3.57.4 \(\int \frac {29229056+446 \log (x)+(-14614529-223 \log (x)) \log (x^2)+(131072+2 \log (x)+(-65536-\log (x)) \log (x^2)) \log (65536+\log (x))}{(65536+\log (x)) \log ^2(x^2)} \, dx\)

Optimal. Leaf size=17 \[ \frac {x (-223-\log (65536+\log (x)))}{\log \left (x^2\right )} \]

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Rubi [F]  time = 0.47, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {29229056+446 \log (x)+(-14614529-223 \log (x)) \log \left (x^2\right )+\left (131072+2 \log (x)+(-65536-\log (x)) \log \left (x^2\right )\right ) \log (65536+\log (x))}{(65536+\log (x)) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(29229056 + 446*Log[x] + (-14614529 - 223*Log[x])*Log[x^2] + (131072 + 2*Log[x] + (-65536 - Log[x])*Log[x^
2])*Log[65536 + Log[x]])/((65536 + Log[x])*Log[x^2]^2),x]

[Out]

(223*x*ExpIntegralEi[Log[x^2]/2])/(2*Sqrt[x^2]) - (223*x)/Log[x^2] - 14614529*Defer[Int][1/((65536 + Log[x])*L
og[x^2]), x] - 223*Defer[Int][Log[x]/((65536 + Log[x])*Log[x^2]), x] + 2*Defer[Int][Log[65536 + Log[x]]/Log[x^
2]^2, x] - Defer[Int][Log[65536 + Log[x]]/Log[x^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {29229056+446 \log (x)-14614529 \log \left (x^2\right )-223 \log (x) \log \left (x^2\right )}{(65536+\log (x)) \log ^2\left (x^2\right )}-\frac {\left (-2+\log \left (x^2\right )\right ) \log (65536+\log (x))}{\log ^2\left (x^2\right )}\right ) \, dx\\ &=\int \frac {29229056+446 \log (x)-14614529 \log \left (x^2\right )-223 \log (x) \log \left (x^2\right )}{(65536+\log (x)) \log ^2\left (x^2\right )} \, dx-\int \frac {\left (-2+\log \left (x^2\right )\right ) \log (65536+\log (x))}{\log ^2\left (x^2\right )} \, dx\\ &=\int \left (\frac {446}{\log ^2\left (x^2\right )}+\frac {-14614529-223 \log (x)}{(65536+\log (x)) \log \left (x^2\right )}\right ) \, dx-\int \left (-\frac {2 \log (65536+\log (x))}{\log ^2\left (x^2\right )}+\frac {\log (65536+\log (x))}{\log \left (x^2\right )}\right ) \, dx\\ &=2 \int \frac {\log (65536+\log (x))}{\log ^2\left (x^2\right )} \, dx+446 \int \frac {1}{\log ^2\left (x^2\right )} \, dx+\int \frac {-14614529-223 \log (x)}{(65536+\log (x)) \log \left (x^2\right )} \, dx-\int \frac {\log (65536+\log (x))}{\log \left (x^2\right )} \, dx\\ &=-\frac {223 x}{\log \left (x^2\right )}+2 \int \frac {\log (65536+\log (x))}{\log ^2\left (x^2\right )} \, dx+223 \int \frac {1}{\log \left (x^2\right )} \, dx+\int \left (-\frac {14614529}{(65536+\log (x)) \log \left (x^2\right )}-\frac {223 \log (x)}{(65536+\log (x)) \log \left (x^2\right )}\right ) \, dx-\int \frac {\log (65536+\log (x))}{\log \left (x^2\right )} \, dx\\ &=-\frac {223 x}{\log \left (x^2\right )}+2 \int \frac {\log (65536+\log (x))}{\log ^2\left (x^2\right )} \, dx-223 \int \frac {\log (x)}{(65536+\log (x)) \log \left (x^2\right )} \, dx-14614529 \int \frac {1}{(65536+\log (x)) \log \left (x^2\right )} \, dx+\frac {(223 x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{2 \sqrt {x^2}}-\int \frac {\log (65536+\log (x))}{\log \left (x^2\right )} \, dx\\ &=\frac {223 x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{2 \sqrt {x^2}}-\frac {223 x}{\log \left (x^2\right )}+2 \int \frac {\log (65536+\log (x))}{\log ^2\left (x^2\right )} \, dx-223 \int \frac {\log (x)}{(65536+\log (x)) \log \left (x^2\right )} \, dx-14614529 \int \frac {1}{(65536+\log (x)) \log \left (x^2\right )} \, dx-\int \frac {\log (65536+\log (x))}{\log \left (x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 24, normalized size = 1.41 \begin {gather*} -\frac {223 x}{\log \left (x^2\right )}-\frac {x \log (65536+\log (x))}{\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(29229056 + 446*Log[x] + (-14614529 - 223*Log[x])*Log[x^2] + (131072 + 2*Log[x] + (-65536 - Log[x])*
Log[x^2])*Log[65536 + Log[x]])/((65536 + Log[x])*Log[x^2]^2),x]

[Out]

(-223*x)/Log[x^2] - (x*Log[65536 + Log[x]])/Log[x^2]

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fricas [A]  time = 1.13, size = 17, normalized size = 1.00 \begin {gather*} -\frac {x \log \left (\log \relax (x) + 65536\right ) + 223 \, x}{2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-log(x)-65536)*log(x^2)+2*log(x)+131072)*log(log(x)+65536)+(-223*log(x)-14614529)*log(x^2)+446*lo
g(x)[phone])/(log(x)+65536)/log(x^2)^2,x, algorithm="fricas")

[Out]

-1/2*(x*log(log(x) + 65536) + 223*x)/log(x)

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giac [A]  time = 0.18, size = 20, normalized size = 1.18 \begin {gather*} -\frac {x \log \left (\log \relax (x) + 65536\right )}{2 \, \log \relax (x)} - \frac {223 \, x}{2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-log(x)-65536)*log(x^2)+2*log(x)+131072)*log(log(x)+65536)+(-223*log(x)-14614529)*log(x^2)+446*lo
g(x)[phone])/(log(x)+65536)/log(x^2)^2,x, algorithm="giac")

[Out]

-1/2*x*log(log(x) + 65536)/log(x) - 223/2*x/log(x)

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maple [C]  time = 0.10, size = 119, normalized size = 7.00

method	result	size
risch	\(-\frac {2 i x \ln \left (\ln \relax (x )+65536\right )}{4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}-\frac {446 i x}{4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-ln(x)-65536)*ln(x^2)+2*ln(x)+131072)*ln(ln(x)+65536)+(-223*ln(x)-14614529)*ln(x^2)+446*ln(x)[phone])
/(ln(x)+65536)/ln(x^2)^2,x,method=_RETURNVERBOSE)

[Out]

-2*I*x/(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*ln(ln(x)+65536)-44
6*I*x/(4*I*ln(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)

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maxima [A]  time = 0.46, size = 17, normalized size = 1.00 \begin {gather*} -\frac {x \log \left (\log \relax (x) + 65536\right ) + 223 \, x}{2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-log(x)-65536)*log(x^2)+2*log(x)+131072)*log(log(x)+65536)+(-223*log(x)-14614529)*log(x^2)+446*lo
g(x)[phone])/(log(x)+65536)/log(x^2)^2,x, algorithm="maxima")

[Out]

-1/2*(x*log(log(x) + 65536) + 223*x)/log(x)

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mupad [B]  time = 4.38, size = 16, normalized size = 0.94 \begin {gather*} -\frac {x\,\left (\ln \left (\ln \relax (x)+65536\right )+223\right )}{\ln \left (x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((446*log(x) - log(x^2)*(223*log(x) + 14614529) + log(log(x) + 65536)*(2*log(x) - log(x^2)*(log(x) + 65536)
+ 131072) + 29229056)/(log(x^2)^2*(log(x) + 65536)),x)

[Out]

-(x*(log(log(x) + 65536) + 223))/log(x^2)

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sympy [A]  time = 0.35, size = 22, normalized size = 1.29 \begin {gather*} - \frac {x \log {\left (\log {\relax (x )} + 65536 \right )}}{2 \log {\relax (x )}} - \frac {223 x}{2 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-ln(x)-65536)*ln(x**2)+2*ln(x)+131072)*ln(ln(x)+65536)+(-223*ln(x)-14614529)*ln(x**2)+446*ln(x)+2
9229056)/(ln(x)+65536)/ln(x**2)**2,x)

[Out]

-x*log(log(x) + 65536)/(2*log(x)) - 223*x/(2*log(x))

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