3.48.73 \(\int (-1+4 x^3+8 x^3 \log (4)+4 x^3 \log ^2(4)) \, dx\)

Optimal. Leaf size=15 \[ 1029-x+x^4 (1+\log (4))^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6} \begin {gather*} x^4 (1+\log (4))^2-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + 4*x^3 + 8*x^3*Log[4] + 4*x^3*Log[4]^2,x]

[Out]

-x + x^4*(1 + Log[4])^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+4 x^3 \log ^2(4)+x^3 (4+8 \log (4))\right ) \, dx\\ &=\int \left (-1+x^3 \left (4+8 \log (4)+4 \log ^2(4)\right )\right ) \, dx\\ &=-x+x^4 (1+\log (4))^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.47 \begin {gather*} -x+x^4+2 x^4 \log (4)+x^4 \log ^2(4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + 4*x^3 + 8*x^3*Log[4] + 4*x^3*Log[4]^2,x]

[Out]

-x + x^4 + 2*x^4*Log[4] + x^4*Log[4]^2

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fricas [A]  time = 0.58, size = 23, normalized size = 1.53 \begin {gather*} 4 \, x^{4} \log \relax (2)^{2} + 4 \, x^{4} \log \relax (2) + x^{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(2)^2+16*x^3*log(2)+4*x^3-1,x, algorithm="fricas")

[Out]

4*x^4*log(2)^2 + 4*x^4*log(2) + x^4 - x

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giac [A]  time = 0.12, size = 23, normalized size = 1.53 \begin {gather*} 4 \, x^{4} \log \relax (2)^{2} + 4 \, x^{4} \log \relax (2) + x^{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(2)^2+16*x^3*log(2)+4*x^3-1,x, algorithm="giac")

[Out]

4*x^4*log(2)^2 + 4*x^4*log(2) + x^4 - x

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maple [A]  time = 0.03, size = 21, normalized size = 1.40

method	result	size
norman	\(\left (4 \ln \relax (2)^{2}+4 \ln \relax (2)+1\right ) x^{4}-x\)	\(21\)
gosper	\(4 x^{4} \ln \relax (2)^{2}+4 x^{4} \ln \relax (2)+x^{4}-x\)	\(24\)
default	\(4 x^{4} \ln \relax (2)^{2}+4 x^{4} \ln \relax (2)+x^{4}-x\)	\(24\)
risch	\(4 x^{4} \ln \relax (2)^{2}+4 x^{4} \ln \relax (2)+x^{4}-x\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x^3*ln(2)^2+16*x^3*ln(2)+4*x^3-1,x,method=_RETURNVERBOSE)

[Out]

(4*ln(2)^2+4*ln(2)+1)*x^4-x

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maxima [A]  time = 0.36, size = 23, normalized size = 1.53 \begin {gather*} 4 \, x^{4} \log \relax (2)^{2} + 4 \, x^{4} \log \relax (2) + x^{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(2)^2+16*x^3*log(2)+4*x^3-1,x, algorithm="maxima")

[Out]

4*x^4*log(2)^2 + 4*x^4*log(2) + x^4 - x

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mupad [B]  time = 0.03, size = 18, normalized size = 1.20 \begin {gather*} x^4\,\left (\ln \left (16\right )+4\,{\ln \relax (2)}^2+1\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x^3*log(2)^2 + 16*x^3*log(2) + 4*x^3 - 1,x)

[Out]

x^4*(log(16) + 4*log(2)^2 + 1) - x

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sympy [A]  time = 0.05, size = 17, normalized size = 1.13 \begin {gather*} x^{4} \left (1 + 4 \log {\relax (2 )}^{2} + 4 \log {\relax (2 )}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x**3*ln(2)**2+16*x**3*ln(2)+4*x**3-1,x)

[Out]

x**4*(1 + 4*log(2)**2 + 4*log(2)) - x

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