3.38.15 \(\int \frac {-3 e^x x+e^{2 x} x^2+(12+4 e^x x^2) \log (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2})+(3 x-e^x x^2) \log ^2(\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2})}{-3 e^x x+e^{2 x} x^2} \, dx\)

Optimal. Leaf size=28 \[ x+e^{-x} \log ^2\left (\frac {\left (e^x-\frac {3}{x}\right )^2}{e^{8/5}}\right ) \]

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Rubi [F]  time = 4.65, 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 {-3 e^x x+e^{2 x} x^2+\left (12+4 e^x x^2\right ) \log \left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )+\left (3 x-e^x x^2\right ) \log ^2\left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )}{-3 e^x x+e^{2 x} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*E^x*x + E^(2*x)*x^2 + (12 + 4*E^x*x^2)*Log[(9 - 6*E^x*x + E^(2*x)*x^2)/(E^(8/5)*x^2)] + (3*x - E^x*x^2
)*Log[(9 - 6*E^x*x + E^(2*x)*x^2)/(E^(8/5)*x^2)]^2)/(-3*E^x*x + E^(2*x)*x^2),x]

[Out]

224/(25*E^x) + x - (72*ExpIntegralEi[-x])/5 - (36*Log[(3 - E^x*x)^2/x^2])/(5*E^x) + (72*Defer[Int][(-3 + E^x*x
)^(-1), x])/5 - (96*Defer[Int][1/(E^x*(-3 + E^x*x)), x])/5 + 12*Log[(3 - E^x*x)^2/x^2]*Defer[Int][1/(E^x*(-3 +
E^x*x)), x] - (96*Defer[Int][1/(E^x*x*(-3 + E^x*x)), x])/5 + 12*Log[(3 - E^x*x)^2/x^2]*Defer[Int][1/(E^x*x*(-
3 + E^x*x)), x] + (72*Defer[Int][x/(-3 + E^x*x), x])/5 - Defer[Int][Log[(-3 + E^x*x)^2/x^2]^2/E^x, x] - 24*Def
er[Int][Defer[Int][1/(E^x*(-3 + E^x*x)), x], x] - 72*Defer[Int][Defer[Int][1/(E^x*(-3 + E^x*x)), x]/(-3 + E^x*
x), x] - 72*Defer[Int][Defer[Int][1/(E^x*(-3 + E^x*x)), x]/(x*(-3 + E^x*x)), x] - 24*Defer[Int][Defer[Int][1/(
E^x*x*(-3 + E^x*x)), x], x] - 72*Defer[Int][Defer[Int][1/(E^x*x*(-3 + E^x*x)), x]/(-3 + E^x*x), x] - 72*Defer[
Int][Defer[Int][1/(E^x*x*(-3 + E^x*x)), x]/(x*(-3 + E^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (3 e^x x-e^{2 x} x^2-\left (12+4 e^x x^2\right ) \log \left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )-\left (3 x-e^x x^2\right ) \log ^2\left (\frac {9-6 e^x x+e^{2 x} x^2}{e^{8/5} x^2}\right )\right )}{x \left (3-e^x x\right )} \, dx\\ &=\int \left (1+\frac {12 e^{-x} (1+x) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{5 x \left (-3+e^x x\right )}-\frac {1}{25} e^{-x} \left (-28+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )\right ) \, dx\\ &=x-\frac {1}{25} \int e^{-x} \left (-28+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \, dx+\frac {12}{5} \int \frac {e^{-x} (1+x) \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{x \left (-3+e^x x\right )} \, dx\\ &=x-\frac {1}{25} \int \left (224 e^{-x}-180 e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )+25 e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \, dx+\frac {12}{5} \int \left (\frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{-3+e^x x}+\frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{x \left (-3+e^x x\right )}\right ) \, dx\\ &=x+\frac {12}{5} \int \frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{-3+e^x x} \, dx+\frac {12}{5} \int \frac {e^{-x} \left (-8+5 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right )}{x \left (-3+e^x x\right )} \, dx+\frac {36}{5} \int e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx-\frac {224}{25} \int e^{-x} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {12}{5} \int \left (-\frac {8 e^{-x}}{-3+e^x x}+\frac {5 e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{-3+e^x x}\right ) \, dx+\frac {12}{5} \int \left (-\frac {8 e^{-x}}{x \left (-3+e^x x\right )}+\frac {5 e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{x \left (-3+e^x x\right )}\right ) \, dx-\frac {36}{5} \int \frac {6 e^{-x}+2 x^2}{3 x-e^x x^2} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )-\frac {36}{5} \int \left (\frac {2 e^{-x}}{x}-\frac {2 (1+x)}{-3+e^x x}\right ) \, dx+12 \int \frac {e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{-3+e^x x} \, dx+12 \int \frac {e^{-x} \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )}{x \left (-3+e^x x\right )} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )-12 \int \frac {2 \left (-3-e^x x^2\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (3-e^x x\right )} \, dx-12 \int \frac {2 \left (-3-e^x x^2\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (3-e^x x\right )} \, dx-\frac {72}{5} \int \frac {e^{-x}}{x} \, dx+\frac {72}{5} \int \frac {1+x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \left (\frac {1}{-3+e^x x}+\frac {x}{-3+e^x x}\right ) \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \frac {\left (-3-e^x x^2\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (3-e^x x\right )} \, dx-24 \int \frac {\left (-3-e^x x^2\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (3-e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx+\frac {3 (1+x) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx+\frac {3 (1+x) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx\right ) \, dx-72 \int \frac {(1+x) \int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )} \, dx-72 \int \frac {(1+x) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx\right ) \, dx-72 \int \left (\frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{-3+e^x x}+\frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx-72 \int \left (\frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{-3+e^x x}+\frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )}\right ) \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ &=\frac {224 e^{-x}}{25}+x-\frac {72 \text {Ei}(-x)}{5}-\frac {36}{5} e^{-x} \log \left (\frac {\left (3-e^x x\right )^2}{x^2}\right )+\frac {72}{5} \int \frac {1}{-3+e^x x} \, dx+\frac {72}{5} \int \frac {x}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{-3+e^x x} \, dx-\frac {96}{5} \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-24 \int \left (\int \frac {e^{-x}}{-3+e^x x} \, dx\right ) \, dx-24 \int \left (\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx\right ) \, dx-72 \int \frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{-3+e^x x} \, dx-72 \int \frac {\int \frac {e^{-x}}{-3+e^x x} \, dx}{x \left (-3+e^x x\right )} \, dx-72 \int \frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{-3+e^x x} \, dx-72 \int \frac {\int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx}{x \left (-3+e^x x\right )} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{-3+e^x x} \, dx+\left (12 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \int \frac {e^{-x}}{x \left (-3+e^x x\right )} \, dx-\int e^{-x} \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 51, normalized size = 1.82 \begin {gather*} \frac {1}{25} e^{-x} \left (64+25 e^x x-80 \log \left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )+25 \log ^2\left (\frac {\left (-3+e^x x\right )^2}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*E^x*x + E^(2*x)*x^2 + (12 + 4*E^x*x^2)*Log[(9 - 6*E^x*x + E^(2*x)*x^2)/(E^(8/5)*x^2)] + (3*x - E
^x*x^2)*Log[(9 - 6*E^x*x + E^(2*x)*x^2)/(E^(8/5)*x^2)]^2)/(-3*E^x*x + E^(2*x)*x^2),x]

[Out]

(64 + 25*E^x*x - 80*Log[(-3 + E^x*x)^2/x^2] + 25*Log[(-3 + E^x*x)^2/x^2]^2)/(25*E^x)

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fricas [A]  time = 0.74, size = 34, normalized size = 1.21 \begin {gather*} {\left (x e^{x} + \log \left (\frac {{\left (x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{x} + 9\right )} e^{\left (-\frac {8}{5}\right )}}{x^{2}}\right )^{2}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*x^2+3*x)*log((exp(x)^2*x^2-6*exp(x)*x+9)/x^2/exp(4/5)^2)^2+(4*exp(x)*x^2+12)*log((exp(x)^2
*x^2-6*exp(x)*x+9)/x^2/exp(4/5)^2)+exp(x)^2*x^2-3*exp(x)*x)/(exp(x)^2*x^2-3*exp(x)*x),x, algorithm="fricas")

[Out]

(x*e^x + log((x^2*e^(2*x) - 6*x*e^x + 9)*e^(-8/5)/x^2)^2)*e^(-x)

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giac [B]  time = 2.10, size = 59, normalized size = 2.11 \begin {gather*} \frac {1}{25} \, {\left (25 \, x e^{x} + 25 \, \log \left (\frac {x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{x} + 9}{x^{2}}\right )^{2} - 80 \, \log \left (\frac {x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{x} + 9}{x^{2}}\right ) + 64\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*x^2+3*x)*log((exp(x)^2*x^2-6*exp(x)*x+9)/x^2/exp(4/5)^2)^2+(4*exp(x)*x^2+12)*log((exp(x)^2
*x^2-6*exp(x)*x+9)/x^2/exp(4/5)^2)+exp(x)^2*x^2-3*exp(x)*x)/(exp(x)^2*x^2-3*exp(x)*x),x, algorithm="giac")

[Out]

1/25*(25*x*e^x + 25*log((x^2*e^(2*x) - 6*x*e^x + 9)/x^2)^2 - 80*log((x^2*e^(2*x) - 6*x*e^x + 9)/x^2) + 64)*e^(
-x)

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maple [C]  time = 0.50, size = 2783, normalized size = 99.39

method	result	size
risch	\(\text {Expression too large to display}\)	\(2783\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp(x)*x^2+3*x)*ln((exp(x)^2*x^2-6*exp(x)*x+9)/x^2/exp(4/5)^2)^2+(4*exp(x)*x^2+12)*ln((exp(x)^2*x^2-6*e
xp(x)*x+9)/x^2/exp(4/5)^2)+exp(x)^2*x^2-3*exp(x)*x)/(exp(x)^2*x^2-3*exp(x)*x),x,method=_RETURNVERBOSE)

[Out]

4*exp(-x)*ln(exp(x)*x-3)^2-2/5*(-5*I*Pi*csgn(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2-5*I*Pi*csgn(I*(exp(x)*x-3)^2)
*csgn(I/x^2*(exp(x)*x-3)^2)^2+10*I*Pi*csgn(I*x)*csgn(I*x^2)^2-5*I*Pi*csgn(I*x^2)^3+5*I*Pi*csgn(I/x^2*(exp(x)*x
-3)^2)^3-5*I*Pi*csgn(I*x)^2*csgn(I*x^2)+5*I*Pi*csgn(I/x^2)*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)-1
0*I*Pi*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^2+5*I*Pi*csgn(I*(exp(x)*x-3)^2)^3+5*I*Pi*csgn(I*(exp(x)*x-3
))^2*csgn(I*(exp(x)*x-3)^2)+16+20*ln(x))*exp(-x)*ln(exp(x)*x-3)+1/100*(256+100*Pi^2*csgn(I*x)*csgn(I*x^2)^2*cs
gn(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2-100*Pi^2*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)*csgn(I*x)*csgn(I
*x^2)^2-50*Pi^2*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)^2*csgn(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)+50*Pi^2
*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2-50*Pi^2*csgn(I*(exp(x)
*x-3))^2*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^3+100*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I*(exp(x)*x
-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2-100*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I/x^2*(exp(x)*x-3)^2)^3+200*I*ln(x)*
Pi*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)-400*I*ln(x)*Pi*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^2-
200*I*ln(x)*Pi*csgn(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2-200*I*ln(x)*Pi*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(
x)*x-3)^2)^2-200*I*Pi*ln(x)*csgn(I*x)^2*csgn(I*x^2)+400*I*Pi*ln(x)*csgn(I*x)*csgn(I*x^2)^2+160*I*Pi*csgn(I/x^2
)*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)+50*Pi^2*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)*csgn
(I*x)^2*csgn(I*x^2)-100*Pi^2*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^2*csgn(I*x)^2*csgn(I*x^2)+200*Pi^2*cs
gn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^2*csgn(I*x)*csgn(I*x^2)^2+100*Pi^2*csgn(I*(exp(x)*x-3))*csgn(I*(exp(
x)*x-3)^2)^3*csgn(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)-100*Pi^2*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^2*csg
n(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2+100*Pi^2*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^2*csgn(I/x^2*(exp(x
)*x-3)^2)^3+50*Pi^2*csgn(I*x^2)^3*csgn(I/x^2)*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)-100*Pi^2*csgn(
I/x^2)*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^4-50*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I/x^2)*csgn(I/
x^2*(exp(x)*x-3)^2)^2-50*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2+50*P
i^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I/x^2*(exp(x)*x-3)^2)^3+200*I*ln(x)*Pi*csgn(I/x^2)*csgn(I*(exp(x)*x-3)^2)*csg
n(I/x^2*(exp(x)*x-3)^2)-25*Pi^2*csgn(I/x^2)^2*csgn(I/x^2*(exp(x)*x-3)^2)^4+50*Pi^2*csgn(I/x^2)*csgn(I/x^2*(exp
(x)*x-3)^2)^5-160*I*Pi*csgn(I*x^2)^3+160*I*Pi*csgn(I*(exp(x)*x-3)^2)^3+160*I*Pi*csgn(I/x^2*(exp(x)*x-3)^2)^3-2
5*Pi^2*csgn(I*(exp(x)*x-3))^4*csgn(I*(exp(x)*x-3)^2)^2+100*Pi^2*csgn(I*(exp(x)*x-3))^3*csgn(I*(exp(x)*x-3)^2)^
3-150*Pi^2*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)^4+100*Pi^2*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2
)^5-25*Pi^2*csgn(I*(exp(x)*x-3)^2)^2*csgn(I/x^2*(exp(x)*x-3)^2)^4+50*Pi^2*csgn(I*(exp(x)*x-3)^2)^3*csgn(I*x^2)
^3+50*Pi^2*csgn(I*(exp(x)*x-3)^2)^4*csgn(I/x^2*(exp(x)*x-3)^2)^2-25*Pi^2*csgn(I*x)^4*csgn(I*x^2)^2+100*Pi^2*cs
gn(I*x)^3*csgn(I*x^2)^3-150*Pi^2*csgn(I*x)^2*csgn(I*x^2)^4+100*Pi^2*csgn(I*x)*csgn(I*x^2)^5+640*ln(x)+400*ln(x
)^2+100*exp(x)*x-25*Pi^2*csgn(I/x^2*(exp(x)*x-3)^2)^6-25*Pi^2*csgn(I*x^2)^6-25*Pi^2*csgn(I*(exp(x)*x-3)^2)^6+5
0*Pi^2*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)*csgn(I*x^2)^3+50*Pi^2*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(
x)*x-3)^2)^2*csgn(I/x^2*(exp(x)*x-3)^2)^2-100*Pi^2*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^2*csgn(I*x^2)^3
-100*Pi^2*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x-3)^2)^3*csgn(I/x^2*(exp(x)*x-3)^2)^2+50*Pi^2*csgn(I*(exp(x)*x-
3)^2)^3*csgn(I*x)^2*csgn(I*x^2)-100*Pi^2*csgn(I*(exp(x)*x-3)^2)^3*csgn(I*x)*csgn(I*x^2)^2-50*Pi^2*csgn(I*(exp(
x)*x-3)^2)^4*csgn(I/x^2)*csgn(I/x^2*(exp(x)*x-3)^2)+50*Pi^2*csgn(I*(exp(x)*x-3)^2)^3*csgn(I/x^2)*csgn(I/x^2*(e
xp(x)*x-3)^2)^2-50*Pi^2*csgn(I*(exp(x)*x-3)^2)^3*csgn(I/x^2*(exp(x)*x-3)^2)^3-50*Pi^2*csgn(I*x^2)^3*csgn(I/x^2
)*csgn(I/x^2*(exp(x)*x-3)^2)^2-50*Pi^2*csgn(I*x^2)^3*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2+50*Pi
^2*csgn(I*x^2)^3*csgn(I/x^2*(exp(x)*x-3)^2)^3-25*Pi^2*csgn(I/x^2)^2*csgn(I*(exp(x)*x-3)^2)^2*csgn(I/x^2*(exp(x
)*x-3)^2)^2+50*Pi^2*csgn(I/x^2)^2*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^3+50*Pi^2*csgn(I/x^2)*csgn
(I*(exp(x)*x-3)^2)^2*csgn(I/x^2*(exp(x)*x-3)^2)^3+50*Pi^2*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^5+
200*I*ln(x)*Pi*csgn(I*(exp(x)*x-3)^2)^3-200*I*ln(x)*Pi*csgn(I*x^2)^3+200*I*ln(x)*Pi*csgn(I/x^2*(exp(x)*x-3)^2)
^3-160*I*Pi*csgn(I*x)^2*csgn(I*x^2)+320*I*Pi*csgn(I*x)*csgn(I*x^2)^2-160*I*Pi*csgn(I/x^2)*csgn(I/x^2*(exp(x)*x
-3)^2)^2+160*I*Pi*csgn(I*(exp(x)*x-3))^2*csgn(I*(exp(x)*x-3)^2)-320*I*Pi*csgn(I*(exp(x)*x-3))*csgn(I*(exp(x)*x
-3)^2)^2-160*I*Pi*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)^2+50*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I/x
^2)*csgn(I*(exp(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2)-100*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I/x^2)*csgn(I*(exp
(x)*x-3)^2)*csgn(I/x^2*(exp(x)*x-3)^2))*exp(-x)

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maxima [A]  time = 0.43, size = 49, normalized size = 1.75 \begin {gather*} \frac {1}{25} \, {\left (25 \, x e^{x} - 40 \, {\left (5 \, \log \relax (x) + 4\right )} \log \left (x e^{x} - 3\right ) + 100 \, \log \left (x e^{x} - 3\right )^{2} + 100 \, \log \relax (x)^{2} + 160 \, \log \relax (x) + 64\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*x^2+3*x)*log((exp(x)^2*x^2-6*exp(x)*x+9)/x^2/exp(4/5)^2)^2+(4*exp(x)*x^2+12)*log((exp(x)^2
*x^2-6*exp(x)*x+9)/x^2/exp(4/5)^2)+exp(x)^2*x^2-3*exp(x)*x)/(exp(x)^2*x^2-3*exp(x)*x),x, algorithm="maxima")

[Out]

1/25*(25*x*e^x - 40*(5*log(x) + 4)*log(x*e^x - 3) + 100*log(x*e^x - 3)^2 + 100*log(x)^2 + 160*log(x) + 64)*e^(
-x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (\frac {{\mathrm {e}}^{-\frac {8}{5}}\,\left (x^2\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^x+9\right )}{x^2}\right )\,\left (4\,x^2\,{\mathrm {e}}^x+12\right )+x^2\,{\mathrm {e}}^{2\,x}+{\ln \left (\frac {{\mathrm {e}}^{-\frac {8}{5}}\,\left (x^2\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^x+9\right )}{x^2}\right )}^2\,\left (3\,x-x^2\,{\mathrm {e}}^x\right )-3\,x\,{\mathrm {e}}^x}{x^2\,{\mathrm {e}}^{2\,x}-3\,x\,{\mathrm {e}}^x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((exp(-8/5)*(x^2*exp(2*x) - 6*x*exp(x) + 9))/x^2)*(4*x^2*exp(x) + 12) + x^2*exp(2*x) + log((exp(-8/5)*
(x^2*exp(2*x) - 6*x*exp(x) + 9))/x^2)^2*(3*x - x^2*exp(x)) - 3*x*exp(x))/(x^2*exp(2*x) - 3*x*exp(x)),x)

[Out]

int((log((exp(-8/5)*(x^2*exp(2*x) - 6*x*exp(x) + 9))/x^2)*(4*x^2*exp(x) + 12) + x^2*exp(2*x) + log((exp(-8/5)*
(x^2*exp(2*x) - 6*x*exp(x) + 9))/x^2)^2*(3*x - x^2*exp(x)) - 3*x*exp(x))/(x^2*exp(2*x) - 3*x*exp(x)), x)

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sympy [A]  time = 0.43, size = 32, normalized size = 1.14 \begin {gather*} x + e^{- x} \log {\left (\frac {x^{2} e^{2 x} - 6 x e^{x} + 9}{x^{2} e^{\frac {8}{5}}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*x**2+3*x)*ln((exp(x)**2*x**2-6*exp(x)*x+9)/x**2/exp(4/5)**2)**2+(4*exp(x)*x**2+12)*ln((exp
(x)**2*x**2-6*exp(x)*x+9)/x**2/exp(4/5)**2)+exp(x)**2*x**2-3*exp(x)*x)/(exp(x)**2*x**2-3*exp(x)*x),x)

[Out]

x + exp(-x)*log((x**2*exp(2*x) - 6*x*exp(x) + 9)*exp(-8/5)/x**2)**2

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