3.38.62 \(\int \frac {6 x^2+6 e^x x^2+(-2 x^2+e^x (-2 x^2+x^3)) \log (x^2)+(16-3 x+e^x (16+x-3 x^2)) \log ^2(x^2)+(-2 x^2-2 e^x x^2+(x^2+e^x x^2) \log (x^2)+(-4-4 e^x) \log ^2(x^2)) \log (x+e^x x)}{9 x^4+9 e^x x^4+(72 x^2-54 x^3+e^x (72 x^2-54 x^3)) \log (x^2)+(144-216 x+81 x^2+e^x (144-216 x+81 x^2)) \log ^2(x^2)+(-6 x^4-6 e^x x^4+(-48 x^2+36 x^3+e^x (-48 x^2+36 x^3)) \log (x^2)+(-96+144 x-54 x^2+e^x (-96+144 x-54 x^2)) \log ^2(x^2)) \log (x+e^x x)+(x^4+e^x x^4+(8 x^2-6 x^3+e^x (8 x^2-6 x^3)) \log (x^2)+(16-24 x+9 x^2+e^x (16-24 x+9 x^2)) \log ^2(x^2)) \log ^2(x+e^x x)} \, dx\)

Optimal. Leaf size=33 \[ \frac {x}{\left (4-3 x+\frac {x^2}{\log \left (x^2\right )}\right ) \left (3-\log \left (x+e^x x\right )\right )} \]

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Rubi [F]  time = 21.55, 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 {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*x^2 + 6*E^x*x^2 + (-2*x^2 + E^x*(-2*x^2 + x^3))*Log[x^2] + (16 - 3*x + E^x*(16 + x - 3*x^2))*Log[x^2]^2
+ (-2*x^2 - 2*E^x*x^2 + (x^2 + E^x*x^2)*Log[x^2] + (-4 - 4*E^x)*Log[x^2]^2)*Log[x + E^x*x])/(9*x^4 + 9*E^x*x^
4 + (72*x^2 - 54*x^3 + E^x*(72*x^2 - 54*x^3))*Log[x^2] + (144 - 216*x + 81*x^2 + E^x*(144 - 216*x + 81*x^2))*L
og[x^2]^2 + (-6*x^4 - 6*E^x*x^4 + (-48*x^2 + 36*x^3 + E^x*(-48*x^2 + 36*x^3))*Log[x^2] + (-96 + 144*x - 54*x^2
+ E^x*(-96 + 144*x - 54*x^2))*Log[x^2]^2)*Log[x + E^x*x] + (x^4 + E^x*x^4 + (8*x^2 - 6*x^3 + E^x*(8*x^2 - 6*x
^3))*Log[x^2] + (16 - 24*x + 9*x^2 + E^x*(16 - 24*x + 9*x^2))*Log[x^2]^2)*Log[x + E^x*x]^2),x]

[Out]

-1/3*Defer[Int][(-3 + Log[(1 + E^x)*x])^(-2), x] - (7*Defer[Int][1/((-4 + 3*x)*(-3 + Log[(1 + E^x)*x])^2), x])
/3 - 4*Defer[Int][1/((-4 + 3*x)^2*(-3 + Log[(1 + E^x)*x])), x] - (128*Defer[Int][1/((-3 + Log[(1 + E^x)*x])*(x
^2 + 4*Log[x^2] - 3*x*Log[x^2])^2), x])/81 - (16*Defer[Int][x/((-3 + Log[(1 + E^x)*x])*(x^2 + 4*Log[x^2] - 3*x
*Log[x^2])^2), x])/27 - 2*Defer[Int][x^2/((-3 + Log[(1 + E^x)*x])*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])^2), x] + D
efer[Int][x^3/((-3 + Log[(1 + E^x)*x])*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])^2), x]/3 - (1024*Defer[Int][1/((-4 +
3*x)^2*(-3 + Log[(1 + E^x)*x])*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])^2), x])/81 - (256*Defer[Int][1/((-4 + 3*x)*(-
3 + Log[(1 + E^x)*x])*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])^2), x])/27 - (8*Defer[Int][1/((-3 + Log[(1 + E^x)*x])^
2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x])/27 + (16*Defer[Int][x/((-3 + Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] -
3*x*Log[x^2])), x])/9 + Defer[Int][x^2/((-3 + Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x]/3 -
(128*Defer[Int][1/((-4 + 3*x)^2*(-3 + Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x])/3 - (320*Def
er[Int][1/((-4 + 3*x)*(-3 + Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x])/27 - Defer[Int][(x*Log
[x^2])/((1 + E^x)*(-3 + Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x] + (4*Defer[Int][Log[x + E^x
*x]/((3 - Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x])/3 + (64*Defer[Int][Log[x + E^x*x]/((4 -
3*x)^2*(3 - Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x])/3 + (16*Defer[Int][Log[x + E^x*x]/((4
- 3*x)*(3 - Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x])/3 + (32*Defer[Int][Log[x + E^x*x]/((-4
+ 3*x)*(3 - Log[(1 + E^x)*x])^2*(x^2 + 4*Log[x^2] - 3*x*Log[x^2])), x])/3 + (8*Defer[Int][Log[x + E^x*x]/((3
- Log[(1 + E^x)*x])^2*(-x^2 - 4*Log[x^2] + 3*x*Log[x^2])), x])/9 + (64*Defer[Int][Log[x + E^x*x]/((4 - 3*x)^2*
(3 - Log[(1 + E^x)*x])^2*(-x^2 - 4*Log[x^2] + 3*x*Log[x^2])), x])/9 + Defer[Int][(x*Log[x + E^x*x])/((3 - Log[
(1 + E^x)*x])^2*(-x^2 - 4*Log[x^2] + 3*x*Log[x^2])), x]/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \left (1+e^x\right ) x^2+\left (-2+e^x (-2+x)\right ) x^2 \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (1+e^x\right ) \log \left (\left (1+e^x\right ) x\right ) \left (-2 x^2+x^2 \log \left (x^2\right )-4 \log ^2\left (x^2\right )\right )}{\left (1+e^x\right ) \left (3-\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+(4-3 x) \log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )}+\frac {6 x^2-2 x^2 \log \left (\left (1+e^x\right ) x\right )-2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )+x^2 \log \left (\left (1+e^x\right ) x\right ) \log \left (x^2\right )+16 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )-3 x^2 \log ^2\left (x^2\right )-4 \log \left (\left (1+e^x\right ) x\right ) \log ^2\left (x^2\right )}{\left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2}\right ) \, dx\\ &=-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx+\int \frac {6 x^2-2 x^2 \log \left (\left (1+e^x\right ) x\right )-2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )+x^2 \log \left (\left (1+e^x\right ) x\right ) \log \left (x^2\right )+16 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )-3 x^2 \log ^2\left (x^2\right )-4 \log \left (\left (1+e^x\right ) x\right ) \log ^2\left (x^2\right )}{\left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2} \, dx\\ &=-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx+\int \frac {6 x^2+(-2+x) x^2 \log \left (x^2\right )+\left (16+x-3 x^2\right ) \log ^2\left (x^2\right )+\log \left (\left (1+e^x\right ) x\right ) \left (-2 x^2+x^2 \log \left (x^2\right )-4 \log ^2\left (x^2\right )\right )}{\left (3-\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+(4-3 x) \log \left (x^2\right )\right )^2} \, dx\\ &=-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx+\int \left (\frac {16+x-3 x^2-4 \log \left (\left (1+e^x\right ) x\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2}+\frac {x^2 \left (-32+48 x-26 x^2+3 x^3\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right ) \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2}+\frac {x^2 \left (-40+8 x+3 x^2+12 \log \left (\left (1+e^x\right ) x\right )-3 x \log \left (\left (1+e^x\right ) x\right )\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )}\right ) \, dx\\ &=\int \frac {16+x-3 x^2-4 \log \left (\left (1+e^x\right ) x\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2} \, dx+\int \frac {x^2 \left (-32+48 x-26 x^2+3 x^3\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right ) \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )^2} \, dx+\int \frac {x^2 \left (-40+8 x+3 x^2+12 \log \left (\left (1+e^x\right ) x\right )-3 x \log \left (\left (1+e^x\right ) x\right )\right )}{(-4+3 x)^2 \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx-\int \frac {x \log \left (x^2\right )}{\left (1+e^x\right ) \left (-3+\log \left (\left (1+e^x\right ) x\right )\right )^2 \left (x^2+4 \log \left (x^2\right )-3 x \log \left (x^2\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 35, normalized size = 1.06 \begin {gather*} -\frac {x \log \left (x^2\right )}{\left (-3+\log \left (\left (1+e^x\right ) x\right )\right ) \left (x^2+(4-3 x) \log \left (x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x^2 + 6*E^x*x^2 + (-2*x^2 + E^x*(-2*x^2 + x^3))*Log[x^2] + (16 - 3*x + E^x*(16 + x - 3*x^2))*Log[
x^2]^2 + (-2*x^2 - 2*E^x*x^2 + (x^2 + E^x*x^2)*Log[x^2] + (-4 - 4*E^x)*Log[x^2]^2)*Log[x + E^x*x])/(9*x^4 + 9*
E^x*x^4 + (72*x^2 - 54*x^3 + E^x*(72*x^2 - 54*x^3))*Log[x^2] + (144 - 216*x + 81*x^2 + E^x*(144 - 216*x + 81*x
^2))*Log[x^2]^2 + (-6*x^4 - 6*E^x*x^4 + (-48*x^2 + 36*x^3 + E^x*(-48*x^2 + 36*x^3))*Log[x^2] + (-96 + 144*x -
54*x^2 + E^x*(-96 + 144*x - 54*x^2))*Log[x^2]^2)*Log[x + E^x*x] + (x^4 + E^x*x^4 + (8*x^2 - 6*x^3 + E^x*(8*x^2
- 6*x^3))*Log[x^2] + (16 - 24*x + 9*x^2 + E^x*(16 - 24*x + 9*x^2))*Log[x^2]^2)*Log[x + E^x*x]^2),x]

[Out]

-((x*Log[x^2])/((-3 + Log[(1 + E^x)*x])*(x^2 + (4 - 3*x)*Log[x^2])))

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fricas [A]  time = 0.95, size = 49, normalized size = 1.48 \begin {gather*} \frac {x \log \left (x^{2}\right )}{3 \, x^{2} - 3 \, {\left (3 \, x - 4\right )} \log \left (x^{2}\right ) - {\left (x^{2} - {\left (3 \, x - 4\right )} \log \left (x^{2}\right )\right )} \log \left (x e^{x} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)-4)*log(x^2)^2+(exp(x)*x^2+x^2)*log(x^2)-2*exp(x)*x^2-2*x^2)*log(exp(x)*x+x)+((-3*x^2+x+
16)*exp(x)-3*x+16)*log(x^2)^2+((x^3-2*x^2)*exp(x)-2*x^2)*log(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x
)+9*x^2-24*x+16)*log(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*log(x^2)+exp(x)*x^4+x^4)*log(exp(x)*x+x)^2+(((
-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96)*log(x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*log(x^2)-6*exp(x)*
x^4-6*x^4)*log(exp(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*log(x^2)^2+((-54*x^3+72*x^2)*exp(x)-54
*x^3+72*x^2)*log(x^2)+9*exp(x)*x^4+9*x^4),x, algorithm="fricas")

[Out]

x*log(x^2)/(3*x^2 - 3*(3*x - 4)*log(x^2) - (x^2 - (3*x - 4)*log(x^2))*log(x*e^x + x))

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giac [B]  time = 3.23, size = 69, normalized size = 2.09 \begin {gather*} -\frac {2 \, x \log \relax (x)}{x^{2} \log \relax (x) - 6 \, x \log \relax (x)^{2} + x^{2} \log \left (e^{x} + 1\right ) - 6 \, x \log \relax (x) \log \left (e^{x} + 1\right ) - 3 \, x^{2} + 18 \, x \log \relax (x) + 8 \, \log \relax (x)^{2} + 8 \, \log \relax (x) \log \left (e^{x} + 1\right ) - 24 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)-4)*log(x^2)^2+(exp(x)*x^2+x^2)*log(x^2)-2*exp(x)*x^2-2*x^2)*log(exp(x)*x+x)+((-3*x^2+x+
16)*exp(x)-3*x+16)*log(x^2)^2+((x^3-2*x^2)*exp(x)-2*x^2)*log(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x
)+9*x^2-24*x+16)*log(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*log(x^2)+exp(x)*x^4+x^4)*log(exp(x)*x+x)^2+(((
-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96)*log(x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*log(x^2)-6*exp(x)*
x^4-6*x^4)*log(exp(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*log(x^2)^2+((-54*x^3+72*x^2)*exp(x)-54
*x^3+72*x^2)*log(x^2)+9*exp(x)*x^4+9*x^4),x, algorithm="giac")

[Out]

-2*x*log(x)/(x^2*log(x) - 6*x*log(x)^2 + x^2*log(e^x + 1) - 6*x*log(x)*log(e^x + 1) - 3*x^2 + 18*x*log(x) + 8*
log(x)^2 + 8*log(x)*log(e^x + 1) - 24*log(x))

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maple [C]  time = 30.77, size = 266, normalized size = 8.06

method	result	size
risch	\(\frac {2 i x \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )\right )}{\left (3 \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-6 \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+3 \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}-4 \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+8 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i x^{2}+12 i x \ln \relax (x )-16 i \ln \relax (x )\right ) \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )^{3}+2 i \ln \relax (x )+2 i \ln \left ({\mathrm e}^{x}+1\right )-6 i\right )}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*exp(x)-4)*ln(x^2)^2+(exp(x)*x^2+x^2)*ln(x^2)-2*exp(x)*x^2-2*x^2)*ln(exp(x)*x+x)+((-3*x^2+x+16)*exp(x
)-3*x+16)*ln(x^2)^2+((x^3-2*x^2)*exp(x)-2*x^2)*ln(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x)+9*x^2-24*
x+16)*ln(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*ln(x^2)+exp(x)*x^4+x^4)*ln(exp(x)*x+x)^2+(((-54*x^2+144*x-
96)*exp(x)-54*x^2+144*x-96)*ln(x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*ln(x^2)-6*exp(x)*x^4-6*x^4)*ln(ex
p(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*ln(x^2)^2+((-54*x^3+72*x^2)*exp(x)-54*x^3+72*x^2)*ln(x^
2)+9*exp(x)*x^4+9*x^4),x,method=_RETURNVERBOSE)

[Out]

2*I*x*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+4*I*ln(x))/(3*Pi*x*csgn(I*x)^2
*csgn(I*x^2)-6*Pi*x*csgn(I*x)*csgn(I*x^2)^2+3*Pi*x*csgn(I*x^2)^3-4*Pi*csgn(I*x)^2*csgn(I*x^2)+8*Pi*csgn(I*x)*c
sgn(I*x^2)^2-4*Pi*csgn(I*x^2)^3-2*I*x^2+12*I*x*ln(x)-16*I*ln(x))/(Pi*csgn(I*x)*csgn(I*(exp(x)+1))*csgn(I*x*(ex
p(x)+1))-Pi*csgn(I*x)*csgn(I*x*(exp(x)+1))^2-Pi*csgn(I*(exp(x)+1))*csgn(I*x*(exp(x)+1))^2+Pi*csgn(I*x*(exp(x)+
1))^3+2*I*ln(x)+2*I*ln(exp(x)+1)-6*I)

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maxima [A]  time = 1.29, size = 56, normalized size = 1.70 \begin {gather*} \frac {2 \, x \log \relax (x)}{2 \, {\left (3 \, x - 4\right )} \log \relax (x)^{2} + 3 \, x^{2} - {\left (x^{2} + 18 \, x - 24\right )} \log \relax (x) - {\left (x^{2} - 2 \, {\left (3 \, x - 4\right )} \log \relax (x)\right )} \log \left (e^{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)-4)*log(x^2)^2+(exp(x)*x^2+x^2)*log(x^2)-2*exp(x)*x^2-2*x^2)*log(exp(x)*x+x)+((-3*x^2+x+
16)*exp(x)-3*x+16)*log(x^2)^2+((x^3-2*x^2)*exp(x)-2*x^2)*log(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x
)+9*x^2-24*x+16)*log(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*log(x^2)+exp(x)*x^4+x^4)*log(exp(x)*x+x)^2+(((
-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96)*log(x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*log(x^2)-6*exp(x)*
x^4-6*x^4)*log(exp(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*log(x^2)^2+((-54*x^3+72*x^2)*exp(x)-54
*x^3+72*x^2)*log(x^2)+9*exp(x)*x^4+9*x^4),x, algorithm="maxima")

[Out]

2*x*log(x)/(2*(3*x - 4)*log(x)^2 + 3*x^2 - (x^2 + 18*x - 24)*log(x) - (x^2 - 2*(3*x - 4)*log(x))*log(e^x + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {6\,x^2\,{\mathrm {e}}^x-\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (2\,x^2-x^3\right )+2\,x^2\right )+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (-3\,x^2+x+16\right )-3\,x+16\right )-\ln \left (x+x\,{\mathrm {e}}^x\right )\,\left (2\,x^2\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left (4\,{\mathrm {e}}^x+4\right )+2\,x^2-\ln \left (x^2\right )\,\left (x^2\,{\mathrm {e}}^x+x^2\right )\right )+6\,x^2}{9\,x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (81\,x^2-216\,x+144\right )-216\,x+81\,x^2+144\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (72\,x^2-54\,x^3\right )+72\,x^2-54\,x^3\right )-\ln \left (x+x\,{\mathrm {e}}^x\right )\,\left (6\,x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (54\,x^2-144\,x+96\right )-144\,x+54\,x^2+96\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (48\,x^2-36\,x^3\right )+48\,x^2-36\,x^3\right )+6\,x^4\right )+{\ln \left (x+x\,{\mathrm {e}}^x\right )}^2\,\left (x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (9\,x^2-24\,x+16\right )-24\,x+9\,x^2+16\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (8\,x^2-6\,x^3\right )+8\,x^2-6\,x^3\right )+x^4\right )+9\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^2*exp(x) - log(x^2)*(exp(x)*(2*x^2 - x^3) + 2*x^2) + log(x^2)^2*(exp(x)*(x - 3*x^2 + 16) - 3*x + 16)
- log(x + x*exp(x))*(2*x^2*exp(x) + log(x^2)^2*(4*exp(x) + 4) + 2*x^2 - log(x^2)*(x^2*exp(x) + x^2)) + 6*x^2)/
(9*x^4*exp(x) + log(x^2)^2*(exp(x)*(81*x^2 - 216*x + 144) - 216*x + 81*x^2 + 144) + log(x^2)*(exp(x)*(72*x^2 -
54*x^3) + 72*x^2 - 54*x^3) - log(x + x*exp(x))*(6*x^4*exp(x) + log(x^2)^2*(exp(x)*(54*x^2 - 144*x + 96) - 144
*x + 54*x^2 + 96) + log(x^2)*(exp(x)*(48*x^2 - 36*x^3) + 48*x^2 - 36*x^3) + 6*x^4) + log(x + x*exp(x))^2*(x^4*
exp(x) + log(x^2)^2*(exp(x)*(9*x^2 - 24*x + 16) - 24*x + 9*x^2 + 16) + log(x^2)*(exp(x)*(8*x^2 - 6*x^3) + 8*x^
2 - 6*x^3) + x^4) + 9*x^4),x)

[Out]

int((6*x^2*exp(x) - log(x^2)*(exp(x)*(2*x^2 - x^3) + 2*x^2) + log(x^2)^2*(exp(x)*(x - 3*x^2 + 16) - 3*x + 16)
- log(x + x*exp(x))*(2*x^2*exp(x) + log(x^2)^2*(4*exp(x) + 4) + 2*x^2 - log(x^2)*(x^2*exp(x) + x^2)) + 6*x^2)/
(9*x^4*exp(x) + log(x^2)^2*(exp(x)*(81*x^2 - 216*x + 144) - 216*x + 81*x^2 + 144) + log(x^2)*(exp(x)*(72*x^2 -
54*x^3) + 72*x^2 - 54*x^3) - log(x + x*exp(x))*(6*x^4*exp(x) + log(x^2)^2*(exp(x)*(54*x^2 - 144*x + 96) - 144
*x + 54*x^2 + 96) + log(x^2)*(exp(x)*(48*x^2 - 36*x^3) + 48*x^2 - 36*x^3) + 6*x^4) + log(x + x*exp(x))^2*(x^4*
exp(x) + log(x^2)^2*(exp(x)*(9*x^2 - 24*x + 16) - 24*x + 9*x^2 + 16) + log(x^2)*(exp(x)*(8*x^2 - 6*x^3) + 8*x^
2 - 6*x^3) + x^4) + 9*x^4), x)

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sympy [B]  time = 0.63, size = 54, normalized size = 1.64 \begin {gather*} - \frac {x \log {\left (x^{2} \right )}}{- 3 x^{2} + 9 x \log {\left (x^{2} \right )} + \left (x^{2} - 3 x \log {\left (x^{2} \right )} + 4 \log {\left (x^{2} \right )}\right ) \log {\left (x e^{x} + x \right )} - 12 \log {\left (x^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*exp(x)-4)*ln(x**2)**2+(exp(x)*x**2+x**2)*ln(x**2)-2*exp(x)*x**2-2*x**2)*ln(exp(x)*x+x)+((-3*x*
*2+x+16)*exp(x)-3*x+16)*ln(x**2)**2+((x**3-2*x**2)*exp(x)-2*x**2)*ln(x**2)+6*exp(x)*x**2+6*x**2)/((((9*x**2-24
*x+16)*exp(x)+9*x**2-24*x+16)*ln(x**2)**2+((-6*x**3+8*x**2)*exp(x)-6*x**3+8*x**2)*ln(x**2)+exp(x)*x**4+x**4)*l
n(exp(x)*x+x)**2+(((-54*x**2+144*x-96)*exp(x)-54*x**2+144*x-96)*ln(x**2)**2+((36*x**3-48*x**2)*exp(x)+36*x**3-
48*x**2)*ln(x**2)-6*exp(x)*x**4-6*x**4)*ln(exp(x)*x+x)+((81*x**2-216*x+144)*exp(x)+81*x**2-216*x+144)*ln(x**2)
**2+((-54*x**3+72*x**2)*exp(x)-54*x**3+72*x**2)*ln(x**2)+9*exp(x)*x**4+9*x**4),x)

[Out]

-x*log(x**2)/(-3*x**2 + 9*x*log(x**2) + (x**2 - 3*x*log(x**2) + 4*log(x**2))*log(x*exp(x) + x) - 12*log(x**2))

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