3.55.42 \(\int \frac {-16-128 x^2+768 x^4+(-2-16 x^2+96 x^4) \log (4)}{x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {2 \left (1-4 x^2\right )^2 (8+\log (4))}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {14} \begin {gather*} 32 x^3 (8+\log (4))-16 x (8+\log (4))+\frac {2 (8+\log (4))}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 - 128*x^2 + 768*x^4 + (-2 - 16*x^2 + 96*x^4)*Log[4])/x^2,x]

[Out]

(2*(8 + Log[4]))/x - 16*x*(8 + Log[4]) + 32*x^3*(8 + Log[4])

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 (-16 (8+\log (4))-\frac {2 (8+\log (4))}{x^2}+96 x^2 (8+\log (4))\right ) \, dx\\ &=\frac {2 (8+\log (4))}{x}-16 x (8+\log (4))+32 x^3 (8+\log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} 2 \left (\frac {1}{x}-8 x+16 x^3\right ) (8+\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 - 128*x^2 + 768*x^4 + (-2 - 16*x^2 + 96*x^4)*Log[4])/x^2,x]

[Out]

2*(x^(-1) - 8*x + 16*x^3)*(8 + Log[4])

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fricas [A]  time = 1.57, size = 32, normalized size = 1.78 \begin {gather*} \frac {4 \, {\left (64 \, x^{4} - 32 \, x^{2} + {\left (16 \, x^{4} - 8 \, x^{2} + 1\right )} \log \relax (2) + 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(96*x^4-16*x^2-2)*log(2)+768*x^4-128*x^2-16)/x^2,x, algorithm="fricas")

[Out]

4*(64*x^4 - 32*x^2 + (16*x^4 - 8*x^2 + 1)*log(2) + 4)/x

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giac [A]  time = 0.21, size = 30, normalized size = 1.67 \begin {gather*} 64 \, x^{3} \log \relax (2) + 256 \, x^{3} - 32 \, x \log \relax (2) - 128 \, x + \frac {4 \, {\left (\log \relax (2) + 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(96*x^4-16*x^2-2)*log(2)+768*x^4-128*x^2-16)/x^2,x, algorithm="giac")

[Out]

64*x^3*log(2) + 256*x^3 - 32*x*log(2) - 128*x + 4*(log(2) + 4)/x

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maple [A]  time = 0.04, size = 21, normalized size = 1.17

method	result	size
default	\(2 \left (8+2 \ln \relax (2)\right ) \left (16 x^{3}-8 x +\frac {1}{x}\right )\)	\(21\)
norman	\(\frac {\left (-32 \ln \relax (2)-128\right ) x^{2}+\left (64 \ln \relax (2)+256\right ) x^{4}+4 \ln \relax (2)+16}{x}\)	\(31\)
gosper	\(\frac {4 \left (2 x \ln \relax (2)+\ln \relax (2)+8 x +4\right ) \left (8 x^{3}-4 x^{2}-2 x +1\right )}{x}\)	\(33\)
risch	\(64 x^{3} \ln \relax (2)+256 x^{3}-32 x \ln \relax (2)-128 x +\frac {4 \ln \relax (2)}{x}+\frac {16}{x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(96*x^4-16*x^2-2)*ln(2)+768*x^4-128*x^2-16)/x^2,x,method=_RETURNVERBOSE)

[Out]

2*(8+2*ln(2))*(16*x^3-8*x+1/x)

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maxima [A]  time = 0.38, size = 26, normalized size = 1.44 \begin {gather*} 64 \, x^{3} {\left (\log \relax (2) + 4\right )} - 32 \, x {\left (\log \relax (2) + 4\right )} + \frac {4 \, {\left (\log \relax (2) + 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(96*x^4-16*x^2-2)*log(2)+768*x^4-128*x^2-16)/x^2,x, algorithm="maxima")

[Out]

64*x^3*(log(2) + 4) - 32*x*(log(2) + 4) + 4*(log(2) + 4)/x

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mupad [B]  time = 0.06, size = 28, normalized size = 1.56 \begin {gather*} \frac {\ln \left (16\right )+16}{x}-x\,\left (32\,\ln \relax (2)+128\right )+x^3\,\left (64\,\ln \relax (2)+256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(2)*(16*x^2 - 96*x^4 + 2) + 128*x^2 - 768*x^4 + 16)/x^2,x)

[Out]

(log(16) + 16)/x - x*(32*log(2) + 128) + x^3*(64*log(2) + 256)

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sympy [A]  time = 0.10, size = 27, normalized size = 1.50 \begin {gather*} x^{3} \left (64 \log {\relax (2 )} + 256\right ) + x \left (-128 - 32 \log {\relax (2 )}\right ) + \frac {4 \log {\relax (2 )} + 16}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(96*x**4-16*x**2-2)*ln(2)+768*x**4-128*x**2-16)/x**2,x)

[Out]

x**3*(64*log(2) + 256) + x*(-128 - 32*log(2)) + (4*log(2) + 16)/x

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