3.96.1 \(\int \frac {2-2 x^4-4 \log (4)+(2+50 x^4) \log ^2(4)}{x^3} \, dx\)

Optimal. Leaf size=27 \[ 4+25 x^2 \log ^2(4)-\frac {\left (-1-x^2+\log (4)\right )^2}{x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {14} \begin {gather*} -\left (x^2 \left (1-25 \log ^2(4)\right )\right )-\frac {(1-\log (4))^2}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 2*x^4 - 4*Log[4] + (2 + 50*x^4)*Log[4]^2)/x^3,x]

[Out]

-((1 - Log[4])^2/x^2) - x^2*(1 - 25*Log[4]^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 (\frac {2 (-1+\log (4))^2}{x^3}+2 x \left (-1+25 \log ^2(4)\right )\right ) \, dx\\ &=-\frac {(1-\log (4))^2}{x^2}-x^2 \left (1-25 \log ^2(4)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.96 \begin {gather*} -x^2-\frac {(-1+\log (4))^2}{x^2}+25 x^2 \log ^2(4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 2*x^4 - 4*Log[4] + (2 + 50*x^4)*Log[4]^2)/x^3,x]

[Out]

-x^2 - (-1 + Log[4])^2/x^2 + 25*x^2*Log[4]^2

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fricas [A]  time = 0.68, size = 27, normalized size = 1.00 \begin {gather*} -\frac {x^{4} - 4 \, {\left (25 \, x^{4} - 1\right )} \log \relax (2)^{2} - 4 \, \log \relax (2) + 1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(50*x^4+2)*log(2)^2-8*log(2)-2*x^4+2)/x^3,x, algorithm="fricas")

[Out]

-(x^4 - 4*(25*x^4 - 1)*log(2)^2 - 4*log(2) + 1)/x^2

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giac [A]  time = 0.14, size = 32, normalized size = 1.19 \begin {gather*} 100 \, x^{2} \log \relax (2)^{2} - x^{2} - \frac {4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(50*x^4+2)*log(2)^2-8*log(2)-2*x^4+2)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 29, normalized size = 1.07

method	result	size
norman	\(\frac {\left (100 \ln \relax (2)^{2}-1\right ) x^{4}-4 \ln \relax (2)^{2}+4 \ln \relax (2)-1}{x^{2}}\)	\(29\)
gosper	\(\frac {100 x^{4} \ln \relax (2)^{2}-x^{4}-4 \ln \relax (2)^{2}+4 \ln \relax (2)-1}{x^{2}}\)	\(31\)
default	\(\left (100 \ln \relax (2)^{2}-1\right ) x^{2}-\frac {4 \ln \relax (2)^{2}-4 \ln \relax (2)+1}{x^{2}}\)	\(31\)
risch	\(100 x^{2} \ln \relax (2)^{2}-x^{2}-\frac {4 \ln \relax (2)^{2}}{x^{2}}+\frac {4 \ln \relax (2)}{x^{2}}-\frac {1}{x^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(50*x^4+2)*ln(2)^2-8*ln(2)-2*x^4+2)/x^3,x,method=_RETURNVERBOSE)

[Out]

((100*ln(2)^2-1)*x^4-4*ln(2)^2+4*ln(2)-1)/x^2

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maxima [A]  time = 0.35, size = 30, normalized size = 1.11 \begin {gather*} {\left (100 \, \log \relax (2)^{2} - 1\right )} x^{2} - \frac {4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(50*x^4+2)*log(2)^2-8*log(2)-2*x^4+2)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.09, size = 30, normalized size = 1.11 \begin {gather*} x^2\,\left (100\,{\ln \relax (2)}^2-1\right )-\frac {4\,{\ln \relax (2)}^2-\ln \left (16\right )+1}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*log(2) - 4*log(2)^2*(50*x^4 + 2) + 2*x^4 - 2)/x^3,x)

[Out]

x^2*(100*log(2)^2 - 1) - (4*log(2)^2 - log(16) + 1)/x^2

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sympy [A]  time = 0.13, size = 29, normalized size = 1.07 \begin {gather*} - x^{2} \left (1 - 100 \log {\relax (2 )}^{2}\right ) - \frac {- 4 \log {\relax (2 )} + 1 + 4 \log {\relax (2 )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(50*x**4+2)*ln(2)**2-8*ln(2)-2*x**4+2)/x**3,x)

[Out]

-x**2*(1 - 100*log(2)**2) - (-4*log(2) + 1 + 4*log(2)**2)/x**2

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