3.54.20 \(\int \frac {1}{2} (-2+e^{x^3+e^{x^2} x^3} (3 x^2+e^{x^2} (3 x^2+2 x^4))) \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{2} e^{\left (1+e^{x^2}\right ) x^3}-x \]

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Rubi [A]  time = 0.08, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 6706} \begin {gather*} \frac {1}{2} e^{x^3+e^{x^2} x^3}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + E^(x^3 + E^x^2*x^3)*(3*x^2 + E^x^2*(3*x^2 + 2*x^4)))/2,x]

[Out]

E^(x^3 + E^x^2*x^3)/2 - x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\frac {1}{2} \int \left (-2+e^{x^3+e^{x^2} x^3} \left (3 x^2+e^{x^2} \left (3 x^2+2 x^4\right )\right )\right ) \, dx\\ &=-x+\frac {1}{2} \int e^{x^3+e^{x^2} x^3} \left (3 x^2+e^{x^2} \left (3 x^2+2 x^4\right )\right ) \, dx\\ &=\frac {1}{2} e^{x^3+e^{x^2} x^3}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{2} e^{x^3+e^{x^2} x^3}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + E^(x^3 + E^x^2*x^3)*(3*x^2 + E^x^2*(3*x^2 + 2*x^4)))/2,x]

[Out]

E^(x^3 + E^x^2*x^3)/2 - x

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fricas [A]  time = 0.68, size = 19, normalized size = 0.90 \begin {gather*} -x + \frac {1}{2} \, e^{\left (x^{3} e^{\left (x^{2}\right )} + x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 1/2*e^(x^3*e^(x^2) + x^3)

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giac [A]  time = 0.14, size = 19, normalized size = 0.90 \begin {gather*} -x + \frac {1}{2} \, e^{\left (x^{3} e^{\left (x^{2}\right )} + x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 1/2*e^(x^3*e^(x^2) + x^3)

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maple [A]  time = 0.07, size = 18, normalized size = 0.86

method	result	size
risch	\(\frac {{\mathrm e}^{x^{3} \left ({\mathrm e}^{x^{2}}+1\right )}}{2}-x\)	\(18\)
default	\(-x +\frac {{\mathrm e}^{x^{3} {\mathrm e}^{x^{2}}+x^{3}}}{2}\)	\(20\)
norman	\(-x +\frac {{\mathrm e}^{x^{3} {\mathrm e}^{x^{2}}+x^{3}}}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*exp(x^3*(exp(x^2)+1))-x

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maxima [A]  time = 0.42, size = 19, normalized size = 0.90 \begin {gather*} -x + \frac {1}{2} \, e^{\left (x^{3} e^{\left (x^{2}\right )} + x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 1/2*e^(x^3*e^(x^2) + x^3)

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mupad [B]  time = 0.12, size = 19, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{x^3\,{\mathrm {e}}^{x^2}}}{2}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^3*exp(x^2) + x^3)*(exp(x^2)*(3*x^2 + 2*x^4) + 3*x^2))/2 - 1,x)

[Out]

(exp(x^3)*exp(x^3*exp(x^2)))/2 - x

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sympy [A]  time = 0.29, size = 15, normalized size = 0.71 \begin {gather*} - x + \frac {e^{x^{3} e^{x^{2}} + x^{3}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((2*x**4+3*x**2)*exp(x**2)+3*x**2)*exp(x**3*exp(x**2)+x**3)-1,x)

[Out]

-x + exp(x**3*exp(x**2) + x**3)/2

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