3.4.76 \(\int \frac {-4 x^2-6 x^3-2 x^4+(10 x+16 x^2+6 x^3) \log (8)+(-6-10 x-4 x^2) \log ^2(8)}{x^7} \, dx\)

Optimal. Leaf size=19 \[ \frac {\left (1+\frac {1}{x}\right )^2 (x-\log (8))^2}{x^4} \]

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Rubi [B]  time = 0.05, antiderivative size = 50, normalized size of antiderivative = 2.63, number of steps used = 2, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {14} \begin {gather*} \frac {\log ^2(8)}{x^6}-\frac {2 (1-\log (8)) \log (8)}{x^5}+\frac {1+\log ^2(8)-4 \log (8)}{x^4}+\frac {2 (1-\log (8))}{x^3}+\frac {1}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^2 - 6*x^3 - 2*x^4 + (10*x + 16*x^2 + 6*x^3)*Log[8] + (-6 - 10*x - 4*x^2)*Log[8]^2)/x^7,x]

[Out]

x^(-2) + (2*(1 - Log[8]))/x^3 - (2*(1 - Log[8])*Log[8])/x^5 + Log[8]^2/x^6 + (1 - 4*Log[8] + Log[8]^2)/x^4

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 (-\frac {2}{x^3}+\frac {6 (-1+\log (8))}{x^4}-\frac {10 (-1+\log (8)) \log (8)}{x^6}-\frac {6 \log ^2(8)}{x^7}-\frac {4 \left (1-4 \log (8)+\log ^2(8)\right )}{x^5}\right ) \, dx\\ &=\frac {1}{x^2}+\frac {2 (1-\log (8))}{x^3}-\frac {2 (1-\log (8)) \log (8)}{x^5}+\frac {\log ^2(8)}{x^6}+\frac {1-4 \log (8)+\log ^2(8)}{x^4}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 51, normalized size = 2.68 \begin {gather*} \frac {10 x^4-20 x^3 (-1+\log (8))+10 \log ^2(8)+5 x^2 (2+\log (8) (-8+\log (64)))+4 x \log (8) (-5+\log (32768))}{10 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 - 6*x^3 - 2*x^4 + (10*x + 16*x^2 + 6*x^3)*Log[8] + (-6 - 10*x - 4*x^2)*Log[8]^2)/x^7,x]

[Out]

(10*x^4 - 20*x^3*(-1 + Log[8]) + 10*Log[8]^2 + 5*x^2*(2 + Log[8]*(-8 + Log[64])) + 4*x*Log[8]*(-5 + Log[32768]
))/(10*x^6)

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fricas [B]  time = 0.49, size = 44, normalized size = 2.32 \begin {gather*} \frac {x^{4} + 2 \, x^{3} + 9 \, {\left (x^{2} + 2 \, x + 1\right )} \log \relax (2)^{2} + x^{2} - 6 \, {\left (x^{3} + 2 \, x^{2} + x\right )} \log \relax (2)}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*(-4*x^2-10*x-6)*log(2)^2+3*(6*x^3+16*x^2+10*x)*log(2)-2*x^4-6*x^3-4*x^2)/x^7,x, algorithm="fricas
")

[Out]

(x^4 + 2*x^3 + 9*(x^2 + 2*x + 1)*log(2)^2 + x^2 - 6*(x^3 + 2*x^2 + x)*log(2))/x^6

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giac [B]  time = 0.32, size = 57, normalized size = 3.00 \begin {gather*} \frac {x^{4} - 6 \, x^{3} \log \relax (2) + 9 \, x^{2} \log \relax (2)^{2} + 2 \, x^{3} - 12 \, x^{2} \log \relax (2) + 18 \, x \log \relax (2)^{2} + x^{2} - 6 \, x \log \relax (2) + 9 \, \log \relax (2)^{2}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*(-4*x^2-10*x-6)*log(2)^2+3*(6*x^3+16*x^2+10*x)*log(2)-2*x^4-6*x^3-4*x^2)/x^7,x, algorithm="giac")

[Out]

(x^4 - 6*x^3*log(2) + 9*x^2*log(2)^2 + 2*x^3 - 12*x^2*log(2) + 18*x*log(2)^2 + x^2 - 6*x*log(2) + 9*log(2)^2)/
x^6

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maple [A]  time = 0.08, size = 25, normalized size = 1.32

method	result	size
gosper	\(\frac {\left (3 x \ln \relax (2)-x^{2}+3 \ln \relax (2)-x \right )^{2}}{x^{6}}\)	\(25\)
norman	\(\frac {x^{4}+\left (18 \ln \relax (2)^{2}-6 \ln \relax (2)\right ) x +\left (-6 \ln \relax (2)+2\right ) x^{3}+\left (9 \ln \relax (2)^{2}-12 \ln \relax (2)+1\right ) x^{2}+9 \ln \relax (2)^{2}}{x^{6}}\)	\(54\)
risch	\(\frac {x^{4}+\left (18 \ln \relax (2)^{2}-6 \ln \relax (2)\right ) x +\left (-6 \ln \relax (2)+2\right ) x^{3}+\left (9 \ln \relax (2)^{2}-12 \ln \relax (2)+1\right ) x^{2}+9 \ln \relax (2)^{2}}{x^{6}}\)	\(54\)
default	\(\frac {1}{x^{2}}-\frac {2 \left (9 \ln \relax (2)-3\right )}{3 x^{3}}+\frac {9 \ln \relax (2)^{2}}{x^{6}}-\frac {-18 \ln \relax (2)^{2}+24 \ln \relax (2)-2}{2 x^{4}}+\frac {6 \ln \relax (2) \left (3 \ln \relax (2)-1\right )}{x^{5}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*(-4*x^2-10*x-6)*ln(2)^2+3*(6*x^3+16*x^2+10*x)*ln(2)-2*x^4-6*x^3-4*x^2)/x^7,x,method=_RETURNVERBOSE)

[Out]

(3*x*ln(2)-x^2+3*ln(2)-x)^2/x^6

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maxima [B]  time = 0.52, size = 55, normalized size = 2.89 \begin {gather*} \frac {x^{4} - 2 \, x^{3} {\left (3 \, \log \relax (2) - 1\right )} + {\left (9 \, \log \relax (2)^{2} - 12 \, \log \relax (2) + 1\right )} x^{2} + 6 \, {\left (3 \, \log \relax (2)^{2} - \log \relax (2)\right )} x + 9 \, \log \relax (2)^{2}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*(-4*x^2-10*x-6)*log(2)^2+3*(6*x^3+16*x^2+10*x)*log(2)-2*x^4-6*x^3-4*x^2)/x^7,x, algorithm="maxima
")

[Out]

(x^4 - 2*x^3*(3*log(2) - 1) + (9*log(2)^2 - 12*log(2) + 1)*x^2 + 6*(3*log(2)^2 - log(2))*x + 9*log(2)^2)/x^6

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mupad [B]  time = 0.08, size = 51, normalized size = 2.68 \begin {gather*} \frac {x^4+\left (2-\ln \left (64\right )\right )\,x^3+\left (9\,{\ln \relax (2)}^2-\ln \left (4096\right )+1\right )\,x^2+\left (18\,{\ln \relax (2)}^2-\ln \left (64\right )\right )\,x+9\,{\ln \relax (2)}^2}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*log(2)^2*(10*x + 4*x^2 + 6) - 3*log(2)*(10*x + 16*x^2 + 6*x^3) + 4*x^2 + 6*x^3 + 2*x^4)/x^7,x)

[Out]

(x^2*(9*log(2)^2 - log(4096) + 1) - x^3*(log(64) - 2) + 9*log(2)^2 - x*(log(64) - 18*log(2)^2) + x^4)/x^6

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sympy [B]  time = 1.50, size = 54, normalized size = 2.84 \begin {gather*} - \frac {- x^{4} + x^{3} \left (-2 + 6 \log {\relax (2 )}\right ) + x^{2} \left (- 9 \log {\relax (2 )}^{2} - 1 + 12 \log {\relax (2 )}\right ) + x \left (- 18 \log {\relax (2 )}^{2} + 6 \log {\relax (2 )}\right ) - 9 \log {\relax (2 )}^{2}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*(-4*x**2-10*x-6)*ln(2)**2+3*(6*x**3+16*x**2+10*x)*ln(2)-2*x**4-6*x**3-4*x**2)/x**7,x)

[Out]

-(-x**4 + x**3*(-2 + 6*log(2)) + x**2*(-9*log(2)**2 - 1 + 12*log(2)) + x*(-18*log(2)**2 + 6*log(2)) - 9*log(2)
**2)/x**6

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