3.26.15 \(\int \frac {-800+100 x^3 \log (3)-2 x^6 \log ^2(3)+(-200 x^6-10 x^9 \log (3)) \log ^2(4)+100 x^{12} \log ^4(4)}{25 x^9} \, dx\) [2515]

Optimal. Leaf size=25 \[ \left (\frac {2}{x^4}-\frac {\log (3)}{5 x}+x^2 \log ^2(4)\right )^2 \]

[Out]

(4*x^2*ln(2)^2+2/x^4-1/5*ln(3)/x)^2

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 2.08, number of steps used = 3, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 14} \begin {gather*} \frac {4}{x^8}-\frac {4 \log (3)}{5 x^5}+x^4 \log ^4(4)+\frac {\log ^2(3)+100 \log ^2(4)}{25 x^2}-\frac {2}{5} x \log (3) \log ^2(4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-800 + 100*x^3*Log[3] - 2*x^6*Log[3]^2 + (-200*x^6 - 10*x^9*Log[3])*Log[4]^2 + 100*x^12*Log[4]^4)/(25*x^9
),x]

[Out]

4/x^8 - (4*Log[3])/(5*x^5) - (2*x*Log[3]*Log[4]^2)/5 + x^4*Log[4]^4 + (Log[3]^2 + 100*Log[4]^2)/(25*x^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 {1}{25} \int \frac {-800+100 x^3 \log (3)-2 x^6 \log ^2(3)+\left (-200 x^6-10 x^9 \log (3)\right ) \log ^2(4)+100 x^{12} \log ^4(4)}{x^9} \, dx\\ &=\frac {1}{25} \int \left (-\frac {800}{x^9}+\frac {100 \log (3)}{x^6}-10 \log (3) \log ^2(4)+100 x^3 \log ^4(4)-\frac {2 \left (\log ^2(3)+100 \log ^2(4)\right )}{x^3}\right ) \, dx\\ &=\frac {4}{x^8}-\frac {4 \log (3)}{5 x^5}-\frac {2}{5} x \log (3) \log ^2(4)+x^4 \log ^4(4)+\frac {\log ^2(3)+100 \log ^2(4)}{25 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
time = 0.02, size = 52, normalized size = 2.08 \begin {gather*} \frac {4}{x^8}-\frac {2}{5} x \log (3) \log ^2(4)+x^4 \log ^4(4)+\frac {\log ^2(3)+100 \log ^2(4)}{25 x^2}-\frac {\log (81)}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-800 + 100*x^3*Log[3] - 2*x^6*Log[3]^2 + (-200*x^6 - 10*x^9*Log[3])*Log[4]^2 + 100*x^12*Log[4]^4)/(
25*x^9),x]

[Out]

4/x^8 - (2*x*Log[3]*Log[4]^2)/5 + x^4*Log[4]^4 + (Log[3]^2 + 100*Log[4]^2)/(25*x^2) - Log[81]/(5*x^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
time = 0.15, size = 50, normalized size = 2.00

method	result	size
risch	\(16 \ln \left (2\right )^{4} x^{4}-\frac {8 \ln \left (2\right )^{2} \ln \left (3\right ) x}{5}+\frac {100+\left (400 \ln \left (2\right )^{2}+\ln \left (3\right )^{2}\right ) x^{6}-20 x^{3} \ln \left (3\right )}{25 x^{8}}\)	\(49\)
default	\(16 \ln \left (2\right )^{4} x^{4}-\frac {8 \ln \left (2\right )^{2} \ln \left (3\right ) x}{5}-\frac {-400 \ln \left (2\right )^{2}-\ln \left (3\right )^{2}}{25 x^{2}}+\frac {4}{x^{8}}-\frac {4 \ln \left (3\right )}{5 x^{5}}\)	\(50\)
norman	\(\frac {4+\left (16 \ln \left (2\right )^{2}+\frac {\ln \left (3\right )^{2}}{25}\right ) x^{6}-\frac {4 x^{3} \ln \left (3\right )}{5}+16 x^{12} \ln \left (2\right )^{4}-\frac {8 \ln \left (2\right )^{2} \ln \left (3\right ) x^{9}}{5}}{x^{8}}\)	\(51\)
gosper	\(\frac {400 x^{12} \ln \left (2\right )^{4}-40 \ln \left (2\right )^{2} \ln \left (3\right ) x^{9}+400 x^{6} \ln \left (2\right )^{2}+x^{6} \ln \left (3\right )^{2}-20 x^{3} \ln \left (3\right )+100}{25 x^{8}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(1600*x^12*ln(2)^4+4*(-10*x^9*ln(3)-200*x^6)*ln(2)^2-2*x^6*ln(3)^2+100*x^3*ln(3)-800)/x^9,x,method=_R
ETURNVERBOSE)

[Out]

16*ln(2)^4*x^4-8/5*ln(2)^2*ln(3)*x-1/25*(-400*ln(2)^2-ln(3)^2)/x^2+4/x^8-4/5*ln(3)/x^5

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Maxima [A]
time = 0.26, size = 48, normalized size = 1.92 \begin {gather*} 16 \, x^{4} \log \left (2\right )^{4} - \frac {8}{5} \, x \log \left (3\right ) \log \left (2\right )^{2} + \frac {{\left (\log \left (3\right )^{2} + 400 \, \log \left (2\right )^{2}\right )} x^{6} - 20 \, x^{3} \log \left (3\right ) + 100}{25 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(1600*x^12*log(2)^4+4*(-10*x^9*log(3)-200*x^6)*log(2)^2-2*x^6*log(3)^2+100*x^3*log(3)-800)/x^9,
x, algorithm="maxima")

[Out]

16*x^4*log(2)^4 - 8/5*x*log(3)*log(2)^2 + 1/25*((log(3)^2 + 400*log(2)^2)*x^6 - 20*x^3*log(3) + 100)/x^8

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Fricas [A]
time = 0.33, size = 50, normalized size = 2.00 \begin {gather*} \frac {400 \, x^{12} \log \left (2\right )^{4} + x^{6} \log \left (3\right )^{2} + 400 \, x^{6} \log \left (2\right )^{2} - 20 \, {\left (2 \, x^{9} \log \left (2\right )^{2} + x^{3}\right )} \log \left (3\right ) + 100}{25 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(1600*x^12*log(2)^4+4*(-10*x^9*log(3)-200*x^6)*log(2)^2-2*x^6*log(3)^2+100*x^3*log(3)-800)/x^9,
x, algorithm="fricas")

[Out]

1/25*(400*x^12*log(2)^4 + x^6*log(3)^2 + 400*x^6*log(2)^2 - 20*(2*x^9*log(2)^2 + x^3)*log(3) + 100)/x^8

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
time = 0.38, size = 53, normalized size = 2.12 \begin {gather*} 16 x^{4} \log {\left (2 \right )}^{4} - \frac {8 x \log {\left (2 \right )}^{2} \log {\left (3 \right )}}{5} + \frac {x^{6} \left (\log {\left (3 \right )}^{2} + 400 \log {\left (2 \right )}^{2}\right ) - 20 x^{3} \log {\left (3 \right )} + 100}{25 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(1600*x**12*ln(2)**4+4*(-10*x**9*ln(3)-200*x**6)*ln(2)**2-2*x**6*ln(3)**2+100*x**3*ln(3)-800)/x
**9,x)

[Out]

16*x**4*log(2)**4 - 8*x*log(2)**2*log(3)/5 + (x**6*(log(3)**2 + 400*log(2)**2) - 20*x**3*log(3) + 100)/(25*x**
8)

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Giac [A]
time = 0.40, size = 50, normalized size = 2.00 \begin {gather*} 16 \, x^{4} \log \left (2\right )^{4} - \frac {8}{5} \, x \log \left (3\right ) \log \left (2\right )^{2} + \frac {x^{6} \log \left (3\right )^{2} + 400 \, x^{6} \log \left (2\right )^{2} - 20 \, x^{3} \log \left (3\right ) + 100}{25 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(1600*x^12*log(2)^4+4*(-10*x^9*log(3)-200*x^6)*log(2)^2-2*x^6*log(3)^2+100*x^3*log(3)-800)/x^9,
x, algorithm="giac")

[Out]

16*x^4*log(2)^4 - 8/5*x*log(3)*log(2)^2 + 1/25*(x^6*log(3)^2 + 400*x^6*log(2)^2 - 20*x^3*log(3) + 100)/x^8

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Mupad [B]
time = 1.50, size = 25, normalized size = 1.00 \begin {gather*} \frac {{\left (20\,{\ln \left (2\right )}^2\,x^6-\ln \left (3\right )\,x^3+10\right )}^2}{25\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x^6*log(3)^2)/25 - 64*x^12*log(2)^4 + (4*log(2)^2*(10*x^9*log(3) + 200*x^6))/25 - 4*x^3*log(3) + 32)/
x^9,x)

[Out]

(20*x^6*log(2)^2 - x^3*log(3) + 10)^2/(25*x^8)

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