = 2.54 \begin {gather*} 16 \, x^{2} \log \left (-e^{x} \log \relax (x) + 2\right )^{2} - 32 \, x^{2} \log \left (-e^{x} \log \relax (x) + 2\right ) \log \left (\log \relax (x)\right ) + 16 \, x^{2} \log \left (\log \relax (x)\right )^{2} - 32 \, x^{2} \log \left (-e^{x} \log \relax (x) + 2\right ) + 32 \, x^{2} \log \left (\log \relax (x)\right ) + 15 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)*log(x)^2-64*x*log(x))*log((-exp(x)*log(x)+2)/log(x))^2+((32*x^2-64*x)*exp(x)*log(x)^2+
128*x*log(x)+64*x)*log((-exp(x)*log(x)+2)/log(x))+(-32*x^2+30*x)*exp(x)*log(x)^2-60*x*log(x)-64*x)/(exp(x)*log
(x)^2-2*log(x)),x, algorithm="giac")

[Out]

16*x^2*log(-e^x*log(x) + 2)^2 - 32*x^2*log(-e^x*log(x) + 2)*log(log(x)) + 16*x^2*log(log(x))^2 - 32*x^2*log(-e
^x*log(x) + 2) + 32*x^2*log(log(x)) + 15*x^2

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maple [C]  time = 0.27, size = 1209, normalized size = 43.18

method	result	size
risch	\(-16 \pi ^{2} x^{2}+32 x^{2} \ln \left (\ln \relax (x )\right )+15 x^{2}+16 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{4}+16 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{4} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )+32 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2}-4 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{4} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )^{2}-16 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2}-16 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-4 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right )^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{4}-8 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{5}-8 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{5} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-32 i \pi \,x^{2} \ln \left (\ln \relax (x )\right )-16 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{3}+16 x^{2} \ln \left (\ln \relax (x )\right )^{2}+32 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2}-4 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{6}-16 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{3}-16 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{4}+16 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{5}+16 x^{2} \ln \left ({\mathrm e}^{x} \ln \relax (x )-2\right )^{2}+32 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2} \ln \left (\ln \relax (x )\right )+8 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right )^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{3} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-4 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right )^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )^{2}-16 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{3} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )+16 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-16 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{3} \ln \left (\ln \relax (x )\right )+\left (-32 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2}+16 i \pi \,x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2}-16 i \pi \,x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )+16 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{3}+16 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )+32 i \pi \,x^{2}-32 x^{2} \ln \left (\ln \relax (x )\right )-32 x^{2}\right ) \ln \left ({\mathrm e}^{x} \ln \relax (x )-2\right )+16 i \pi \,x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \ln \left (\ln \relax (x )\right )-32 i \pi \,x^{2}-16 i \pi \,x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2}-16 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )+8 \pi ^{2} x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{3} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )^{2}+16 i \pi \,x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )-16 i \pi \,x^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2} \ln \left (\ln \relax (x )\right )-16 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x} \ln \relax (x )-2\right )}{\ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \ln \left (\ln \relax (x )\right )\)	\(1209\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x*exp(x)*ln(x)^2-64*x*ln(x))*ln((-exp(x)*ln(x)+2)/ln(x))^2+((32*x^2-64*x)*exp(x)*ln(x)^2+128*x*ln(x)+
64*x)*ln((-exp(x)*ln(x)+2)/ln(x))+(-32*x^2+30*x)*exp(x)*ln(x)^2-60*x*ln(x)-64*x)/(exp(x)*ln(x)^2-2*ln(x)),x,me
thod=_RETURNVERBOSE)

[Out]

-16*Pi^2*x^2+32*x^2*ln(ln(x))+15*x^2+16*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))^4+16*
Pi^2*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^4*csgn(I/ln(x))+32*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2-4*Pi^2*x^
2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^4*csgn(I/ln(x))^2-16*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*l
n(x)-2))^2-16*Pi^2*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2*csgn(I/ln(x))-4*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))^2*cs
gn(I/ln(x)*(exp(x)*ln(x)-2))^4-8*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))^5-8*Pi^2*x^2
*csgn(I/ln(x)*(exp(x)*ln(x)-2))^5*csgn(I/ln(x))-32*I*x^2*Pi+16*x^2*ln(ln(x))^2-32*I*Pi*x^2*ln(ln(x))+16*x^2*ln
(exp(x)*ln(x)-2)^2+16*I*Pi*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))*csgn(I/ln(x))*ln(ln(x))
+(-32*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2+16*I*Pi*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x
)-2))^2-16*I*Pi*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))*csgn(I/ln(x))+16*I*Pi*x^2*csgn(I/l
n(x)*(exp(x)*ln(x)-2))^3+16*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2*csgn(I/ln(x))+32*I*x^2*Pi-32*x^2*ln(ln(x
))-32*x^2)*ln(exp(x)*ln(x)-2)+32*Pi^2*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2-4*Pi^2*x^2*csgn(I/ln(x)*(exp(x)*ln(
x)-2))^6-16*Pi^2*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^3-16*Pi^2*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^4+16*Pi^2*x^2
*csgn(I/ln(x)*(exp(x)*ln(x)-2))^5+8*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))^3*csgn(I/
ln(x))^2-16*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^3-16*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2*csgn(I/ln(x
))+32*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2*ln(ln(x))+8*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))^2*csgn(I/ln(x)*(
exp(x)*ln(x)-2))^3*csgn(I/ln(x))-4*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2*csgn(I
/ln(x))^2-16*Pi^2*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))^3*csgn(I/ln(x))+16*Pi^2*x^2*csgn
(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))*csgn(I/ln(x))-16*I*Pi*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/
ln(x)*(exp(x)*ln(x)-2))^2-16*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^3*ln(ln(x))-16*I*Pi*x^2*csgn(I*(exp(x)*ln
(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2*ln(ln(x))-16*I*Pi*x^2*csgn(I/ln(x)*(exp(x)*ln(x)-2))^2*csgn(I/ln(x))*
ln(ln(x))+16*I*Pi*x^2*csgn(I*(exp(x)*ln(x)-2))*csgn(I/ln(x)*(exp(x)*ln(x)-2))*csgn(I/ln(x))

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maxima [B]  time = 0.42, size = 62, normalized size = 2.21 \begin {gather*} 16 \, x^{2} \log \left (-e^{x} \log \relax (x) + 2\right )^{2} + 16 \, x^{2} \log \left (\log \relax (x)\right )^{2} + 32 \, x^{2} \log \left (\log \relax (x)\right ) + 15 \, x^{2} - 32 \, {\left (x^{2} \log \left (\log \relax (x)\right ) + x^{2}\right )} \log \left (-e^{x} \log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)*log(x)^2-64*x*log(x))*log((-exp(x)*log(x)+2)/log(x))^2+((32*x^2-64*x)*exp(x)*log(x)^2+
128*x*log(x)+64*x)*log((-exp(x)*log(x)+2)/log(x))+(-32*x^2+30*x)*exp(x)*log(x)^2-60*x*log(x)-64*x)/(exp(x)*log
(x)^2-2*log(x)),x, algorithm="maxima")

[Out]

16*x^2*log(-e^x*log(x) + 2)^2 + 16*x^2*log(log(x))^2 + 32*x^2*log(log(x)) + 15*x^2 - 32*(x^2*log(log(x)) + x^2
)*log(-e^x*log(x) + 2)

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mupad [B]  time = 5.59, size = 46, normalized size = 1.64 \begin {gather*} 16\,x^2\,{\ln \left (-\frac {{\mathrm {e}}^x\,\ln \relax (x)-2}{\ln \relax (x)}\right )}^2-32\,x^2\,\ln \left (-\frac {{\mathrm {e}}^x\,\ln \relax (x)-2}{\ln \relax (x)}\right )+15\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((64*x + log(-(exp(x)*log(x) - 2)/log(x))^2*(64*x*log(x) - 32*x*exp(x)*log(x)^2) + 60*x*log(x) - log(-(exp(
x)*log(x) - 2)/log(x))*(64*x + 128*x*log(x) - exp(x)*log(x)^2*(64*x - 32*x^2)) - exp(x)*log(x)^2*(30*x - 32*x^
2))/(2*log(x) - exp(x)*log(x)^2),x)

[Out]

16*x^2*log(-(exp(x)*log(x) - 2)/log(x))^2 - 32*x^2*log(-(exp(x)*log(x) - 2)/log(x)) + 15*x^2

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sympy [B]  time = 0.96, size = 42, normalized size = 1.50 \begin {gather*} 16 x^{2} \log {\left (\frac {- e^{x} \log {\relax (x )} + 2}{\log {\relax (x )}} \right )}^{2} - 32 x^{2} \log {\left (\frac {- e^{x} \log {\relax (x )} + 2}{\log {\relax (x )}} \right )} + 15 x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)*ln(x)**2-64*x*ln(x))*ln((-exp(x)*ln(x)+2)/ln(x))**2+((32*x**2-64*x)*exp(x)*ln(x)**2+12
8*x*ln(x)+64*x)*ln((-exp(x)*ln(x)+2)/ln(x))+(-32*x**2+30*x)*exp(x)*ln(x)**2-60*x*ln(x)-64*x)/(exp(x)*ln(x)**2-
2*ln(x)),x)

[Out]

16*x**2*log((-exp(x)*log(x) + 2)/log(x))**2 - 32*x**2*log((-exp(x)*log(x) + 2)/log(x)) + 15*x**2

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