3.90.27 \(\int \frac {2 x^4+(120+60 x-x^3) \log ^2(2)}{x^3 \log ^2(2)} \, dx\) [8927]

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

[Out]

x^2/ln(2)^2-exp(3)-x+4-12*(5*x+5)/x^2

________________________________________________________________________________________

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*}

________________________________________________________________________________________

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

________________________________________________________________________________________

Maple [A]
time = 0.40, size = 35, normalized size = 1.35

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

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)^2*(x^2-x*ln(2)^2-60*ln(2)^2/x^2-60*ln(2)^2/x)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 34, normalized size = 1.31 \begin {gather*} -\frac {x \log \left (2\right )^{2} - x^{2} + \frac {60 \, {\left (x \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )}}{x^{2}}}{\log \left (2\right )^{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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^{4} - {\left (x^{3} + 60 \, x + 60\right )} \log \left (2\right )^{2}}{x^{2} \log \left (2\right )^{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)

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 34, normalized size = 1.31 \begin {gather*} \frac {x^{2} - x \log {\left (2 \right )}^{2} + \frac {- 60 x \log {\left (2 \right )}^{2} - 60 \log {\left (2 \right )}^{2}}{x^{2}}}{\log {\left (2 \right )}^{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

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 34, normalized size = 1.31 \begin {gather*} -\frac {x \log \left (2\right )^{2} - x^{2} + \frac {60 \, {\left (x \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )}}{x^{2}}}{\log \left (2\right )^{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

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^2}{{\ln \left (2\right )}^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

________________________________________________________________________________________