3.58.32 \(\int \frac {9-9 e^x+9 x+(36+58 x+6 x^2+e^x (-36-22 x-2 x^2)) \log (\frac {1}{5} (x^2-e^x x^2+x^3))+(9-9 e^x+9 x) \log ^2(\frac {1}{5} (x^2-e^x x^2+x^3))}{-x^2+e^x x^2-x^3+(-2 x^2+2 e^x x^2-2 x^3) \log ^2(\frac {1}{5} (x^2-e^x x^2+x^3))+(-x^2+e^x x^2-x^3) \log ^4(\frac {1}{5} (x^2-e^x x^2+x^3))} \, dx\)

Optimal. Leaf size=30 \[ \frac {1+\frac {9}{x}}{1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )} \]

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Rubi [F]  time = 10.52, 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 {9-9 e^x+9 x+\left (36+58 x+6 x^2+e^x \left (-36-22 x-2 x^2\right )\right ) \log \left (\frac {1}{5} \left (x^2-e^x x^2+x^3\right )\right )+\left (9-9 e^x+9 x\right ) \log ^2\left (\frac {1}{5} \left (x^2-e^x x^2+x^3\right )\right )}{-x^2+e^x x^2-x^3+\left (-2 x^2+2 e^x x^2-2 x^3\right ) \log ^2\left (\frac {1}{5} \left (x^2-e^x x^2+x^3\right )\right )+\left (-x^2+e^x x^2-x^3\right ) \log ^4\left (\frac {1}{5} \left (x^2-e^x x^2+x^3\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 - 9*E^x + 9*x + (36 + 58*x + 6*x^2 + E^x*(-36 - 22*x - 2*x^2))*Log[(x^2 - E^x*x^2 + x^3)/5] + (9 - 9*E^
x + 9*x)*Log[(x^2 - E^x*x^2 + x^3)/5]^2)/(-x^2 + E^x*x^2 - x^3 + (-2*x^2 + 2*E^x*x^2 - 2*x^3)*Log[(x^2 - E^x*x
^2 + x^3)/5]^2 + (-x^2 + E^x*x^2 - x^3)*Log[(x^2 - E^x*x^2 + x^3)/5]^4),x]

[Out]

-2*Defer[Int][Log[(x^2*(1 - E^x + x))/5]/(1 + Log[(x^2*(1 - E^x + x))/5]^2)^2, x] - 18*Defer[Int][Log[(x^2*(1
- E^x + x))/5]/((-1 + E^x - x)*(1 + Log[(x^2*(1 - E^x + x))/5]^2)^2), x] - 36*Defer[Int][Log[(x^2*(1 - E^x + x
))/5]/(x^2*(1 + Log[(x^2*(1 - E^x + x))/5]^2)^2), x] - 22*Defer[Int][Log[(x^2*(1 - E^x + x))/5]/(x*(1 + Log[(x
^2*(1 - E^x + x))/5]^2)^2), x] - 2*Defer[Int][(x*Log[(x^2*(1 - E^x + x))/5])/((-1 + E^x - x)*(1 + Log[(x^2*(1
- E^x + x))/5]^2)^2), x] - 9*Defer[Int][1/(x^2*(1 + Log[(x^2*(1 - E^x + x))/5]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 \left (-1+e^x-x\right )+2 (9+x) \left (-2-3 x+e^x (2+x)\right ) \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )+9 \left (-1+e^x-x\right ) \log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1-e^x+x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\\ &=\int \left (-\frac {2 (9+x) \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}+\frac {-9-36 \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )-22 x \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )-2 x^2 \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )-9 \log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {(9+x) \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\right )+\int \frac {-9-36 \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )-22 x \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )-2 x^2 \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )-9 \log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\\ &=-\left (2 \int \left (\frac {9 \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}+\frac {x \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}\right ) \, dx\right )+\int \frac {-9-2 \left (18+11 x+x^2\right ) \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )-9 \log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {x \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\right )-18 \int \frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx+\int \left (-\frac {2 \left (18+11 x+x^2\right ) \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}-\frac {9}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\right )-2 \int \frac {\left (18+11 x+x^2\right ) \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx-9 \int \frac {1}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )} \, dx-18 \int \frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {x \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\right )-2 \int \left (\frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}+\frac {18 \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}+\frac {11 \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2}\right ) \, dx-9 \int \frac {1}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )} \, dx-18 \int \frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\right )-2 \int \frac {x \log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx-9 \int \frac {1}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )} \, dx-18 \int \frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{\left (-1+e^x-x\right ) \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx-22 \int \frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx-36 \int \frac {\log \left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )}{x^2 \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 29, normalized size = 0.97 \begin {gather*} \frac {9+x}{x \left (1+\log ^2\left (\frac {1}{5} x^2 \left (1-e^x+x\right )\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 - 9*E^x + 9*x + (36 + 58*x + 6*x^2 + E^x*(-36 - 22*x - 2*x^2))*Log[(x^2 - E^x*x^2 + x^3)/5] + (9
- 9*E^x + 9*x)*Log[(x^2 - E^x*x^2 + x^3)/5]^2)/(-x^2 + E^x*x^2 - x^3 + (-2*x^2 + 2*E^x*x^2 - 2*x^3)*Log[(x^2 -
E^x*x^2 + x^3)/5]^2 + (-x^2 + E^x*x^2 - x^3)*Log[(x^2 - E^x*x^2 + x^3)/5]^4),x]

[Out]

(9 + x)/(x*(1 + Log[(x^2*(1 - E^x + x))/5]^2))

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fricas [A]  time = 0.74, size = 31, normalized size = 1.03 \begin {gather*} \frac {x + 9}{x \log \left (\frac {1}{5} \, x^{3} - \frac {1}{5} \, x^{2} e^{x} + \frac {1}{5} \, x^{2}\right )^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(x)+9*x+9)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+((-2*x^2-22*x-36)*exp(x)+6*x^2+58*x+36)*lo
g(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)-9*exp(x)+9*x+9)/((exp(x)*x^2-x^3-x^2)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^
4+(2*exp(x)*x^2-2*x^3-2*x^2)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+exp(x)*x^2-x^3-x^2),x, algorithm="fricas")

[Out]

(x + 9)/(x*log(1/5*x^3 - 1/5*x^2*e^x + 1/5*x^2)^2 + x)

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giac [A]  time = 30.22, size = 31, normalized size = 1.03 \begin {gather*} \frac {x + 9}{x \log \left (\frac {1}{5} \, x^{3} - \frac {1}{5} \, x^{2} e^{x} + \frac {1}{5} \, x^{2}\right )^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(x)+9*x+9)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+((-2*x^2-22*x-36)*exp(x)+6*x^2+58*x+36)*lo
g(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)-9*exp(x)+9*x+9)/((exp(x)*x^2-x^3-x^2)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^
4+(2*exp(x)*x^2-2*x^3-2*x^2)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+exp(x)*x^2-x^3-x^2),x, algorithm="giac")

[Out]

(x + 9)/(x*log(1/5*x^3 - 1/5*x^2*e^x + 1/5*x^2)^2 + x)

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maple [C]  time = 13.87, size = 1503, normalized size = 50.10

method	result	size
risch	\(-\frac {4 \left (x +9\right )}{x \left (-4-4 \ln \relax (5)^{2}-16 \ln \relax (x )^{2}+8 \ln \relax (5) \ln \left (1-{\mathrm e}^{x}+x \right )-16 \ln \relax (x ) \ln \left (1-{\mathrm e}^{x}+x \right )+4 i \ln \left (1-{\mathrm e}^{x}+x \right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}+\pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{6}+16 \ln \relax (5) \ln \relax (x )+\pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}+4 i \ln \left (1-{\mathrm e}^{x}+x \right ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right ) \mathrm {csgn}\left (i x^{2}\right )-4 i \ln \relax (5) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right ) \mathrm {csgn}\left (i x^{2}\right )+8 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right ) \mathrm {csgn}\left (i x^{2}\right )-16 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-4 i \ln \left (1-{\mathrm e}^{x}+x \right ) \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3}+4 i \ln \relax (5) \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3}-8 i \ln \relax (x ) \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3}+\pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{2}+6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}+\pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{5} \mathrm {csgn}\left (i x^{2}\right )+\pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{4}-2 \pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{5}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3} \mathrm {csgn}\left (i x^{2}\right )-4 i \ln \relax (5) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}-4 i \ln \relax (5) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+8 i \ln \relax (5) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \ln \left (1-{\mathrm e}^{x}+x \right )^{2}+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{4} \mathrm {csgn}\left (i x^{2}\right )+8 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-4 i \ln \relax (5) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}+8 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}+4 i \ln \left (1-{\mathrm e}^{x}+x \right ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )+8 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+4 i \ln \left (1-{\mathrm e}^{x}+x \right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-8 i \ln \left (1-{\mathrm e}^{x}+x \right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+4 i \ln \relax (5) \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-8 i \ln \relax (x ) \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 i \ln \left (1-{\mathrm e}^{x}+x \right ) \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x -1\right ) x^{2}\right )^{3}\right )}\)	\(1503\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*exp(x)+9*x+9)*ln(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+((-2*x^2-22*x-36)*exp(x)+6*x^2+58*x+36)*ln(-1/5*e
xp(x)*x^2+1/5*x^3+1/5*x^2)-9*exp(x)+9*x+9)/((exp(x)*x^2-x^3-x^2)*ln(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^4+(2*exp(
x)*x^2-2*x^3-2*x^2)*ln(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+exp(x)*x^2-x^3-x^2),x,method=_RETURNVERBOSE)

[Out]

-4*(x+9)/x/(-4+Pi^2*csgn(I*x)^4*csgn(I*x^2)^2-4*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3+6*Pi^2*csgn(I*x)^2*csgn(I*x^2)^
4-4*Pi^2*csgn(I*x)*csgn(I*x^2)^5-4*ln(5)^2-16*ln(x)^2+8*ln(5)*ln(1-exp(x)+x)-16*ln(x)*ln(1-exp(x)+x)+16*ln(5)*
ln(x)+4*I*ln(1-exp(x)+x)*Pi*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)*csgn(I*x^2)-4*I*ln(5)*Pi*csgn(I*(exp
(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)*csgn(I*x^2)+8*I*ln(x)*Pi*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)*csgn
(I*x^2)+Pi^2*csgn(I*x^2)^6-4*I*ln(5)*Pi*csgn(I*x^2)^3-2*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I*(exp(x)-x-1)*x^2)^
3+4*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I*(exp(x)-x-1)*x^2)^3-4*Pi^2*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^
2)^4*csgn(I*x^2)+8*I*ln(x)*Pi*csgn(I*x^2)^3+4*I*ln(1-exp(x)+x)*Pi*csgn(I*x^2)^3+Pi^2*csgn(I*(exp(x)-x-1)*x^2)^
4*csgn(I*x^2)^2-2*Pi^2*csgn(I*x^2)^4*csgn(I*(exp(x)-x-1)*x^2)^2-4*I*ln(5)*Pi*csgn(I*(exp(x)-x-1))*csgn(I*(exp(
x)-x-1)*x^2)^2+8*I*ln(x)*Pi*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)^2+4*I*ln(1-exp(x)+x)*Pi*csgn(I*(exp(
x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)^2+2*Pi^2*csgn(I*x)^2*csgn(I*x^2)^2*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*
x^2)-4*Pi^2*csgn(I*x)*csgn(I*x^2)^3*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)+8*I*ln(x)*Pi*csgn(I*x)^2*csg
n(I*x^2)-16*I*ln(x)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(1-exp(x)+x)*Pi*csgn(I*x)^2*csgn(I*x^2)-8*I*ln(1-exp(x)+x
)*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi^2*csgn(I*(exp(x)-x-1)*x^2)^6-2*Pi^2*csgn(I*x^2)^3*csgn(I*(exp(x)-x-1)*x^2)^3+2
*Pi^2*csgn(I*(exp(x)-x-1)*x^2)^5*csgn(I*x^2)+4*I*ln(5)*Pi*csgn(I*(exp(x)-x-1)*x^2)^2*csgn(I*x^2)-8*I*ln(x)*Pi*
csgn(I*(exp(x)-x-1)*x^2)^2*csgn(I*x^2)-4*I*ln(1-exp(x)+x)*Pi*csgn(I*(exp(x)-x-1)*x^2)^2*csgn(I*x^2)+2*Pi^2*csg
n(I*x^2)^3*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)^2+2*Pi^2*csgn(I*(exp(x)-x-1))^2*csgn(I*(exp(x)-x-1)*x
^2)^3*csgn(I*x^2)-4*ln(1-exp(x)+x)^2-4*I*ln(1-exp(x)+x)*Pi*csgn(I*(exp(x)-x-1)*x^2)^3+4*I*ln(5)*Pi*csgn(I*(exp
(x)-x-1)*x^2)^3+2*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)^2-4*Pi^2*csgn(I*x
)*csgn(I*x^2)^2*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)^2-8*I*ln(x)*Pi*csgn(I*(exp(x)-x-1)*x^2)^3+Pi^2*c
sgn(I*(exp(x)-x-1))^2*csgn(I*(exp(x)-x-1)*x^2)^4-2*Pi^2*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)^5+Pi^2*c
sgn(I*(exp(x)-x-1))^2*csgn(I*(exp(x)-x-1)*x^2)^2*csgn(I*x^2)^2-2*Pi^2*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)
*x^2)^3*csgn(I*x^2)^2+2*Pi^2*csgn(I*x^2)^4*csgn(I*(exp(x)-x-1))*csgn(I*(exp(x)-x-1)*x^2)-2*Pi^2*csgn(I*x)^2*cs
gn(I*x^2)^2*csgn(I*(exp(x)-x-1)*x^2)^2+4*Pi^2*csgn(I*x)*csgn(I*x^2)^3*csgn(I*(exp(x)-x-1)*x^2)^2-4*I*ln(5)*Pi*
csgn(I*x)^2*csgn(I*x^2)+8*I*ln(5)*Pi*csgn(I*x)*csgn(I*x^2)^2)

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maxima [B]  time = 0.97, size = 61, normalized size = 2.03 \begin {gather*} \frac {x + 9}{x \log \left (x - e^{x} + 1\right )^{2} - 4 \, x \log \relax (5) \log \relax (x) + 4 \, x \log \relax (x)^{2} + {\left (\log \relax (5)^{2} + 1\right )} x - 2 \, {\left (x \log \relax (5) - 2 \, x \log \relax (x)\right )} \log \left (x - e^{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(x)+9*x+9)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+((-2*x^2-22*x-36)*exp(x)+6*x^2+58*x+36)*lo
g(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)-9*exp(x)+9*x+9)/((exp(x)*x^2-x^3-x^2)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^
4+(2*exp(x)*x^2-2*x^3-2*x^2)*log(-1/5*exp(x)*x^2+1/5*x^3+1/5*x^2)^2+exp(x)*x^2-x^3-x^2),x, algorithm="maxima")

[Out]

(x + 9)/(x*log(x - e^x + 1)^2 - 4*x*log(5)*log(x) + 4*x*log(x)^2 + (log(5)^2 + 1)*x - 2*(x*log(5) - 2*x*log(x)
)*log(x - e^x + 1))

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mupad [B]  time = 3.89, size = 51, normalized size = 1.70 \begin {gather*} \frac {\left (x+9\right )\,\left (x-x\,{\mathrm {e}}^x+x^2\right )}{x^2\,\left ({\ln \left (\frac {x^2}{5}-\frac {x^2\,{\mathrm {e}}^x}{5}+\frac {x^3}{5}\right )}^2+1\right )\,\left (x-{\mathrm {e}}^x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x - 9*exp(x) + log(x^2/5 - (x^2*exp(x))/5 + x^3/5)^2*(9*x - 9*exp(x) + 9) + log(x^2/5 - (x^2*exp(x))/5
+ x^3/5)*(58*x - exp(x)*(22*x + 2*x^2 + 36) + 6*x^2 + 36) + 9)/(log(x^2/5 - (x^2*exp(x))/5 + x^3/5)^4*(x^2 -
x^2*exp(x) + x^3) - x^2*exp(x) + x^2 + x^3 + log(x^2/5 - (x^2*exp(x))/5 + x^3/5)^2*(2*x^2 - 2*x^2*exp(x) + 2*x
^3)),x)

[Out]

((x + 9)*(x - x*exp(x) + x^2))/(x^2*(log(x^2/5 - (x^2*exp(x))/5 + x^3/5)^2 + 1)*(x - exp(x) + 1))

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sympy [A]  time = 0.29, size = 27, normalized size = 0.90 \begin {gather*} \frac {x + 9}{x \log {\left (\frac {x^{3}}{5} - \frac {x^{2} e^{x}}{5} + \frac {x^{2}}{5} \right )}^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*exp(x)+9*x+9)*ln(-1/5*exp(x)*x**2+1/5*x**3+1/5*x**2)**2+((-2*x**2-22*x-36)*exp(x)+6*x**2+58*x+3
6)*ln(-1/5*exp(x)*x**2+1/5*x**3+1/5*x**2)-9*exp(x)+9*x+9)/((exp(x)*x**2-x**3-x**2)*ln(-1/5*exp(x)*x**2+1/5*x**
3+1/5*x**2)**4+(2*exp(x)*x**2-2*x**3-2*x**2)*ln(-1/5*exp(x)*x**2+1/5*x**3+1/5*x**2)**2+exp(x)*x**2-x**3-x**2),
x)

[Out]

(x + 9)/(x*log(x**3/5 - x**2*exp(x)/5 + x**2/5)**2 + x)

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