3.36.36 \(\int \frac {e^{-3+e^{4+x+e^5 (4+x)}} (2 x^2 \log (1-x)+(-2 x+2 x^2+e^{4+x+e^5 (4+x)} (-x^2+x^3+e^5 (-x^2+x^3))) \log ^2(1-x))}{-1+x} \, dx\)

Optimal. Leaf size=27 \[ e^{-3+e^{\left (1+e^5\right ) (4+x)}} x^2 \log ^2(1-x) \]

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Rubi [F]  time = 11.90, 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 {e^{-3+e^{4+x+e^5 (4+x)}} \left (2 x^2 \log (1-x)+\left (-2 x+2 x^2+e^{4+x+e^5 (4+x)} \left (-x^2+x^3+e^5 \left (-x^2+x^3\right )\right )\right ) \log ^2(1-x)\right )}{-1+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*ExpIntegralEi[E^(4 + x + E^5*(4 + x))]*Log[1 - x])/(E^3*(1 + E^5)) + 2*Log[1 - x]*Defer[Int][E^(-3 + E^(4 +
x + E^5*(4 + x)))/(-1 + x), x] + 2*Log[1 - x]*Defer[Int][E^(-3 + E^(4 + x + E^5*(4 + x)))*x, x] - (2*Defer[In
t][ExpIntegralEi[E^(4*(1 + E^5) + (1 + E^5)*x)]/(-1 + x), x])/(E^3*(1 + E^5)) + 2*Defer[Int][E^(-3 + E^(4 + x
+ E^5*(4 + x)))*Log[1 - x]^2, x] + (1 + E^5)*Defer[Int][E^(1 + E^((1 + E^5)*(4 + x)) + x + E^5*(4 + x))*Log[1
- x]^2, x] - 2*Defer[Int][E^(-3 + E^(4 + x + E^5*(4 + x)))*(1 - x)*Log[1 - x]^2, x] - 2*(1 + E^5)*Defer[Int][E
^(1 + E^((1 + E^5)*(4 + x)) + x + E^5*(4 + x))*(1 - x)*Log[1 - x]^2, x] + (1 + E^5)*Defer[Int][E^(1 + E^((1 +
E^5)*(4 + x)) + x + E^5*(4 + x))*(1 - x)^2*Log[1 - x]^2, x] - 2*Defer[Int][Defer[Int][E^(-3 + E^((1 + E^5)*(4
+ x)))/(-1 + x), x]/(-1 + x), x] - 2*Defer[Int][Defer[Int][E^(-3 + E^((1 + E^5)*(4 + x)))*x, x]/(-1 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\exp \left (-3+e^{4+x+e^5 (4+x)}+4 \left (1+e^5\right )+\left (1+e^5\right ) x\right ) (1+e) \left (1-e+e^2-e^3+e^4\right ) x^2 \log ^2(1-x)+\frac {2 e^{-3+e^{4+x+e^5 (4+x)}} x \log (1-x) (x-\log (1-x)+x \log (1-x))}{-1+x}\right ) \, dx\\ &=2 \int \frac {e^{-3+e^{4+x+e^5 (4+x)}} x \log (1-x) (x-\log (1-x)+x \log (1-x))}{-1+x} \, dx+\left (1+e^5\right ) \int \exp \left (-3+e^{4+x+e^5 (4+x)}+4 \left (1+e^5\right )+\left (1+e^5\right ) x\right ) x^2 \log ^2(1-x) \, dx\\ &=2 \int \left (\frac {e^{-3+e^{4+x+e^5 (4+x)}} x^2 \log (1-x)}{-1+x}+e^{-3+e^{4+x+e^5 (4+x)}} x \log ^2(1-x)\right ) \, dx+\left (1+e^5\right ) \int \exp \left (e^{4+x+e^5 (4+x)}-3 \left (1-\frac {4}{3} \left (1+e^5\right )\right )+\left (1+e^5\right ) x\right ) x^2 \log ^2(1-x) \, dx\\ &=2 \int \frac {e^{-3+e^{4+x+e^5 (4+x)}} x^2 \log (1-x)}{-1+x} \, dx+2 \int e^{-3+e^{4+x+e^5 (4+x)}} x \log ^2(1-x) \, dx+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} x^2 \log ^2(1-x) \, dx\\ &=\frac {2 \text {Ei}\left (e^{4+x+e^5 (4+x)}\right ) \log (1-x)}{e^3 \left (1+e^5\right )}+2 \int \left (e^{-3+e^{4+x+e^5 (4+x)}} \log ^2(1-x)-e^{-3+e^{4+x+e^5 (4+x)}} (1-x) \log ^2(1-x)\right ) \, dx-2 \int \frac {-\frac {\text {Ei}\left (e^{\left (1+e^5\right ) (4+x)}\right )}{e^3+e^8}-\int \frac {e^{-3+e^{\left (1+e^5\right ) (4+x)}}}{-1+x} \, dx-\int e^{-3+e^{\left (1+e^5\right ) (4+x)}} x \, dx}{1-x} \, dx+\left (1+e^5\right ) \int \left (e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} \log ^2(1-x)-2 e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x) \log ^2(1-x)+e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x)^2 \log ^2(1-x)\right ) \, dx+(2 \log (1-x)) \int \frac {e^{-3+e^{4+x+e^5 (4+x)}}}{-1+x} \, dx+(2 \log (1-x)) \int e^{-3+e^{4+x+e^5 (4+x)}} x \, dx\\ &=\frac {2 \text {Ei}\left (e^{4+x+e^5 (4+x)}\right ) \log (1-x)}{e^3 \left (1+e^5\right )}+2 \int e^{-3+e^{4+x+e^5 (4+x)}} \log ^2(1-x) \, dx-2 \int e^{-3+e^{4+x+e^5 (4+x)}} (1-x) \log ^2(1-x) \, dx-2 \int \left (\frac {-\text {Ei}\left (e^{\left (1+e^5\right ) (4+x)}\right )-e^3 \left (1+e^5\right ) \int \frac {e^{-3+e^{\left (1+e^5\right ) (4+x)}}}{-1+x} \, dx}{e^3 \left (1+e^5\right ) (1-x)}+\frac {\int e^{-3+e^{\left (1+e^5\right ) (4+x)}} x \, dx}{-1+x}\right ) \, dx+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} \log ^2(1-x) \, dx+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x)^2 \log ^2(1-x) \, dx-\left (2 \left (1+e^5\right )\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x) \log ^2(1-x) \, dx+(2 \log (1-x)) \int \frac {e^{-3+e^{4+x+e^5 (4+x)}}}{-1+x} \, dx+(2 \log (1-x)) \int e^{-3+e^{4+x+e^5 (4+x)}} x \, dx\\ &=\frac {2 \text {Ei}\left (e^{4+x+e^5 (4+x)}\right ) \log (1-x)}{e^3 \left (1+e^5\right )}+2 \int e^{-3+e^{4+x+e^5 (4+x)}} \log ^2(1-x) \, dx-2 \int e^{-3+e^{4+x+e^5 (4+x)}} (1-x) \log ^2(1-x) \, dx-2 \int \frac {\int e^{-3+e^{\left (1+e^5\right ) (4+x)}} x \, dx}{-1+x} \, dx-\frac {2 \int \frac {-\text {Ei}\left (e^{\left (1+e^5\right ) (4+x)}\right )-e^3 \left (1+e^5\right ) \int \frac {e^{-3+e^{\left (1+e^5\right ) (4+x)}}}{-1+x} \, dx}{1-x} \, dx}{e^3 \left (1+e^5\right )}+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} \log ^2(1-x) \, dx+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x)^2 \log ^2(1-x) \, dx-\left (2 \left (1+e^5\right )\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x) \log ^2(1-x) \, dx+(2 \log (1-x)) \int \frac {e^{-3+e^{4+x+e^5 (4+x)}}}{-1+x} \, dx+(2 \log (1-x)) \int e^{-3+e^{4+x+e^5 (4+x)}} x \, dx\\ &=\frac {2 \text {Ei}\left (e^{4+x+e^5 (4+x)}\right ) \log (1-x)}{e^3 \left (1+e^5\right )}+2 \int e^{-3+e^{4+x+e^5 (4+x)}} \log ^2(1-x) \, dx-2 \int e^{-3+e^{4+x+e^5 (4+x)}} (1-x) \log ^2(1-x) \, dx-2 \int \frac {\int e^{-3+e^{\left (1+e^5\right ) (4+x)}} x \, dx}{-1+x} \, dx-\frac {2 \int \left (\frac {\text {Ei}\left (e^{4 \left (1+e^5\right )+\left (1+e^5\right ) x}\right )}{-1+x}+\frac {e^3 (1+e) \left (1-e+e^2-e^3+e^4\right ) \int \frac {e^{-3+e^{\left (1+e^5\right ) (4+x)}}}{-1+x} \, dx}{-1+x}\right ) \, dx}{e^3 \left (1+e^5\right )}+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} \log ^2(1-x) \, dx+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x)^2 \log ^2(1-x) \, dx-\left (2 \left (1+e^5\right )\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x) \log ^2(1-x) \, dx+(2 \log (1-x)) \int \frac {e^{-3+e^{4+x+e^5 (4+x)}}}{-1+x} \, dx+(2 \log (1-x)) \int e^{-3+e^{4+x+e^5 (4+x)}} x \, dx\\ &=\frac {2 \text {Ei}\left (e^{4+x+e^5 (4+x)}\right ) \log (1-x)}{e^3 \left (1+e^5\right )}+2 \int e^{-3+e^{4+x+e^5 (4+x)}} \log ^2(1-x) \, dx-2 \int e^{-3+e^{4+x+e^5 (4+x)}} (1-x) \log ^2(1-x) \, dx-2 \int \frac {\int \frac {e^{-3+e^{\left (1+e^5\right ) (4+x)}}}{-1+x} \, dx}{-1+x} \, dx-2 \int \frac {\int e^{-3+e^{\left (1+e^5\right ) (4+x)}} x \, dx}{-1+x} \, dx-\frac {2 \int \frac {\text {Ei}\left (e^{4 \left (1+e^5\right )+\left (1+e^5\right ) x}\right )}{-1+x} \, dx}{e^3 \left (1+e^5\right )}+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} \log ^2(1-x) \, dx+\left (1+e^5\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x)^2 \log ^2(1-x) \, dx-\left (2 \left (1+e^5\right )\right ) \int e^{1+e^{\left (1+e^5\right ) (4+x)}+x+e^5 (4+x)} (1-x) \log ^2(1-x) \, dx+(2 \log (1-x)) \int \frac {e^{-3+e^{4+x+e^5 (4+x)}}}{-1+x} \, dx+(2 \log (1-x)) \int e^{-3+e^{4+x+e^5 (4+x)}} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 9.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-3+e^{4+x+e^5 (4+x)}} \left (2 x^2 \log (1-x)+\left (-2 x+2 x^2+e^{4+x+e^5 (4+x)} \left (-x^2+x^3+e^5 \left (-x^2+x^3\right )\right )\right ) \log ^2(1-x)\right )}{-1+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 25, normalized size = 0.93 \begin {gather*} x^{2} e^{\left (e^{\left ({\left (x + 4\right )} e^{5} + x + 4\right )} - 3\right )} \log \left (-x + 1\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(e^((x + 4)*e^5 + x + 4) - 3)*log(-x + 1)^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{2} \log \left (-x + 1\right ) + {\left (2 \, x^{2} + {\left (x^{3} - x^{2} + {\left (x^{3} - x^{2}\right )} e^{5}\right )} e^{\left ({\left (x + 4\right )} e^{5} + x + 4\right )} - 2 \, x\right )} \log \left (-x + 1\right )^{2}\right )} e^{\left (e^{\left ({\left (x + 4\right )} e^{5} + x + 4\right )} - 3\right )}}{x - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^2*log(-x + 1) + (2*x^2 + (x^3 - x^2 + (x^3 - x^2)*e^5)*e^((x + 4)*e^5 + x + 4) - 2*x)*log(-x +
1)^2)*e^(e^((x + 4)*e^5 + x + 4) - 3)/(x - 1), x)

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maple [A]  time = 0.23, size = 25, normalized size = 0.93

method	result	size
risch	\(\ln \left (1-x \right )^{2} x^{2} {\mathrm e}^{{\mathrm e}^{\left (4+x \right ) \left ({\mathrm e}^{5}+1\right )}-3}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1-x)^2*x^2*exp(exp((4+x)*(exp(5)+1))-3)

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maxima [A]  time = 0.69, size = 27, normalized size = 1.00 \begin {gather*} x^{2} e^{\left (e^{\left (x e^{5} + x + 4 \, e^{5} + 4\right )} - 3\right )} \log \left (-x + 1\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(e^(x*e^5 + x + 4*e^5 + 4) - 3)*log(-x + 1)^2

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mupad [B]  time = 0.25, size = 30, normalized size = 1.11 \begin {gather*} x^2\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{{\mathrm {e}}^{4\,{\mathrm {e}}^5}\,{\mathrm {e}}^4\,{\mathrm {e}}^{x\,{\mathrm {e}}^5}\,{\mathrm {e}}^x}\,{\ln \left (1-x\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*exp(-3)*exp(exp(4*exp(5))*exp(4)*exp(x*exp(5))*exp(x))*log(1 - x)^2

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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