3.93.74 \(\int \frac {4 e^4 x^3+e^{4+\frac {4}{x^2}} (8-x^2)}{x^2} \, dx\)

Optimal. Leaf size=20 \[ 15+e^4 x \left (-e^{\frac {4}{x^2}}+2 x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14, 2288} \begin {gather*} 2 e^4 x^2-e^{\frac {4}{x^2}+4} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^4*x^3 + E^(4 + 4/x^2)*(8 - x^2))/x^2,x]

[Out]

-(E^(4 + 4/x^2)*x) + 2*E^4*x^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4 e^4 x-\frac {e^{4+\frac {4}{x^2}} \left (-8+x^2\right )}{x^2}\right ) \, dx\\ &=2 e^4 x^2-\int \frac {e^{4+\frac {4}{x^2}} \left (-8+x^2\right )}{x^2} \, dx\\ &=-e^{4+\frac {4}{x^2}} x+2 e^4 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} e^4 \left (-e^{\frac {4}{x^2}} x+2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^4*x^3 + E^(4 + 4/x^2)*(8 - x^2))/x^2,x]

[Out]

E^4*(-(E^(4/x^2)*x) + 2*x^2)

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fricas [A]  time = 0.97, size = 22, normalized size = 1.10 \begin {gather*} 2 \, x^{2} e^{4} - x e^{\left (\frac {4 \, {\left (x^{2} + 1\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+8)*exp(4)*exp(4/x^2)+4*x^3*exp(4))/x^2,x, algorithm="fricas")

[Out]

2*x^2*e^4 - x*e^(4*(x^2 + 1)/x^2)

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giac [A]  time = 0.14, size = 22, normalized size = 1.10 \begin {gather*} 2 \, x^{2} e^{4} - x e^{\left (\frac {4 \, {\left (x^{2} + 1\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+8)*exp(4)*exp(4/x^2)+4*x^3*exp(4))/x^2,x, algorithm="giac")

[Out]

2*x^2*e^4 - x*e^(4*(x^2 + 1)/x^2)

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maple [A]  time = 0.08, size = 20, normalized size = 1.00

method	result	size
derivativedivides	\(2 x^{2} {\mathrm e}^{4}-x \,{\mathrm e}^{4} {\mathrm e}^{\frac {4}{x^{2}}}\)	\(20\)
default	\(2 x^{2} {\mathrm e}^{4}-x \,{\mathrm e}^{4} {\mathrm e}^{\frac {4}{x^{2}}}\)	\(20\)
risch	\(2 x^{2} {\mathrm e}^{4}-x \,{\mathrm e}^{\frac {4 x^{2}+4}{x^{2}}}\)	\(23\)
norman	\(\frac {2 x^{3} {\mathrm e}^{4}-x^{2} {\mathrm e}^{4} {\mathrm e}^{\frac {4}{x^{2}}}}{x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+8)*exp(4)*exp(4/x^2)+4*x^3*exp(4))/x^2,x,method=_RETURNVERBOSE)

[Out]

2*x^2*exp(4)-exp(4)*exp(1/x^2)^4*x

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maxima [C]  time = 0.40, size = 56, normalized size = 2.80 \begin {gather*} -x \sqrt {-\frac {1}{x^{2}}} e^{4} \Gamma \left (-\frac {1}{2}, -\frac {4}{x^{2}}\right ) + 2 \, x^{2} e^{4} - \frac {2 \, \sqrt {\pi } {\left (\operatorname {erf}\left (2 \, \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )} e^{4}}{x \sqrt {-\frac {1}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+8)*exp(4)*exp(4/x^2)+4*x^3*exp(4))/x^2,x, algorithm="maxima")

[Out]

-x*sqrt(-1/x^2)*e^4*gamma(-1/2, -4/x^2) + 2*x^2*e^4 - 2*sqrt(pi)*(erf(2*sqrt(-1/x^2)) - 1)*e^4/(x*sqrt(-1/x^2)
)

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mupad [B]  time = 7.50, size = 16, normalized size = 0.80 \begin {gather*} x\,{\mathrm {e}}^4\,\left (2\,x-{\mathrm {e}}^{\frac {4}{x^2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*exp(4) - exp(4)*exp(4/x^2)*(x^2 - 8))/x^2,x)

[Out]

x*exp(4)*(2*x - exp(4/x^2))

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sympy [A]  time = 0.16, size = 19, normalized size = 0.95 \begin {gather*} 2 x^{2} e^{4} - x e^{4} e^{\frac {4}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+8)*exp(4)*exp(4/x**2)+4*x**3*exp(4))/x**2,x)

[Out]

2*x**2*exp(4) - x*exp(4)*exp(4/x**2)

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