3.88.52 \(\int \frac {1}{4} (2+e^{21} (-1-2 x)-e^{21} \log (x)) \, dx\)

Optimal. Leaf size=18 \[ \frac {1}{4} \left (2 x-e^{21} x (x+\log (x))\right ) \]

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Rubi [B]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 2.11, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 2295} \begin {gather*} -\frac {1}{16} e^{21} (2 x+1)^2+\frac {e^{21} x}{4}+\frac {x}{2}-\frac {1}{4} e^{21} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^21*(-1 - 2*x) - E^21*Log[x])/4,x]

[Out]

x/2 + (E^21*x)/4 - (E^21*(1 + 2*x)^2)/16 - (E^21*x*Log[x])/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (2+e^{21} (-1-2 x)-e^{21} \log (x)\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{16} e^{21} (1+2 x)^2-\frac {1}{4} e^{21} \int \log (x) \, dx\\ &=\frac {x}{2}+\frac {e^{21} x}{4}-\frac {1}{16} e^{21} (1+2 x)^2-\frac {1}{4} e^{21} x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.44 \begin {gather*} \frac {x}{2}-\frac {e^{21} x^2}{4}-\frac {1}{4} e^{21} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^21*(-1 - 2*x) - E^21*Log[x])/4,x]

[Out]

x/2 - (E^21*x^2)/4 - (E^21*x*Log[x])/4

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fricas [A]  time = 0.60, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, x^{2} e^{21} - \frac {1}{4} \, x e^{21} \log \relax (x) + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/4*exp(21)*log(x)+1/4*(-2*x-1)*exp(21)+1/2,x, algorithm="fricas")

[Out]

-1/4*x^2*e^21 - 1/4*x*e^21*log(x) + 1/2*x

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giac [A]  time = 0.17, size = 25, normalized size = 1.39 \begin {gather*} -\frac {1}{4} \, {\left (x^{2} + x\right )} e^{21} - \frac {1}{4} \, {\left (x \log \relax (x) - x\right )} e^{21} + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/4*exp(21)*log(x)+1/4*(-2*x-1)*exp(21)+1/2,x, algorithm="giac")

[Out]

-1/4*(x^2 + x)*e^21 - 1/4*(x*log(x) - x)*e^21 + 1/2*x

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maple [A]  time = 0.03, size = 19, normalized size = 1.06

method	result	size
norman	\(\frac {x}{2}-\frac {{\mathrm e}^{21} x^{2}}{4}-\frac {\ln \relax (x ) {\mathrm e}^{21} x}{4}\)	\(19\)
risch	\(\frac {x}{2}-\frac {{\mathrm e}^{21} x^{2}}{4}-\frac {\ln \relax (x ) {\mathrm e}^{21} x}{4}\)	\(19\)
default	\(\frac {x}{2}+\frac {{\mathrm e}^{21} \left (-x^{2}-x \right )}{4}-\frac {\ln \relax (x ) {\mathrm e}^{21} x}{4}+\frac {x \,{\mathrm e}^{21}}{4}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/4*exp(21)*ln(x)+1/4*(-2*x-1)*exp(21)+1/2,x,method=_RETURNVERBOSE)

[Out]

1/2*x-1/4*exp(21)*x^2-1/4*ln(x)*exp(21)*x

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maxima [A]  time = 0.34, size = 25, normalized size = 1.39 \begin {gather*} -\frac {1}{4} \, {\left (x^{2} + x\right )} e^{21} - \frac {1}{4} \, {\left (x \log \relax (x) - x\right )} e^{21} + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/4*exp(21)*log(x)+1/4*(-2*x-1)*exp(21)+1/2,x, algorithm="maxima")

[Out]

-1/4*(x^2 + x)*e^21 - 1/4*(x*log(x) - x)*e^21 + 1/2*x

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mupad [B]  time = 5.62, size = 14, normalized size = 0.78 \begin {gather*} -\frac {x\,\left (x\,{\mathrm {e}}^{21}+{\mathrm {e}}^{21}\,\ln \relax (x)-2\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2 - (exp(21)*(2*x + 1))/4 - (exp(21)*log(x))/4,x)

[Out]

-(x*(x*exp(21) + exp(21)*log(x) - 2))/4

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sympy [A]  time = 0.10, size = 20, normalized size = 1.11 \begin {gather*} - \frac {x^{2} e^{21}}{4} - \frac {x e^{21} \log {\relax (x )}}{4} + \frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/4*exp(21)*ln(x)+1/4*(-2*x-1)*exp(21)+1/2,x)

[Out]

-x**2*exp(21)/4 - x*exp(21)*log(x)/4 + x/2

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