\(\int (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)) \, dx\) [1549]

Optimal result

Integrand size = 32, antiderivative size = 18 \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx=e^{x^3}+x \left (-3+(3-\log (3))^4\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2240} \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx=e^{x^3}+x \left (78+\log ^4(3)-12 \log ^3(3)+54 \log ^2(3)-108 \log (3)\right ) \]

[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 2240

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx=e^{x^3}+78 x-108 x \log (3)+54 x \log ^2(3)-12 x \log ^3(3)+x \log ^4(3) \]

[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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.83 \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx=x \log \left (3\right )^{4} - 12 \, x \log \left (3\right )^{3} + 54 \, x \log \left (3\right )^{2} - 108 \, x \log \left (3\right ) + 78 \, x + e^{\left (x^{3}\right )} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx=x \left (- 108 \log {\left (3 \right )} - 12 \log {\left (3 \right )}^{3} + \log {\left (3 \right )}^{4} + 54 \log {\left (3 \right )}^{2} + 78\right ) + e^{x^{3}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.83 \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx=x \log \left (3\right )^{4} - 12 \, x \log \left (3\right )^{3} + 54 \, x \log \left (3\right )^{2} - 108 \, x \log \left (3\right ) + 78 \, x + e^{\left (x^{3}\right )} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.83 \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx=x \log \left (3\right )^{4} - 12 \, x \log \left (3\right )^{3} + 54 \, x \log \left (3\right )^{2} - 108 \, x \log \left (3\right ) + 78 \, x + e^{\left (x^{3}\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 8.76 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \left (78+3 e^{x^3} x^2-108 \log (3)+54 \log ^2(3)-12 \log ^3(3)+\log ^4(3)\right ) \, dx={\mathrm {e}}^{x^3}+x\,\left (54\,{\ln \left (3\right )}^2-108\,\ln \left (3\right )-12\,{\ln \left (3\right )}^3+{\ln \left (3\right )}^4+78\right ) \]

[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)