\(\int \frac {1+(3+5 x^4) \log ^2(2)+\log ^2(2) \log (x^{1+\frac {1}{\log ^2(2)}})}{\log ^2(2)} \, dx\) [8870]

Optimal result

Integrand size = 33, antiderivative size = 16 \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=x \left (2+x^4+\log \left (x^{1+\frac {1}{\log ^2(2)}}\right )\right ) \]

[Out]

(x^4+ln(x*exp(ln(x)/ln(2)^2))+2)*x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(16)=32\).

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {12, 2332} \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=x \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )+x^5+3 x+\frac {x}{\log ^2(2)}-x \left (1+\frac {1}{\log ^2(2)}\right ) \]

[In]

Int[(1 + (3 + 5*x^4)*Log[2]^2 + Log[2]^2*Log[x^(1 + Log[2]^(-2))])/Log[2]^2,x]

[Out]

3*x + x^5 - x*(1 + Log[2]^(-2)) + x/Log[2]^2 + x*Log[x^(1 + 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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )\right ) \, dx}{\log ^2(2)} \\ & = \frac {x}{\log ^2(2)}+\int \left (3+5 x^4\right ) \, dx+\int \log \left (x^{1+\frac {1}{\log ^2(2)}}\right ) \, dx \\ & = 3 x+x^5-x \left (1+\frac {1}{\log ^2(2)}\right )+\frac {x}{\log ^2(2)}+x \log \left (x^{1+\frac {1}{\log ^2(2)}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=x \left (2+x^4+\log \left (x^{1+\frac {1}{\log ^2(2)}}\right )\right ) \]

[In]

Integrate[(1 + (3 + 5*x^4)*Log[2]^2 + Log[2]^2*Log[x^(1 + Log[2]^(-2))])/Log[2]^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19

method	result	size
parts	\(x^{5}+2 x +\ln \left (x \,x^{\frac {1}{\ln \left (2\right )^{2}}}\right ) x\)	\(19\)
default	\(\frac {\ln \left (2\right )^{2} \left (x^{5}+3 x \right )+\ln \left (2\right )^{2} \ln \left (x \,x^{\frac {1}{\ln \left (2\right )^{2}}}\right ) x -x \ln \left (2\right )^{2}}{\ln \left (2\right )^{2}}\)	\(41\)
parallelrisch	\(\frac {\frac {x^{5} \ln \left (2\right )^{12}+2 \ln \left (2\right )^{12} x -x \ln \left (2\right )^{10}+\ln \left (2\right )^{12} \ln \left (x \,{\mathrm e}^{\frac {\ln \left (x \right )}{\ln \left (2\right )^{2}}}\right ) x}{\ln \left (2\right )^{10}}+x}{\ln \left (2\right )^{2}}\)	\(53\)
risch	\(x \ln \left (x^{\frac {1}{\ln \left (2\right )^{2}}}\right )+x \ln \left (x \right )+\frac {i \pi x \,\operatorname {csgn}\left (i x^{\frac {1}{\ln \left (2\right )^{2}}}\right ) \operatorname {csgn}\left (i x \,x^{\frac {1}{\ln \left (2\right )^{2}}}\right )^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i x^{\frac {1}{\ln \left (2\right )^{2}}}\right ) \operatorname {csgn}\left (i x \,x^{\frac {1}{\ln \left (2\right )^{2}}}\right ) \operatorname {csgn}\left (i x \right )}{2}-\frac {i \pi x \operatorname {csgn}\left (i x \,x^{\frac {1}{\ln \left (2\right )^{2}}}\right )^{3}}{2}+\frac {i \pi x \operatorname {csgn}\left (i x \,x^{\frac {1}{\ln \left (2\right )^{2}}}\right )^{2} \operatorname {csgn}\left (i x \right )}{2}+x^{5}+2 x\)	\(121\)

[In]

int((ln(2)^2*ln(x*exp(ln(x)/ln(2)^2))+(5*x^4+3)*ln(2)^2+1)/ln(2)^2,x,method=_RETURNVERBOSE)

[Out]

x^5+2*x+ln(x*x^(1/ln(2)^2))*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=\frac {{\left (x^{5} + 2 \, x\right )} \log \left (2\right )^{2} + {\left (x \log \left (2\right )^{2} + x\right )} \log \left (x\right )}{\log \left (2\right )^{2}} \]

[In]

integrate((log(2)^2*log(x*exp(log(x)/log(2)^2))+(5*x^4+3)*log(2)^2+1)/log(2)^2,x, algorithm="fricas")

[Out]

((x^5 + 2*x)*log(2)^2 + (x*log(2)^2 + x)*log(x))/log(2)^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).

Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=\frac {x^{5} \log {\left (2 \right )}^{2} + x \log {\left (2 \right )}^{2} \log {\left (x x^{\frac {1}{\log {\left (2 \right )}^{2}}} \right )} + 2 x \log {\left (2 \right )}^{2}}{\log {\left (2 \right )}^{2}} \]

[In]

integrate((ln(2)**2*ln(x*exp(ln(x)/ln(2)**2))+(5*x**4+3)*ln(2)**2+1)/ln(2)**2,x)

[Out]

(x**5*log(2)**2 + x*log(2)**2*log(x*x**(log(2)**(-2))) + 2*x*log(2)**2)/log(2)**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (16) = 32\).

Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.19 \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=-\frac {x {\left (\frac {1}{\log \left (2\right )^{2}} + 1\right )} \log \left (2\right )^{2} - x \log \left (2\right )^{2} \log \left (x^{\frac {1}{\log \left (2\right )^{2}} + 1}\right ) - {\left (x^{5} + 3 \, x\right )} \log \left (2\right )^{2} - x}{\log \left (2\right )^{2}} \]

[In]

integrate((log(2)^2*log(x*exp(log(x)/log(2)^2))+(5*x^4+3)*log(2)^2+1)/log(2)^2,x, algorithm="maxima")

[Out]

-(x*(1/log(2)^2 + 1)*log(2)^2 - x*log(2)^2*log(x^(1/log(2)^2 + 1)) - (x^5 + 3*x)*log(2)^2 - x)/log(2)^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.38 \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=\frac {{\left (x \log \left (x\right ) - x\right )} {\left (\frac {1}{\log \left (2\right )^{2}} + 1\right )} \log \left (2\right )^{2} + {\left (x^{5} + 3 \, x\right )} \log \left (2\right )^{2} + x}{\log \left (2\right )^{2}} \]

[In]

integrate((log(2)^2*log(x*exp(log(x)/log(2)^2))+(5*x^4+3)*log(2)^2+1)/log(2)^2,x, algorithm="giac")

[Out]

((x*log(x) - x)*(1/log(2)^2 + 1)*log(2)^2 + (x^5 + 3*x)*log(2)^2 + x)/log(2)^2

Mupad [B] (verification not implemented)

Time = 12.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1+\left (3+5 x^4\right ) \log ^2(2)+\log ^2(2) \log \left (x^{1+\frac {1}{\log ^2(2)}}\right )}{\log ^2(2)} \, dx=x\,\left (\ln \left (x^{\frac {1}{{\ln \left (2\right )}^2}+1}\right )+x^4+2\right ) \]

[In]

int((log(2)^2*log(x*x^(1/log(2)^2)) + log(2)^2*(5*x^4 + 3) + 1)/log(2)^2,x)

[Out]

x*(log(x^(1/log(2)^2 + 1)) + x^4 + 2)