3.70.84 \(\int \frac {-30+e^{\frac {1}{20} (40+e^{-1+x^2} (-1+2 x))} (10 e^x x+e^{-1+x+x^2} (x-x^2+2 x^3))}{10 x} \, dx\)

Optimal. Leaf size=26 \[ e^{2-\frac {1}{20} e^{-1+x^2} (1-2 x)+x}-3 \log (x) \]

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Rubi [F]  time = 1.38, 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 {-30+e^{\frac {1}{20} \left (40+e^{-1+x^2} (-1+2 x)\right )} \left (10 e^x x+e^{-1+x+x^2} \left (x-x^2+2 x^3\right )\right )}{10 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-30 + E^((40 + E^(-1 + x^2)*(-1 + 2*x))/20)*(10*E^x*x + E^(-1 + x + x^2)*(x - x^2 + 2*x^3)))/(10*x),x]

[Out]

-3*Log[x] + Defer[Int][E^(2 + x + (E^(-1 + x^2)*(-1 + 2*x))/20), x] + Defer[Int][E^(1 + x + x^2 + (E^(-1 + x^2
)*(-1 + 2*x))/20), x]/10 - Defer[Int][E^(1 + x + x^2 + (E^(-1 + x^2)*(-1 + 2*x))/20)*x, x]/10 + Defer[Int][E^(
1 + x + x^2 + (E^(-1 + x^2)*(-1 + 2*x))/20)*x^2, x]/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{10} \int \frac {-30+e^{\frac {1}{20} \left (40+e^{-1+x^2} (-1+2 x)\right )} \left (10 e^x x+e^{-1+x+x^2} \left (x-x^2+2 x^3\right )\right )}{x} \, dx\\ &=\frac {1}{10} \int \left (-\frac {30}{x}+e^{1+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \left (10 e+e^{x^2}-e^{x^2} x+2 e^{x^2} x^2\right )\right ) \, dx\\ &=-3 \log (x)+\frac {1}{10} \int e^{1+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \left (10 e+e^{x^2}-e^{x^2} x+2 e^{x^2} x^2\right ) \, dx\\ &=-3 \log (x)+\frac {1}{10} \int e^{1+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \left (10 e+e^{x^2} \left (1-x+2 x^2\right )\right ) \, dx\\ &=-3 \log (x)+\frac {1}{10} \int \left (10 e^{2+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)}+e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \left (1-x+2 x^2\right )\right ) \, dx\\ &=-3 \log (x)+\frac {1}{10} \int e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \left (1-x+2 x^2\right ) \, dx+\int e^{2+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \, dx\\ &=-3 \log (x)+\frac {1}{10} \int \left (e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)}-e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)} x+2 e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)} x^2\right ) \, dx+\int e^{2+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \, dx\\ &=-3 \log (x)+\frac {1}{10} \int e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \, dx-\frac {1}{10} \int e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)} x \, dx+\frac {1}{5} \int e^{1+x+x^2+\frac {1}{20} e^{-1+x^2} (-1+2 x)} x^2 \, dx+\int e^{2+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 26, normalized size = 1.00 \begin {gather*} e^{2+x+\frac {1}{20} e^{-1+x^2} (-1+2 x)}-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-30 + E^((40 + E^(-1 + x^2)*(-1 + 2*x))/20)*(10*E^x*x + E^(-1 + x + x^2)*(x - x^2 + 2*x^3)))/(10*x)
,x]

[Out]

E^(2 + x + (E^(-1 + x^2)*(-1 + 2*x))/20) - 3*Log[x]

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fricas [A]  time = 1.06, size = 43, normalized size = 1.65 \begin {gather*} -{\left (3 \, e^{\left (x^{2} - 1\right )} \log \relax (x) - e^{\left (x^{2} + \frac {1}{20} \, {\left (2 \, x - 1\right )} e^{\left (x^{2} - 1\right )} + x + 1\right )}\right )} e^{\left (-x^{2} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(((2*x^3-x^2+x)*exp(x)*exp(x^2-1)+10*exp(x)*x)*exp(1/20*(2*x-1)*exp(x^2-1)+2)-30)/x,x, algorith
m="fricas")

[Out]

-(3*e^(x^2 - 1)*log(x) - e^(x^2 + 1/20*(2*x - 1)*e^(x^2 - 1) + x + 1))*e^(-x^2 + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (2 \, x^{3} - x^{2} + x\right )} e^{\left (x^{2} + x - 1\right )} + 10 \, x e^{x}\right )} e^{\left (\frac {1}{20} \, {\left (2 \, x - 1\right )} e^{\left (x^{2} - 1\right )} + 2\right )} - 30}{10 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(((2*x^3-x^2+x)*exp(x)*exp(x^2-1)+10*exp(x)*x)*exp(1/20*(2*x-1)*exp(x^2-1)+2)-30)/x,x, algorith
m="giac")

[Out]

integrate(1/10*(((2*x^3 - x^2 + x)*e^(x^2 + x - 1) + 10*x*e^x)*e^(1/20*(2*x - 1)*e^(x^2 - 1) + 2) - 30)/x, x)

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maple [A]  time = 0.10, size = 31, normalized size = 1.19

method	result	size
risch	\(-3 \ln \relax (x )+{\mathrm e}^{x +\frac {{\mathrm e}^{\left (x -1\right ) \left (x +1\right )} x}{10}-\frac {{\mathrm e}^{\left (x -1\right ) \left (x +1\right )}}{20}+2}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/10*(((2*x^3-x^2+x)*exp(x)*exp(x^2-1)+10*exp(x)*x)*exp(1/20*(2*x-1)*exp(x^2-1)+2)-30)/x,x,method=_RETURNV
ERBOSE)

[Out]

-3*ln(x)+exp(x+1/10*exp((x-1)*(x+1))*x-1/20*exp((x-1)*(x+1))+2)

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maxima [A]  time = 0.53, size = 26, normalized size = 1.00 \begin {gather*} e^{\left (\frac {1}{10} \, x e^{\left (x^{2} - 1\right )} + x - \frac {1}{20} \, e^{\left (x^{2} - 1\right )} + 2\right )} - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(((2*x^3-x^2+x)*exp(x)*exp(x^2-1)+10*exp(x)*x)*exp(1/20*(2*x-1)*exp(x^2-1)+2)-30)/x,x, algorith
m="maxima")

[Out]

e^(1/10*x*e^(x^2 - 1) + x - 1/20*e^(x^2 - 1) + 2) - 3*log(x)

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mupad [B]  time = 4.12, size = 26, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{x-\frac {{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-1}}{20}+\frac {x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-1}}{10}+2}-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp((exp(x^2 - 1)*(2*x - 1))/20 + 2)*(10*x*exp(x) + exp(x^2 - 1)*exp(x)*(x - x^2 + 2*x^3)))/10 - 3)/x,x)

[Out]

exp(x - (exp(x^2)*exp(-1))/20 + (x*exp(x^2)*exp(-1))/10 + 2) - 3*log(x)

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sympy [A]  time = 48.25, size = 24, normalized size = 0.92 \begin {gather*} e^{x} e^{\left (\frac {x}{10} - \frac {1}{20}\right ) e^{x^{2} - 1} + 2} - 3 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(((2*x**3-x**2+x)*exp(x)*exp(x**2-1)+10*exp(x)*x)*exp(1/20*(2*x-1)*exp(x**2-1)+2)-30)/x,x)

[Out]

exp(x)*exp((x/10 - 1/20)*exp(x**2 - 1) + 2) - 3*log(x)

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