3.69.51 \(\int e^{(4 x+3 x^2+4 x^3) \log (2)-3 x \log (2) \log (4)} ((4+6 x+12 x^2) \log (2)-3 \log (2) \log (4)) \, dx\)

Optimal. Leaf size=20 \[ 2^{x \left (4 \left (1+x^2\right )-3 (-x+\log (4))\right )} \]

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Rubi [A]  time = 0.09, antiderivative size = 26, normalized size of antiderivative = 1.30, number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6706} \begin {gather*} 2^{4 x^3+3 x^2+4 x} e^{-3 x \log (2) \log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((4*x + 3*x^2 + 4*x^3)*Log[2] - 3*x*Log[2]*Log[4])*((4 + 6*x + 12*x^2)*Log[2] - 3*Log[2]*Log[4]),x]

[Out]

2^(4*x + 3*x^2 + 4*x^3)/E^(3*x*Log[2]*Log[4])

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2^{4 x+3 x^2+4 x^3} e^{-3 x \log (2) \log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.48, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\left (4 x+3 x^2+4 x^3\right ) \log (2)-3 x \log (2) \log (4)} \left (\left (4+6 x+12 x^2\right ) \log (2)-3 \log (2) \log (4)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^((4*x + 3*x^2 + 4*x^3)*Log[2] - 3*x*Log[2]*Log[4])*((4 + 6*x + 12*x^2)*Log[2] - 3*Log[2]*Log[4]),x
]

[Out]

Integrate[E^((4*x + 3*x^2 + 4*x^3)*Log[2] - 3*x*Log[2]*Log[4])*((4 + 6*x + 12*x^2)*Log[2] - 3*Log[2]*Log[4]),
x]

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fricas [A]  time = 0.56, size = 26, normalized size = 1.30 \begin {gather*} e^{\left (-6 \, x \log \relax (2)^{2} + {\left (4 \, x^{3} + 3 \, x^{2} + 4 \, x\right )} \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(2)^2+(12*x^2+6*x+4)*log(2))*exp(-6*x*log(2)^2+(4*x^3+3*x^2+4*x)*log(2)),x, algorithm="fricas
")

[Out]

e^(-6*x*log(2)^2 + (4*x^3 + 3*x^2 + 4*x)*log(2))

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giac [A]  time = 0.25, size = 28, normalized size = 1.40 \begin {gather*} e^{\left (4 \, x^{3} \log \relax (2) + 3 \, x^{2} \log \relax (2) - 6 \, x \log \relax (2)^{2} + 4 \, x \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(2)^2+(12*x^2+6*x+4)*log(2))*exp(-6*x*log(2)^2+(4*x^3+3*x^2+4*x)*log(2)),x, algorithm="giac")

[Out]

e^(4*x^3*log(2) + 3*x^2*log(2) - 6*x*log(2)^2 + 4*x*log(2))

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maple [A]  time = 0.04, size = 24, normalized size = 1.20

method	result	size
risch	\(2^{x \left (4 x^{2}+3 x +4\right )} {\mathrm e}^{-6 x \ln \relax (2)^{2}}\)	\(24\)
derivativedivides	\({\mathrm e}^{-6 x \ln \relax (2)^{2}+\left (4 x^{3}+3 x^{2}+4 x \right ) \ln \relax (2)}\)	\(27\)
default	\({\mathrm e}^{-6 x \ln \relax (2)^{2}+\left (4 x^{3}+3 x^{2}+4 x \right ) \ln \relax (2)}\)	\(27\)
norman	\({\mathrm e}^{-6 x \ln \relax (2)^{2}+\left (4 x^{3}+3 x^{2}+4 x \right ) \ln \relax (2)}\)	\(27\)
gosper	\({\mathrm e}^{4 x^{3} \ln \relax (2)-6 x \ln \relax (2)^{2}+3 x^{2} \ln \relax (2)+4 x \ln \relax (2)}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*ln(2)^2+(12*x^2+6*x+4)*ln(2))*exp(-6*x*ln(2)^2+(4*x^3+3*x^2+4*x)*ln(2)),x,method=_RETURNVERBOSE)

[Out]

2^(x*(4*x^2+3*x+4))*exp(-6*x*ln(2)^2)

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maxima [A]  time = 0.53, size = 28, normalized size = 1.40 \begin {gather*} e^{\left (4 \, x^{3} \log \relax (2) + 3 \, x^{2} \log \relax (2) - 6 \, x \log \relax (2)^{2} + 4 \, x \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*log(2)^2+(12*x^2+6*x+4)*log(2))*exp(-6*x*log(2)^2+(4*x^3+3*x^2+4*x)*log(2)),x, algorithm="maxima
")

[Out]

e^(4*x^3*log(2) + 3*x^2*log(2) - 6*x*log(2)^2 + 4*x*log(2))

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mupad [B]  time = 4.28, size = 28, normalized size = 1.40 \begin {gather*} 2^{4\,x}\,2^{3\,x^2}\,2^{4\,x^3}\,{\mathrm {e}}^{-6\,x\,{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(log(2)*(4*x + 3*x^2 + 4*x^3) - 6*x*log(2)^2)*(log(2)*(6*x + 12*x^2 + 4) - 6*log(2)^2),x)

[Out]

2^(4*x)*2^(3*x^2)*2^(4*x^3)*exp(-6*x*log(2)^2)

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sympy [A]  time = 0.15, size = 26, normalized size = 1.30 \begin {gather*} e^{- 6 x \log {\relax (2 )}^{2} + \left (4 x^{3} + 3 x^{2} + 4 x\right ) \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*ln(2)**2+(12*x**2+6*x+4)*ln(2))*exp(-6*x*ln(2)**2+(4*x**3+3*x**2+4*x)*ln(2)),x)

[Out]

exp(-6*x*log(2)**2 + (4*x**3 + 3*x**2 + 4*x)*log(2))

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