3.85.56 $\int \frac{-4x^2+{\log }^2(2)}{4x^2}dx$

Optimal. Leaf size=19 $$-\frac{19}{12}-\frac{{(x+\frac{\log (2)}{2})}^2}{x}$$

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 2, integrand size = 17, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.118, Rules used = {12, 14} $$\begin{array}{c}-x-\frac{{\log }^2(2)}{4x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-4*x^2 + Log[2]^2)/(4*x^2),x]

[Out]

-x - Log[2]^2/(4*x)

Rule 12

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{4}\int \frac{-4x^2+{\log }^2(2)}{x^2}dx\\ & =\frac{1}{4}\int (-4+\frac{{\log }^2(2)}{x^2})dx\\ & =-x-\frac{{\log }^2(2)}{4x}\end{array}\end{array}$$

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.79 $$\begin{array}{c}-x-\frac{{\log }^2(2)}{4x}\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

-x - Log[2]^2/(4*x)

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fricas [A]  time = 1.01, size = 15, normalized size = 0.79 $$\begin{array}{c}-\frac{4x^2+\log (2{)}^2}{4x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*x^2 + log(2)^2)/x

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giac [A]  time = 0.12, size = 13, normalized size = 0.68 $$\begin{array}{c}-x-\frac{\log (2{)}^2}{4x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 1/4*log(2)^2/x

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maple [A]  time = 0.03, size = 14, normalized size = 0.74

method	result	size
default	$-x-\frac{\ln (2{)}^2}{4x}$	$14$
risch	$-x-\frac{\ln (2{)}^2}{4x}$	$14$
gosper	$-\frac{\ln (2{)}^2+4x^2}{4x}$	$16$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(ln(2)^2-4*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.48, size = 13, normalized size = 0.68 $$\begin{array}{c}-x-\frac{\log (2{)}^2}{4x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - 1/4*log(2)^2/x

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mupad [B]  time = 0.03, size = 13, normalized size = 0.68 $$\begin{array}{c}-x-\frac{{\ln (2)}^2}{4x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)^2/4 - x^2)/x^2,x)

[Out]

- x - log(2)^2/(4*x)

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sympy [A]  time = 0.08, size = 10, normalized size = 0.53 $$\begin{array}{c}-x-\frac{\log {(2)}^2}{4x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(2)**2/(4*x)

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