3.21.25 \(\int \frac {-16 x+x^3+(16+x) \log (\frac {4}{\log (2)})}{x^3} \, dx\)

Optimal. Leaf size=21 \[ x-\frac {(8+x) \left (-2 x+\log \left (\frac {4}{\log (2)}\right )\right )}{x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.38, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14} \begin {gather*} -\frac {8 \log \left (\frac {4}{\log (2)}\right )}{x^2}+x+\frac {16-\log \left (\frac {4}{\log (2)}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16*x + x^3 + (16 + x)*Log[4/Log[2]])/x^3,x]

[Out]

x + (16 - Log[4/Log[2]])/x - (8*Log[4/Log[2]])/x^2

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {-16+\log \left (\frac {4}{\log (2)}\right )}{x^2}+\frac {16 \log \left (\frac {4}{\log (2)}\right )}{x^3}\right ) \, dx\\ &=x+\frac {16-\log \left (\frac {4}{\log (2)}\right )}{x}-\frac {8 \log \left (\frac {4}{\log (2)}\right )}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.38 \begin {gather*} x+\frac {16-\log \left (\frac {4}{\log (2)}\right )}{x}-\frac {8 \log \left (\frac {4}{\log (2)}\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x + x^3 + (16 + x)*Log[4/Log[2]])/x^3,x]

[Out]

x + (16 - Log[4/Log[2]])/x - (8*Log[4/Log[2]])/x^2

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fricas [A]  time = 0.92, size = 23, normalized size = 1.10 \begin {gather*} \frac {x^{3} - {\left (x + 8\right )} \log \left (\frac {4}{\log \relax (2)}\right ) + 16 \, x}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*log(4/log(2))+x^3-16*x)/x^3,x, algorithm="fricas")

[Out]

(x^3 - (x + 8)*log(4/log(2)) + 16*x)/x^2

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giac [A]  time = 0.22, size = 29, normalized size = 1.38 \begin {gather*} x - \frac {x \log \left (\frac {4}{\log \relax (2)}\right ) - 16 \, x + 8 \, \log \left (\frac {4}{\log \relax (2)}\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*log(4/log(2))+x^3-16*x)/x^3,x, algorithm="giac")

[Out]

x - (x*log(4/log(2)) - 16*x + 8*log(4/log(2)))/x^2

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maple [A]  time = 0.05, size = 28, normalized size = 1.33

method	result	size
risch	\(x +\frac {\left (-2 \ln \relax (2)+\ln \left (\ln \relax (2)\right )+16\right ) x -16 \ln \relax (2)+8 \ln \left (\ln \relax (2)\right )}{x^{2}}\)	\(28\)
default	\(x -\frac {\ln \left (\frac {4}{\ln \relax (2)}\right )-16}{x}-\frac {8 \ln \left (\frac {4}{\ln \relax (2)}\right )}{x^{2}}\)	\(29\)
norman	\(\frac {x^{3}+\left (-2 \ln \relax (2)+\ln \left (\ln \relax (2)\right )+16\right ) x -16 \ln \relax (2)+8 \ln \left (\ln \relax (2)\right )}{x^{2}}\)	\(29\)
gosper	\(-\frac {-x^{3}+\ln \left (\frac {4}{\ln \relax (2)}\right ) x +8 \ln \left (\frac {4}{\ln \relax (2)}\right )-16 x}{x^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x+16)*ln(4/ln(2))+x^3-16*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

x+((-2*ln(2)+ln(ln(2))+16)*x-16*ln(2)+8*ln(ln(2)))/x^2

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maxima [A]  time = 0.85, size = 28, normalized size = 1.33 \begin {gather*} x - \frac {x {\left (\log \left (\frac {4}{\log \relax (2)}\right ) - 16\right )} + 8 \, \log \left (\frac {4}{\log \relax (2)}\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*log(4/log(2))+x^3-16*x)/x^3,x, algorithm="maxima")

[Out]

x - (x*(log(4/log(2)) - 16) + 8*log(4/log(2)))/x^2

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mupad [B]  time = 0.09, size = 26, normalized size = 1.24 \begin {gather*} x+\frac {\ln \left (\frac {{\ln \relax (2)}^8}{65536}\right )-x\,\left (\ln \left (\frac {4}{\ln \relax (2)}\right )-16\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4/log(2))*(x + 16) - 16*x + x^3)/x^3,x)

[Out]

x + (log(log(2)^8/65536) - x*(log(4/log(2)) - 16))/x^2

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sympy [A]  time = 0.21, size = 29, normalized size = 1.38 \begin {gather*} x + \frac {x \left (- 2 \log {\relax (2 )} + \log {\left (\log {\relax (2 )} \right )} + 16\right ) - 16 \log {\relax (2 )} + 8 \log {\left (\log {\relax (2 )} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*ln(4/ln(2))+x**3-16*x)/x**3,x)

[Out]

x + (x*(-2*log(2) + log(log(2)) + 16) - 16*log(2) + 8*log(log(2)))/x**2

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