3.90.27 \(\int \frac {2 x^4+(120+60 x-x^3) \log ^2(2)}{x^3 \log ^2(2)} \, dx\)

Optimal. Leaf size=26 \[ 4-e^3-x-\frac {60 (1+x)}{x^2}+\frac {x^2}{\log ^2(2)} \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 14} \begin {gather*} -\frac {60}{x^2}+\frac {x^2}{\log ^2(2)}-x-\frac {60}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^4 + (120 + 60*x - x^3)*Log[2]^2)/(x^3*Log[2]^2),x]

[Out]

-60/x^2 - 60/x - x + x^2/Log[2]^2

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 {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {2 x^4+\left (120+60 x-x^3\right ) \log ^2(2)}{x^3} \, dx}{\log ^2(2)}\\ &=\frac {\int \left (2 x-\log ^2(2)+\frac {120 \log ^2(2)}{x^3}+\frac {60 \log ^2(2)}{x^2}\right ) \, dx}{\log ^2(2)}\\ &=-\frac {60}{x^2}-\frac {60}{x}-x+\frac {x^2}{\log ^2(2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 0.85 \begin {gather*} -\frac {60}{x^2}-\frac {60}{x}-x+\frac {x^2}{\log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^4 + (120 + 60*x - x^3)*Log[2]^2)/(x^3*Log[2]^2),x]

[Out]

-60/x^2 - 60/x - x + x^2/Log[2]^2

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fricas [A]  time = 0.62, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^{4} - {\left (x^{3} + 60 \, x + 60\right )} \log \relax (2)^{2}}{x^{2} \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3+60*x+120)*log(2)^2+2*x^4)/x^3/log(2)^2,x, algorithm="fricas")

[Out]

(x^4 - (x^3 + 60*x + 60)*log(2)^2)/(x^2*log(2)^2)

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giac [A]  time = 0.22, size = 34, normalized size = 1.31 \begin {gather*} -\frac {x \log \relax (2)^{2} - x^{2} + \frac {60 \, {\left (x \log \relax (2)^{2} + \log \relax (2)^{2}\right )}}{x^{2}}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3+60*x+120)*log(2)^2+2*x^4)/x^3/log(2)^2,x, algorithm="giac")

[Out]

-(x*log(2)^2 - x^2 + 60*(x*log(2)^2 + log(2)^2)/x^2)/log(2)^2

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maple [A]  time = 0.05, size = 34, normalized size = 1.31

method	result	size
norman	\(\frac {\frac {x^{4}}{\ln \relax (2)}-60 x \ln \relax (2)-x^{3} \ln \relax (2)-60 \ln \relax (2)}{x^{2} \ln \relax (2)}\)	\(34\)
default	\(\frac {x^{2}-x \ln \relax (2)^{2}-\frac {60 \ln \relax (2)^{2}}{x^{2}}-\frac {60 \ln \relax (2)^{2}}{x}}{\ln \relax (2)^{2}}\)	\(35\)
risch	\(-x +\frac {x^{2}}{\ln \relax (2)^{2}}+\frac {-60 x \ln \relax (2)^{2}-60 \ln \relax (2)^{2}}{\ln \relax (2)^{2} x^{2}}\)	\(35\)
gosper	\(-\frac {-x^{4}+x^{3} \ln \relax (2)^{2}+60 x \ln \relax (2)^{2}+60 \ln \relax (2)^{2}}{\ln \relax (2)^{2} x^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3+60*x+120)*ln(2)^2+2*x^4)/x^3/ln(2)^2,x,method=_RETURNVERBOSE)

[Out]

(1/ln(2)*x^4-60*x*ln(2)-x^3*ln(2)-60*ln(2))/x^2/ln(2)

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maxima [A]  time = 0.35, size = 34, normalized size = 1.31 \begin {gather*} -\frac {x \log \relax (2)^{2} - x^{2} + \frac {60 \, {\left (x \log \relax (2)^{2} + \log \relax (2)^{2}\right )}}{x^{2}}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3+60*x+120)*log(2)^2+2*x^4)/x^3/log(2)^2,x, algorithm="maxima")

[Out]

-(x*log(2)^2 - x^2 + 60*(x*log(2)^2 + log(2)^2)/x^2)/log(2)^2

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mupad [B]  time = 0.06, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^2}{{\ln \relax (2)}^2}-x-\frac {60\,x+60}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)^2*(60*x - x^3 + 120) + 2*x^4)/(x^3*log(2)^2),x)

[Out]

x^2/log(2)^2 - x - (60*x + 60)/x^2

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sympy [A]  time = 0.11, size = 34, normalized size = 1.31 \begin {gather*} \frac {x^{2} - x \log {\relax (2 )}^{2} + \frac {- 60 x \log {\relax (2 )}^{2} - 60 \log {\relax (2 )}^{2}}{x^{2}}}{\log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3+60*x+120)*ln(2)**2+2*x**4)/x**3/ln(2)**2,x)

[Out]

(x**2 - x*log(2)**2 + (-60*x*log(2)**2 - 60*log(2)**2)/x**2)/log(2)**2

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