3.16.49 \(\int (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)) \, dx\)

Optimal. Leaf size=18 \[ e^{x^3}+x \left (-3+(3-\log (3))^4\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.67, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2209} \begin {gather*} e^{x^3}+x \left (78+\log ^4(3)-12 \log ^3(3)+54 \log ^2(3)-108 \log (3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[78 + 3*E^x^3*x^2 - 108*Log[3] + 54*Log[3]^2 - 12*Log[3]^3 + Log[3]^4,x]

[Out]

E^x^3 + x*(78 - 108*Log[3] + 54*Log[3]^2 - 12*Log[3]^3 + Log[3]^4)

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x \left (78-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right )+3 \int e^{x^3} x^2 \, dx\\ &=e^{x^3}+x \left (78-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.89 \begin {gather*} e^{x^3}+78 x-108 x \log (3)+54 x \log ^2(3)-12 x \log ^3(3)+x \log ^4(3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[78 + 3*E^x^3*x^2 - 108*Log[3] + 54*Log[3]^2 - 12*Log[3]^3 + Log[3]^4,x]

[Out]

E^x^3 + 78*x - 108*x*Log[3] + 54*x*Log[3]^2 - 12*x*Log[3]^3 + x*Log[3]^4

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fricas [B]  time = 0.74, size = 33, normalized size = 1.83 \begin {gather*} x \log \relax (3)^{4} - 12 \, x \log \relax (3)^{3} + 54 \, x \log \relax (3)^{2} - 108 \, x \log \relax (3) + 78 \, x + e^{\left (x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*exp(x^3)+log(3)^4-12*log(3)^3+54*log(3)^2-108*log(3)+78,x, algorithm="fricas")

[Out]

x*log(3)^4 - 12*x*log(3)^3 + 54*x*log(3)^2 - 108*x*log(3) + 78*x + e^(x^3)

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giac [B]  time = 0.16, size = 33, normalized size = 1.83 \begin {gather*} x \log \relax (3)^{4} - 12 \, x \log \relax (3)^{3} + 54 \, x \log \relax (3)^{2} - 108 \, x \log \relax (3) + 78 \, x + e^{\left (x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*exp(x^3)+log(3)^4-12*log(3)^3+54*log(3)^2-108*log(3)+78,x, algorithm="giac")

[Out]

x*log(3)^4 - 12*x*log(3)^3 + 54*x*log(3)^2 - 108*x*log(3) + 78*x + e^(x^3)

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maple [A]  time = 0.03, size = 30, normalized size = 1.67

method	result	size
norman	\(\left (\ln \relax (3)^{4}-12 \ln \relax (3)^{3}+54 \ln \relax (3)^{2}-108 \ln \relax (3)+78\right ) x +{\mathrm e}^{x^{3}}\)	\(30\)
default	\(78 x +x \ln \relax (3)^{4}+54 x \ln \relax (3)^{2}-12 x \ln \relax (3)^{3}+{\mathrm e}^{x^{3}}-108 x \ln \relax (3)\)	\(34\)
risch	\(78 x +x \ln \relax (3)^{4}+54 x \ln \relax (3)^{2}-12 x \ln \relax (3)^{3}+{\mathrm e}^{x^{3}}-108 x \ln \relax (3)\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x^2*exp(x^3)+ln(3)^4-12*ln(3)^3+54*ln(3)^2-108*ln(3)+78,x,method=_RETURNVERBOSE)

[Out]

(ln(3)^4-12*ln(3)^3+54*ln(3)^2-108*ln(3)+78)*x+exp(x^3)

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maxima [B]  time = 0.47, size = 33, normalized size = 1.83 \begin {gather*} x \log \relax (3)^{4} - 12 \, x \log \relax (3)^{3} + 54 \, x \log \relax (3)^{2} - 108 \, x \log \relax (3) + 78 \, x + e^{\left (x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x^2*exp(x^3)+log(3)^4-12*log(3)^3+54*log(3)^2-108*log(3)+78,x, algorithm="maxima")

[Out]

x*log(3)^4 - 12*x*log(3)^3 + 54*x*log(3)^2 - 108*x*log(3) + 78*x + e^(x^3)

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mupad [B]  time = 1.00, size = 29, normalized size = 1.61 \begin {gather*} {\mathrm {e}}^{x^3}+x\,\left (54\,{\ln \relax (3)}^2-108\,\ln \relax (3)-12\,{\ln \relax (3)}^3+{\ln \relax (3)}^4+78\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*x^2*exp(x^3) - 108*log(3) + 54*log(3)^2 - 12*log(3)^3 + log(3)^4 + 78,x)

[Out]

exp(x^3) + x*(54*log(3)^2 - 108*log(3) - 12*log(3)^3 + log(3)^4 + 78)

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sympy [B]  time = 0.09, size = 31, normalized size = 1.72 \begin {gather*} x \left (- 108 \log {\relax (3 )} - 12 \log {\relax (3 )}^{3} + \log {\relax (3 )}^{4} + 54 \log {\relax (3 )}^{2} + 78\right ) + e^{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*x**2*exp(x**3)+ln(3)**4-12*ln(3)**3+54*ln(3)**2-108*ln(3)+78,x)

[Out]

x*(-108*log(3) - 12*log(3)**3 + log(3)**4 + 54*log(3)**2 + 78) + exp(x**3)

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