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

Optimal. Leaf size=23 \[ \frac {e+2 x+x^2+\log \left (\frac {1}{3 e^5 x}\right )}{x} \]

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Rubi [A]  time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {14, 2304} \begin {gather*} x+\frac {1+e}{x}-\frac {1}{x}-\frac {-\log \left (\frac {1}{x}\right )+5+\log (3)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + (1 + E)/x + x - (5 + Log[3] - Log[x^(-1)])/x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.17 \begin {gather*} -\frac {5}{x}+\frac {e}{x}+x-\frac {\log (3)}{x}+\frac {\log \left (\frac {1}{x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-5/x + E/x + x - Log[3]/x + Log[x^(-1)]/x

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fricas [A]  time = 0.59, size = 18, normalized size = 0.78 \begin {gather*} \frac {x^{2} + e + \log \left (\frac {e^{\left (-5\right )}}{3 \, x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 + e + log(1/3*e^(-5)/x))/x

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giac [A]  time = 0.14, size = 21, normalized size = 0.91 \begin {gather*} x {\left (\frac {e}{x^{2}} - \frac {\log \left (3 \, x e^{5}\right )}{x^{2}} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(e/x^2 - log(3*x*e^5)/x^2 + 1)

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maple [A]  time = 0.06, size = 21, normalized size = 0.91

method	result	size
norman	\(\frac {x^{2}+{\mathrm e}+\ln \left (\frac {{\mathrm e}^{-5}}{3 x}\right )}{x}\)	\(21\)
derivativedivides	\(x +\frac {{\mathrm e}}{x}+\frac {\ln \left (\frac {{\mathrm e}^{-5}}{3 x}\right )}{x}\)	\(23\)
default	\(x +\frac {{\mathrm e}}{x}+\frac {\ln \left (\frac {{\mathrm e}^{-5}}{3 x}\right )}{x}\)	\(23\)
risch	\(\frac {\ln \left (\frac {{\mathrm e}^{-5}}{3 x}\right )}{x}+\frac {{\mathrm e}+x^{2}}{x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2+exp(1)+ln(1/3/x/exp(5)))/x

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maxima [A]  time = 0.34, size = 36, normalized size = 1.57 \begin {gather*} {\left (\frac {e^{\left (-5\right )} \log \left (\frac {e^{\left (-5\right )}}{3 \, x}\right )}{x} - \frac {e^{\left (-5\right )}}{x}\right )} e^{5} + x + \frac {e}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.83, size = 17, normalized size = 0.74 \begin {gather*} x+\frac {\ln \left (\frac {{\mathrm {e}}^{-5}}{3\,x}\right )+\mathrm {e}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + (log(exp(-5)/(3*x)) + exp(1))/x

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sympy [A]  time = 0.14, size = 17, normalized size = 0.74 \begin {gather*} x + \frac {\log {\left (\frac {1}{3 x e^{5}} \right )}}{x} + \frac {e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(exp(-5)/(3*x))/x + E/x

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