3.84.73 \(\int (e^{\frac {512+(-96+96 x+32 x^2) \log ^2(3)}{\log ^2(3)}} (-96-64 x)+2 x) \, dx\)

Optimal. Leaf size=24 \[ -e^{32 \left (-3+3 x+x^2+\frac {16}{\log ^2(3)}\right )}+x^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2244, 2236} \begin {gather*} x^2-e^{32 x^2+96 x-32 \left (3-\frac {16}{\log ^2(3)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((512 + (-96 + 96*x + 32*x^2)*Log[3]^2)/Log[3]^2)*(-96 - 64*x) + 2*x,x]

[Out]

-E^(96*x + 32*x^2 - 32*(3 - 16/Log[3]^2)) + x^2

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2244

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x^2+\int e^{\frac {512+\left (-96+96 x+32 x^2\right ) \log ^2(3)}{\log ^2(3)}} (-96-64 x) \, dx\\ &=x^2+\int e^{96 x+32 x^2-32 \left (3-\frac {16}{\log ^2(3)}\right )} (-96-64 x) \, dx\\ &=-e^{96 x+32 x^2-32 \left (3-\frac {16}{\log ^2(3)}\right )}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 31, normalized size = 1.29 \begin {gather*} -e^{96 x+32 x^2-\frac {32 \left (-16+3 \log ^2(3)\right )}{\log ^2(3)}}+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((512 + (-96 + 96*x + 32*x^2)*Log[3]^2)/Log[3]^2)*(-96 - 64*x) + 2*x,x]

[Out]

-E^(96*x + 32*x^2 - (32*(-16 + 3*Log[3]^2))/Log[3]^2) + x^2

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fricas [A]  time = 1.04, size = 28, normalized size = 1.17 \begin {gather*} x^{2} - e^{\left (\frac {32 \, {\left ({\left (x^{2} + 3 \, x - 3\right )} \log \relax (3)^{2} + 16\right )}}{\log \relax (3)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-96)*exp(((32*x^2+96*x-96)*log(3)^2+512)/log(3)^2)+2*x,x, algorithm="fricas")

[Out]

x^2 - e^(32*((x^2 + 3*x - 3)*log(3)^2 + 16)/log(3)^2)

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giac [A]  time = 0.14, size = 23, normalized size = 0.96 \begin {gather*} x^{2} - e^{\left (32 \, x^{2} + 96 \, x + \frac {512}{\log \relax (3)^{2}} - 96\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-96)*exp(((32*x^2+96*x-96)*log(3)^2+512)/log(3)^2)+2*x,x, algorithm="giac")

[Out]

x^2 - e^(32*x^2 + 96*x + 512/log(3)^2 - 96)

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maple [A]  time = 0.04, size = 24, normalized size = 1.00

method	result	size
default	\(x^{2}-{\mathrm e}^{96 x +\frac {512}{\ln \relax (3)^{2}}-96+32 x^{2}}\)	\(24\)
norman	\(x^{2}-{\mathrm e}^{\frac {\left (32 x^{2}+96 x -96\right ) \ln \relax (3)^{2}+512}{\ln \relax (3)^{2}}}\)	\(30\)
risch	\(x^{2}-{\mathrm e}^{\frac {32 x^{2} \ln \relax (3)^{2}+96 x \ln \relax (3)^{2}-96 \ln \relax (3)^{2}+512}{\ln \relax (3)^{2}}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-64*x-96)*exp(((32*x^2+96*x-96)*ln(3)^2+512)/ln(3)^2)+2*x,x,method=_RETURNVERBOSE)

[Out]

x^2-exp(96*x+512/ln(3)^2-96+32*x^2)

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maxima [A]  time = 0.36, size = 28, normalized size = 1.17 \begin {gather*} x^{2} - e^{\left (\frac {32 \, {\left ({\left (x^{2} + 3 \, x - 3\right )} \log \relax (3)^{2} + 16\right )}}{\log \relax (3)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-96)*exp(((32*x^2+96*x-96)*log(3)^2+512)/log(3)^2)+2*x,x, algorithm="maxima")

[Out]

x^2 - e^(32*((x^2 + 3*x - 3)*log(3)^2 + 16)/log(3)^2)

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mupad [B]  time = 0.14, size = 25, normalized size = 1.04 \begin {gather*} x^2-{\mathrm {e}}^{96\,x}\,{\mathrm {e}}^{-96}\,{\mathrm {e}}^{\frac {512}{{\ln \relax (3)}^2}}\,{\mathrm {e}}^{32\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - exp((log(3)^2*(96*x + 32*x^2 - 96) + 512)/log(3)^2)*(64*x + 96),x)

[Out]

x^2 - exp(96*x)*exp(-96)*exp(512/log(3)^2)*exp(32*x^2)

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sympy [A]  time = 0.11, size = 26, normalized size = 1.08 \begin {gather*} x^{2} - e^{\frac {\left (32 x^{2} + 96 x - 96\right ) \log {\relax (3 )}^{2} + 512}{\log {\relax (3 )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x-96)*exp(((32*x**2+96*x-96)*ln(3)**2+512)/ln(3)**2)+2*x,x)

[Out]

x**2 - exp(((32*x**2 + 96*x - 96)*log(3)**2 + 512)/log(3)**2)

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