3.101.73 \(\int \frac {-x^2-x^3+x^4+x^5+(-2 x-2 x^2+2 x^3+2 x^4) \log (4)+(-1-x+x^2+x^3) \log ^2(4)+e^x (-2 x^2-4 x^3-2 x^4+(-4 x-8 x^2-4 x^3) \log (4)+(-2-4 x-2 x^2) \log ^2(4))+(2 x^2+6 x^3+4 x^4+(2 x+8 x^2+6 x^3) \log (4)+(2 x+2 x^2) \log ^2(4)+e^x (4 x^2+12 x^3+8 x^4+(4 x+16 x^2+12 x^3) \log (4)+(4 x+4 x^2) \log ^2(4))) \log (e^{-x} (x+2 e^x x))}{(x+2 e^x x) \log ^2(e^{-x} (x+2 e^x x))} \, dx\)

Optimal. Leaf size=26 \[ \frac {(1+x)^2 (x+\log (4))^2}{\log \left (2 x+e^{-x} x\right )} \]

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Rubi [F]  time = 7.14, 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 {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 - x^3 + x^4 + x^5 + (-2*x - 2*x^2 + 2*x^3 + 2*x^4)*Log[4] + (-1 - x + x^2 + x^3)*Log[4]^2 + E^x*(-2*
x^2 - 4*x^3 - 2*x^4 + (-4*x - 8*x^2 - 4*x^3)*Log[4] + (-2 - 4*x - 2*x^2)*Log[4]^2) + (2*x^2 + 6*x^3 + 4*x^4 +
(2*x + 8*x^2 + 6*x^3)*Log[4] + (2*x + 2*x^2)*Log[4]^2 + E^x*(4*x^2 + 12*x^3 + 8*x^4 + (4*x + 16*x^2 + 12*x^3)*
Log[4] + (4*x + 4*x^2)*Log[4]^2))*Log[(x + 2*E^x*x)/E^x])/((x + 2*E^x*x)*Log[(x + 2*E^x*x)/E^x]^2),x]

[Out]

-2*Log[4]*(1 + Log[4])*Defer[Int][Log[2*x + x/E^x]^(-2), x] + Log[4]^2*Defer[Int][1/((1 + 2*E^x)*Log[2*x + x/E
^x]^2), x] - Log[4]^2*Defer[Int][1/(x*Log[2*x + x/E^x]^2), x] - 2*Log[4]*Defer[Int][x/Log[2*x + x/E^x]^2, x] -
(1 + Log[4])^2*Defer[Int][x/Log[2*x + x/E^x]^2, x] + 2*Log[4]*(1 + Log[4])*Defer[Int][x/((1 + 2*E^x)*Log[2*x
+ x/E^x]^2), x] - 2*(1 + Log[4])*Defer[Int][x^2/Log[2*x + x/E^x]^2, x] + (1 + Log[4]^2 + Log[256])*Defer[Int][
x^2/((1 + 2*E^x)*Log[2*x + x/E^x]^2), x] - Defer[Int][x^3/Log[2*x + x/E^x]^2, x] + 2*(1 + Log[4])*Defer[Int][x
^3/((1 + 2*E^x)*Log[2*x + x/E^x]^2), x] + Defer[Int][x^4/((1 + 2*E^x)*Log[2*x + x/E^x]^2), x] + 2*Log[4]*(1 +
Log[4])*Defer[Int][Log[2*x + x/E^x]^(-1), x] + 4*Log[4]*Defer[Int][x/Log[2*x + x/E^x], x] + 2*(1 + Log[4])^2*D
efer[Int][x/Log[2*x + x/E^x], x] + 6*(1 + Log[4])*Defer[Int][x^2/Log[2*x + x/E^x], x] + 4*Defer[Int][x^3/Log[2
*x + x/E^x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(1+x) (x+\log (4)) \left ((1+x) \left (-1-2 e^x+x\right ) (x+\log (4))+2 \left (1+2 e^x\right ) x (1+2 x+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx\\ &=\int \left (\frac {(1+x)^2 (x+\log (4))^2}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )}+\frac {(1+x) (x+\log (4)) \left (-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{x \log ^2\left (2 x+e^{-x} x\right )}\right ) \, dx\\ &=\int \frac {(1+x)^2 (x+\log (4))^2}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \frac {(1+x) (x+\log (4)) \left (-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{x \log ^2\left (2 x+e^{-x} x\right )} \, dx\\ &=\int \left (\frac {x^4}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )}+\frac {\log ^2(4)}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )}+\frac {2 x^3 (1+\log (4))}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )}+\frac {2 x \log (4) (1+\log (4))}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )}+\frac {x^2 \left (1+\log ^2(4)+\log (256)\right )}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )}\right ) \, dx+\int \frac {(1+x) (x+\log (4)) \left (-((1+x) (x+\log (4)))+2 x (1+2 x+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{x \log ^2\left (2 x+e^{-x} x\right )} \, dx\\ &=\log ^2(4) \int \frac {1}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 (1+\log (4))) \int \frac {x^3}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 \log (4) (1+\log (4))) \int \frac {x}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\left (1+\log ^2(4)+\log (256)\right ) \int \frac {x^2}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \left (\frac {x \left (-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{\log ^2\left (2 x+e^{-x} x\right )}+\frac {\log (4) \left (-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{x \log ^2\left (2 x+e^{-x} x\right )}+\frac {(1+\log (4)) \left (-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{\log ^2\left (2 x+e^{-x} x\right )}\right ) \, dx+\int \frac {x^4}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx\\ &=\log (4) \int \frac {-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )}{x \log ^2\left (2 x+e^{-x} x\right )} \, dx+\log ^2(4) \int \frac {1}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(1+\log (4)) \int \frac {-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )}{\log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 (1+\log (4))) \int \frac {x^3}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 \log (4) (1+\log (4))) \int \frac {x}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\left (1+\log ^2(4)+\log (256)\right ) \int \frac {x^2}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \frac {x^4}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \frac {x \left (-x^2-\log (4)-x (1+\log (4))+4 x^2 \log \left (\left (2+e^{-x}\right ) x\right )+2 x (1+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{\log ^2\left (2 x+e^{-x} x\right )} \, dx\\ &=\log (4) \int \frac {-((1+x) (x+\log (4)))+2 x (1+2 x+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )}{x \log ^2\left (2 x+e^{-x} x\right )} \, dx+\log ^2(4) \int \frac {1}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(1+\log (4)) \int \frac {-((1+x) (x+\log (4)))+2 x (1+2 x+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )}{\log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 (1+\log (4))) \int \frac {x^3}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 \log (4) (1+\log (4))) \int \frac {x}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\left (1+\log ^2(4)+\log (256)\right ) \int \frac {x^2}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \frac {x^4}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \frac {x \left (-((1+x) (x+\log (4)))+2 x (1+2 x+\log (4)) \log \left (\left (2+e^{-x}\right ) x\right )\right )}{\log ^2\left (2 x+e^{-x} x\right )} \, dx\\ &=\log (4) \int \left (-\frac {x}{\log ^2\left (2 x+e^{-x} x\right )}-\frac {\log (4)}{x \log ^2\left (2 x+e^{-x} x\right )}-\frac {1+\log (4)}{\log ^2\left (2 x+e^{-x} x\right )}+\frac {4 x}{\log \left (2 x+e^{-x} x\right )}+\frac {2 (1+\log (4))}{\log \left (2 x+e^{-x} x\right )}\right ) \, dx+\log ^2(4) \int \frac {1}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(1+\log (4)) \int \left (-\frac {x^2}{\log ^2\left (2 x+e^{-x} x\right )}-\frac {\log (4)}{\log ^2\left (2 x+e^{-x} x\right )}-\frac {x (1+\log (4))}{\log ^2\left (2 x+e^{-x} x\right )}+\frac {4 x^2}{\log \left (2 x+e^{-x} x\right )}+\frac {2 x (1+\log (4))}{\log \left (2 x+e^{-x} x\right )}\right ) \, dx+(2 (1+\log (4))) \int \frac {x^3}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 \log (4) (1+\log (4))) \int \frac {x}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\left (1+\log ^2(4)+\log (256)\right ) \int \frac {x^2}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \left (-\frac {x^3}{\log ^2\left (2 x+e^{-x} x\right )}-\frac {x \log (4)}{\log ^2\left (2 x+e^{-x} x\right )}-\frac {x^2 (1+\log (4))}{\log ^2\left (2 x+e^{-x} x\right )}+\frac {4 x^3}{\log \left (2 x+e^{-x} x\right )}+\frac {2 x^2 (1+\log (4))}{\log \left (2 x+e^{-x} x\right )}\right ) \, dx+\int \frac {x^4}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx\\ &=4 \int \frac {x^3}{\log \left (2 x+e^{-x} x\right )} \, dx+2 \left ((-1-\log (4)) \int \frac {x^2}{\log ^2\left (2 x+e^{-x} x\right )} \, dx\right )-2 \left (\log (4) \int \frac {x}{\log ^2\left (2 x+e^{-x} x\right )} \, dx\right )+(4 \log (4)) \int \frac {x}{\log \left (2 x+e^{-x} x\right )} \, dx+\log ^2(4) \int \frac {1}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx-\log ^2(4) \int \frac {1}{x \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 (1+\log (4))) \int \frac {x^3}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 (1+\log (4))) \int \frac {x^2}{\log \left (2 x+e^{-x} x\right )} \, dx+(4 (1+\log (4))) \int \frac {x^2}{\log \left (2 x+e^{-x} x\right )} \, dx-2 \left ((\log (4) (1+\log (4))) \int \frac {1}{\log ^2\left (2 x+e^{-x} x\right )} \, dx\right )+(2 \log (4) (1+\log (4))) \int \frac {x}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx+(2 \log (4) (1+\log (4))) \int \frac {1}{\log \left (2 x+e^{-x} x\right )} \, dx-(1+\log (4))^2 \int \frac {x}{\log ^2\left (2 x+e^{-x} x\right )} \, dx+\left (2 (1+\log (4))^2\right ) \int \frac {x}{\log \left (2 x+e^{-x} x\right )} \, dx+\left (1+\log ^2(4)+\log (256)\right ) \int \frac {x^2}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx-\int \frac {x^3}{\log ^2\left (2 x+e^{-x} x\right )} \, dx+\int \frac {x^4}{\left (1+2 e^x\right ) \log ^2\left (2 x+e^{-x} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.01, size = 24, normalized size = 0.92 \begin {gather*} \frac {(1+x)^2 (x+\log (4))^2}{\log \left (\left (2+e^{-x}\right ) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 - x^3 + x^4 + x^5 + (-2*x - 2*x^2 + 2*x^3 + 2*x^4)*Log[4] + (-1 - x + x^2 + x^3)*Log[4]^2 + E^
x*(-2*x^2 - 4*x^3 - 2*x^4 + (-4*x - 8*x^2 - 4*x^3)*Log[4] + (-2 - 4*x - 2*x^2)*Log[4]^2) + (2*x^2 + 6*x^3 + 4*
x^4 + (2*x + 8*x^2 + 6*x^3)*Log[4] + (2*x + 2*x^2)*Log[4]^2 + E^x*(4*x^2 + 12*x^3 + 8*x^4 + (4*x + 16*x^2 + 12
*x^3)*Log[4] + (4*x + 4*x^2)*Log[4]^2))*Log[(x + 2*E^x*x)/E^x])/((x + 2*E^x*x)*Log[(x + 2*E^x*x)/E^x]^2),x]

[Out]

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

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fricas [B]  time = 0.47, size = 56, normalized size = 2.15 \begin {gather*} \frac {x^{4} + 2 \, x^{3} + 4 \, {\left (x^{2} + 2 \, x + 1\right )} \log \relax (2)^{2} + x^{2} + 4 \, {\left (x^{3} + 2 \, x^{2} + x\right )} \log \relax (2)}{\log \left ({\left (2 \, x e^{x} + x\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(4*x^2+4*x)*log(2)^2+2*(12*x^3+16*x^2+4*x)*log(2)+8*x^4+12*x^3+4*x^2)*exp(x)+4*(2*x^2+2*x)*log(
2)^2+2*(6*x^3+8*x^2+2*x)*log(2)+4*x^4+6*x^3+2*x^2)*log((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*log(2)^2+2*(-4
*x^3-8*x^2-4*x)*log(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*log(2)^2+2*(2*x^4+2*x^3-2*x^2-2*x)*log(2)+x^5
+x^4-x^3-x^2)/(2*exp(x)*x+x)/log((2*exp(x)*x+x)/exp(x))^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.36, size = 69, normalized size = 2.65 \begin {gather*} \frac {x^{4} + 4 \, x^{3} \log \relax (2) + 4 \, x^{2} \log \relax (2)^{2} + 2 \, x^{3} + 8 \, x^{2} \log \relax (2) + 8 \, x \log \relax (2)^{2} + x^{2} + 4 \, x \log \relax (2) + 4 \, \log \relax (2)^{2}}{\log \left ({\left (2 \, x e^{x} + x\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(4*x^2+4*x)*log(2)^2+2*(12*x^3+16*x^2+4*x)*log(2)+8*x^4+12*x^3+4*x^2)*exp(x)+4*(2*x^2+2*x)*log(
2)^2+2*(6*x^3+8*x^2+2*x)*log(2)+4*x^4+6*x^3+2*x^2)*log((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*log(2)^2+2*(-4
*x^3-8*x^2-4*x)*log(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*log(2)^2+2*(2*x^4+2*x^3-2*x^2-2*x)*log(2)+x^5
+x^4-x^3-x^2)/(2*exp(x)*x+x)/log((2*exp(x)*x+x)/exp(x))^2,x, algorithm="giac")

[Out]

(x^4 + 4*x^3*log(2) + 4*x^2*log(2)^2 + 2*x^3 + 8*x^2*log(2) + 8*x*log(2)^2 + x^2 + 4*x*log(2) + 4*log(2)^2)/lo
g((2*x*e^x + x)*e^(-x))

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maple [C]  time = 0.26, size = 289, normalized size = 11.12

method	result	size
risch	\(\frac {8 x^{2} \ln \relax (2)^{2}+8 x^{3} \ln \relax (2)+2 x^{4}+16 x \ln \relax (2)^{2}+16 x^{2} \ln \relax (2)+4 x^{3}+8 \ln \relax (2)^{2}+8 x \ln \relax (2)+2 x^{2}}{2 \ln \relax (2)+2 \ln \relax (x )-2 \ln \left ({\mathrm e}^{x}\right )+2 \ln \left (\frac {1}{2}+{\mathrm e}^{x}\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )^{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right )-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )+i \pi \,\mathrm {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right )^{3}-i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )^{3}}\)	\(289\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*(4*x^2+4*x)*ln(2)^2+2*(12*x^3+16*x^2+4*x)*ln(2)+8*x^4+12*x^3+4*x^2)*exp(x)+4*(2*x^2+2*x)*ln(2)^2+2*(6
*x^3+8*x^2+2*x)*ln(2)+4*x^4+6*x^3+2*x^2)*ln((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*ln(2)^2+2*(-4*x^3-8*x^2-4
*x)*ln(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*ln(2)^2+2*(2*x^4+2*x^3-2*x^2-2*x)*ln(2)+x^5+x^4-x^3-x^2)/(
2*exp(x)*x+x)/ln((2*exp(x)*x+x)/exp(x))^2,x,method=_RETURNVERBOSE)

[Out]

2*(4*x^2*ln(2)^2+4*x^3*ln(2)+x^4+8*x*ln(2)^2+8*x^2*ln(2)+2*x^3+4*ln(2)^2+4*x*ln(2)+x^2)/(2*ln(2)+2*ln(x)-2*ln(
exp(x))+2*ln(1/2+exp(x))+I*Pi*csgn(I*x)*csgn(I*x*(1/2+exp(x))*exp(-x))^2+I*Pi*csgn(I*exp(-x)*(1/2+exp(x)))*csg
n(I*x*(1/2+exp(x))*exp(-x))^2+I*Pi*csgn(I*exp(-x))*csgn(I*exp(-x)*(1/2+exp(x)))^2-I*Pi*csgn(I*x)*csgn(I*exp(-x
)*(1/2+exp(x)))*csgn(I*x*(1/2+exp(x))*exp(-x))-I*Pi*csgn(I*exp(-x))*csgn(I*(1/2+exp(x)))*csgn(I*exp(-x)*(1/2+e
xp(x)))+I*Pi*csgn(I*(1/2+exp(x)))*csgn(I*exp(-x)*(1/2+exp(x)))^2-I*Pi*csgn(I*x*(1/2+exp(x))*exp(-x))^3-I*Pi*cs
gn(I*exp(-x)*(1/2+exp(x)))^3)

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maxima [B]  time = 0.57, size = 68, normalized size = 2.62 \begin {gather*} -\frac {x^{4} + 2 \, x^{3} {\left (2 \, \log \relax (2) + 1\right )} + {\left (4 \, \log \relax (2)^{2} + 8 \, \log \relax (2) + 1\right )} x^{2} + 4 \, {\left (2 \, \log \relax (2)^{2} + \log \relax (2)\right )} x + 4 \, \log \relax (2)^{2}}{x - \log \relax (x) - \log \left (2 \, e^{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(4*x^2+4*x)*log(2)^2+2*(12*x^3+16*x^2+4*x)*log(2)+8*x^4+12*x^3+4*x^2)*exp(x)+4*(2*x^2+2*x)*log(
2)^2+2*(6*x^3+8*x^2+2*x)*log(2)+4*x^4+6*x^3+2*x^2)*log((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*log(2)^2+2*(-4
*x^3-8*x^2-4*x)*log(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*log(2)^2+2*(2*x^4+2*x^3-2*x^2-2*x)*log(2)+x^5
+x^4-x^3-x^2)/(2*exp(x)*x+x)/log((2*exp(x)*x+x)/exp(x))^2,x, algorithm="maxima")

[Out]

-(x^4 + 2*x^3*(2*log(2) + 1) + (4*log(2)^2 + 8*log(2) + 1)*x^2 + 4*(2*log(2)^2 + log(2))*x + 4*log(2)^2)/(x -
log(x) - log(2*e^x + 1))

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mupad [B]  time = 6.93, size = 259, normalized size = 9.96 \begin {gather*} x^3\,\left (12\,\ln \relax (2)+6\right )+x^2\,\left (16\,\ln \relax (2)+8\,{\ln \relax (2)}^2+2\right )+\frac {4\,x^2\,{\ln \relax (2)}^2+4\,x\,\ln \relax (2)+8\,x\,{\ln \relax (2)}^2+8\,x^2\,\ln \relax (2)+4\,x^3\,\ln \relax (2)+4\,{\ln \relax (2)}^2+x^2+2\,x^3+x^4-\frac {2\,x\,\ln \left ({\mathrm {e}}^{-x}\,\left (x+2\,x\,{\mathrm {e}}^x\right )\right )\,\left (2\,{\mathrm {e}}^x+1\right )\,\left (x+1\right )\,\left (x+\ln \relax (4)+x\,\ln \left (64\right )+4\,{\ln \relax (2)}^2+2\,x^2\right )}{2\,{\mathrm {e}}^x-x+1}}{\ln \left ({\mathrm {e}}^{-x}\,\left (x+2\,x\,{\mathrm {e}}^x\right )\right )}+x\,\left (\ln \left (16\right )+8\,{\ln \relax (2)}^2\right )+4\,x^4-\frac {2\,\left (8\,x^2\,{\ln \relax (2)}^2+4\,x^3\,{\ln \relax (2)}^2-4\,x^4\,{\ln \relax (2)}^2+4\,x^2\,\ln \relax (2)+14\,x^3\,\ln \relax (2)+4\,x^4\,\ln \relax (2)-6\,x^5\,\ln \relax (2)+2\,x^3+5\,x^4+x^5-2\,x^6\right )}{\left (x-2\right )\,\left (2\,{\mathrm {e}}^x-x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(2*log(2)*(4*x + 8*x^2 + 4*x^3) + 4*log(2)^2*(4*x + 2*x^2 + 2) + 2*x^2 + 4*x^3 + 2*x^4) - log(exp
(-x)*(x + 2*x*exp(x)))*(exp(x)*(2*log(2)*(4*x + 16*x^2 + 12*x^3) + 4*log(2)^2*(4*x + 4*x^2) + 4*x^2 + 12*x^3 +
8*x^4) + 2*log(2)*(2*x + 8*x^2 + 6*x^3) + 4*log(2)^2*(2*x + 2*x^2) + 2*x^2 + 6*x^3 + 4*x^4) + 4*log(2)^2*(x -
x^2 - x^3 + 1) + 2*log(2)*(2*x + 2*x^2 - 2*x^3 - 2*x^4) + x^2 + x^3 - x^4 - x^5)/(log(exp(-x)*(x + 2*x*exp(x)
))^2*(x + 2*x*exp(x))),x)

[Out]

x^3*(12*log(2) + 6) + x^2*(16*log(2) + 8*log(2)^2 + 2) + (4*x^2*log(2)^2 + 4*x*log(2) + 8*x*log(2)^2 + 8*x^2*l
og(2) + 4*x^3*log(2) + 4*log(2)^2 + x^2 + 2*x^3 + x^4 - (2*x*log(exp(-x)*(x + 2*x*exp(x)))*(2*exp(x) + 1)*(x +
1)*(x + log(4) + x*log(64) + 4*log(2)^2 + 2*x^2))/(2*exp(x) - x + 1))/log(exp(-x)*(x + 2*x*exp(x))) + x*(log(
16) + 8*log(2)^2) + 4*x^4 - (2*(8*x^2*log(2)^2 + 4*x^3*log(2)^2 - 4*x^4*log(2)^2 + 4*x^2*log(2) + 14*x^3*log(2
) + 4*x^4*log(2) - 6*x^5*log(2) + 2*x^3 + 5*x^4 + x^5 - 2*x^6))/((x - 2)*(2*exp(x) - x + 1))

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sympy [B]  time = 0.34, size = 73, normalized size = 2.81 \begin {gather*} \frac {x^{4} + 2 x^{3} + 4 x^{3} \log {\relax (2 )} + x^{2} + 4 x^{2} \log {\relax (2 )}^{2} + 8 x^{2} \log {\relax (2 )} + 4 x \log {\relax (2 )} + 8 x \log {\relax (2 )}^{2} + 4 \log {\relax (2 )}^{2}}{\log {\left (\left (2 x e^{x} + x\right ) e^{- x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(4*x**2+4*x)*ln(2)**2+2*(12*x**3+16*x**2+4*x)*ln(2)+8*x**4+12*x**3+4*x**2)*exp(x)+4*(2*x**2+2*x
)*ln(2)**2+2*(6*x**3+8*x**2+2*x)*ln(2)+4*x**4+6*x**3+2*x**2)*ln((2*exp(x)*x+x)/exp(x))+(4*(-2*x**2-4*x-2)*ln(2
)**2+2*(-4*x**3-8*x**2-4*x)*ln(2)-2*x**4-4*x**3-2*x**2)*exp(x)+4*(x**3+x**2-x-1)*ln(2)**2+2*(2*x**4+2*x**3-2*x
**2-2*x)*ln(2)+x**5+x**4-x**3-x**2)/(2*exp(x)*x+x)/ln((2*exp(x)*x+x)/exp(x))**2,x)

[Out]

(x**4 + 2*x**3 + 4*x**3*log(2) + x**2 + 4*x**2*log(2)**2 + 8*x**2*log(2) + 4*x*log(2) + 8*x*log(2)**2 + 4*log(
2)**2)/log((2*x*exp(x) + x)*exp(-x))

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