3.47.19 \(\int \frac {-75-500 x-565 x^2+36 x^4+(-600-1920 x+440 x^2) \log (4)+(-1200+320 x-16 x^2) \log ^2(4)}{x^4} \, dx\)

Optimal. Leaf size=30 \[ 4+\frac {\left (1-x-\frac {(-5+x) (1+x+4 (x+\log (4)))}{x}\right )^2}{x} \]

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 1, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {14} \begin {gather*} \frac {25 (1+\log (256))^2}{x^3}+\frac {10 \left (25-16 \log ^2(4)+96 \log (4)\right )}{x^2}+36 x+\frac {565+16 \log ^2(4)-440 \log (4)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-75 - 500*x - 565*x^2 + 36*x^4 + (-600 - 1920*x + 440*x^2)*Log[4] + (-1200 + 320*x - 16*x^2)*Log[4]^2)/x^
4,x]

[Out]

36*x + (10*(25 + 96*Log[4] - 16*Log[4]^2))/x^2 + (565 - 440*Log[4] + 16*Log[4]^2)/x + (25*(1 + Log[256])^2)/x^
3

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} &=\int \left (36+\frac {-565+440 \log (4)-16 \log ^2(4)}{x^2}+\frac {20 \left (-25-96 \log (4)+16 \log ^2(4)\right )}{x^3}-\frac {75 (1+\log (256))^2}{x^4}\right ) \, dx\\ &=36 x+\frac {10 \left (25+96 \log (4)-16 \log ^2(4)\right )}{x^2}+\frac {565-440 \log (4)+16 \log ^2(4)}{x}+\frac {25 (1+\log (256))^2}{x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 53, normalized size = 1.77 \begin {gather*} \frac {565}{x}+36 x-\frac {440 \log (4)}{x}+\frac {16 \log ^2(4)}{x}-\frac {10 \left (-25-96 \log (4)+16 \log ^2(4)\right )}{x^2}+\frac {25 (1+\log (256))^2}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-75 - 500*x - 565*x^2 + 36*x^4 + (-600 - 1920*x + 440*x^2)*Log[4] + (-1200 + 320*x - 16*x^2)*Log[4]
^2)/x^4,x]

[Out]

565/x + 36*x - (440*Log[4])/x + (16*Log[4]^2)/x - (10*(-25 - 96*Log[4] + 16*Log[4]^2))/x^2 + (25*(1 + Log[256]
)^2)/x^3

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fricas [A]  time = 0.79, size = 47, normalized size = 1.57 \begin {gather*} \frac {36 \, x^{4} + 64 \, {\left (x^{2} - 10 \, x + 25\right )} \log \relax (2)^{2} + 565 \, x^{2} - 80 \, {\left (11 \, x^{2} - 24 \, x - 5\right )} \log \relax (2) + 250 \, x + 25}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-16*x^2+320*x-1200)*log(2)^2+2*(440*x^2-1920*x-600)*log(2)+36*x^4-565*x^2-500*x-75)/x^4,x, algor
ithm="fricas")

[Out]

(36*x^4 + 64*(x^2 - 10*x + 25)*log(2)^2 + 565*x^2 - 80*(11*x^2 - 24*x - 5)*log(2) + 250*x + 25)/x^3

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giac [B]  time = 0.14, size = 56, normalized size = 1.87 \begin {gather*} 36 \, x + \frac {64 \, x^{2} \log \relax (2)^{2} - 880 \, x^{2} \log \relax (2) - 640 \, x \log \relax (2)^{2} + 565 \, x^{2} + 1920 \, x \log \relax (2) + 1600 \, \log \relax (2)^{2} + 250 \, x + 400 \, \log \relax (2) + 25}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-16*x^2+320*x-1200)*log(2)^2+2*(440*x^2-1920*x-600)*log(2)+36*x^4-565*x^2-500*x-75)/x^4,x, algor
ithm="giac")

[Out]

36*x + (64*x^2*log(2)^2 - 880*x^2*log(2) - 640*x*log(2)^2 + 565*x^2 + 1920*x*log(2) + 1600*log(2)^2 + 250*x +
400*log(2) + 25)/x^3

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maple [A]  time = 0.07, size = 51, normalized size = 1.70

method	result	size
risch	\(36 x +\frac {\left (64 \ln \relax (2)^{2}-880 \ln \relax (2)+565\right ) x^{2}+\left (-640 \ln \relax (2)^{2}+1920 \ln \relax (2)+250\right ) x +1600 \ln \relax (2)^{2}+400 \ln \relax (2)+25}{x^{3}}\)	\(51\)
norman	\(\frac {\left (-640 \ln \relax (2)^{2}+1920 \ln \relax (2)+250\right ) x +\left (64 \ln \relax (2)^{2}-880 \ln \relax (2)+565\right ) x^{2}+36 x^{4}+1600 \ln \relax (2)^{2}+400 \ln \relax (2)+25}{x^{3}}\)	\(52\)
default	\(36 x -\frac {-64 \ln \relax (2)^{2}+880 \ln \relax (2)-565}{x}-\frac {1280 \ln \relax (2)^{2}-3840 \ln \relax (2)-500}{2 x^{2}}-\frac {-4800 \ln \relax (2)^{2}-1200 \ln \relax (2)-75}{3 x^{3}}\)	\(56\)
gosper	\(\frac {64 x^{2} \ln \relax (2)^{2}+36 x^{4}-640 x \ln \relax (2)^{2}-880 x^{2} \ln \relax (2)+1600 \ln \relax (2)^{2}+1920 x \ln \relax (2)+565 x^{2}+400 \ln \relax (2)+250 x +25}{x^{3}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(-16*x^2+320*x-1200)*ln(2)^2+2*(440*x^2-1920*x-600)*ln(2)+36*x^4-565*x^2-500*x-75)/x^4,x,method=_RETURN
VERBOSE)

[Out]

36*x+((64*ln(2)^2-880*ln(2)+565)*x^2+(-640*ln(2)^2+1920*ln(2)+250)*x+1600*ln(2)^2+400*ln(2)+25)/x^3

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maxima [A]  time = 0.36, size = 51, normalized size = 1.70 \begin {gather*} 36 \, x + \frac {{\left (64 \, \log \relax (2)^{2} - 880 \, \log \relax (2) + 565\right )} x^{2} - 10 \, {\left (64 \, \log \relax (2)^{2} - 192 \, \log \relax (2) - 25\right )} x + 1600 \, \log \relax (2)^{2} + 400 \, \log \relax (2) + 25}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-16*x^2+320*x-1200)*log(2)^2+2*(440*x^2-1920*x-600)*log(2)+36*x^4-565*x^2-500*x-75)/x^4,x, algor
ithm="maxima")

[Out]

36*x + ((64*log(2)^2 - 880*log(2) + 565)*x^2 - 10*(64*log(2)^2 - 192*log(2) - 25)*x + 1600*log(2)^2 + 400*log(
2) + 25)/x^3

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mupad [B]  time = 3.19, size = 47, normalized size = 1.57 \begin {gather*} 36\,x+\frac {25\,{\left (\ln \left (256\right )+1\right )}^2}{x^3}+\frac {64\,{\ln \relax (2)}^2-880\,\ln \relax (2)+565}{x}+\frac {1920\,\ln \relax (2)-640\,{\ln \relax (2)}^2+250}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(500*x + 2*log(2)*(1920*x - 440*x^2 + 600) + 4*log(2)^2*(16*x^2 - 320*x + 1200) + 565*x^2 - 36*x^4 + 75)/
x^4,x)

[Out]

36*x + (25*(log(256) + 1)^2)/x^3 + (64*log(2)^2 - 880*log(2) + 565)/x + (1920*log(2) - 640*log(2)^2 + 250)/x^2

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sympy [B]  time = 0.59, size = 51, normalized size = 1.70 \begin {gather*} 36 x + \frac {x^{2} \left (- 880 \log {\relax (2 )} + 64 \log {\relax (2 )}^{2} + 565\right ) + x \left (- 640 \log {\relax (2 )}^{2} + 250 + 1920 \log {\relax (2 )}\right ) + 25 + 400 \log {\relax (2 )} + 1600 \log {\relax (2 )}^{2}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-16*x**2+320*x-1200)*ln(2)**2+2*(440*x**2-1920*x-600)*ln(2)+36*x**4-565*x**2-500*x-75)/x**4,x)

[Out]

36*x + (x**2*(-880*log(2) + 64*log(2)**2 + 565) + x*(-640*log(2)**2 + 250 + 1920*log(2)) + 25 + 400*log(2) + 1
600*log(2)**2)/x**3

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