3.34.87 \(\int \frac {-3+e^4 (9-4 x^2)}{15 e^4 x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac {4+\frac {-3+\frac {1}{e^4}}{x^2}}{15 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 14} \begin {gather*} \frac {4}{15 x}-\frac {3-\frac {1}{e^4}}{15 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + E^4*(9 - 4*x^2))/(15*E^4*x^4),x]

[Out]

-1/15*(3 - E^(-4))/x^3 + 4/(15*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 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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-3+e^4 \left (9-4 x^2\right )}{x^4} \, dx}{15 e^4}\\ &=\frac {\int \left (\frac {3 \left (-1+3 e^4\right )}{x^4}-\frac {4 e^4}{x^2}\right ) \, dx}{15 e^4}\\ &=-\frac {3-\frac {1}{e^4}}{15 x^3}+\frac {4}{15 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.28 \begin {gather*} \frac {1+e^4 \left (-3+4 x^2\right )}{15 e^4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + E^4*(9 - 4*x^2))/(15*E^4*x^4),x]

[Out]

(1 + E^4*(-3 + 4*x^2))/(15*E^4*x^3)

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fricas [A]  time = 0.52, size = 19, normalized size = 1.06 \begin {gather*} \frac {{\left ({\left (4 \, x^{2} - 3\right )} e^{4} + 1\right )} e^{\left (-4\right )}}{15 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-4*x^2+9)*exp(2)^2-3)/x^4/exp(2)^2,x, algorithm="fricas")

[Out]

1/15*((4*x^2 - 3)*e^4 + 1)*e^(-4)/x^3

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giac [A]  time = 0.15, size = 20, normalized size = 1.11 \begin {gather*} \frac {{\left (4 \, x^{2} e^{4} - 3 \, e^{4} + 1\right )} e^{\left (-4\right )}}{15 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-4*x^2+9)*exp(2)^2-3)/x^4/exp(2)^2,x, algorithm="giac")

[Out]

1/15*(4*x^2*e^4 - 3*e^4 + 1)*e^(-4)/x^3

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maple [A]  time = 0.04, size = 21, normalized size = 1.17

method	result	size
risch	\(\frac {\left (4 x^{2} {\mathrm e}^{4}-3 \,{\mathrm e}^{4}+1\right ) {\mathrm e}^{-4}}{15 x^{3}}\)	\(21\)
default	\(\frac {{\mathrm e}^{-4} \left (\frac {4 \,{\mathrm e}^{4}}{x}-\frac {9 \,{\mathrm e}^{4}-3}{3 x^{3}}\right )}{15}\)	\(26\)
gosper	\(\frac {\left (4 x^{2} {\mathrm e}^{4}-3 \,{\mathrm e}^{4}+1\right ) {\mathrm e}^{-4}}{15 x^{3}}\)	\(27\)
norman	\(\frac {\left (\frac {4 x^{2} {\mathrm e}^{2}}{15}-\frac {\left (3 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-2}}{15}\right ) {\mathrm e}^{-2}}{x^{3}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/15*((-4*x^2+9)*exp(2)^2-3)/x^4/exp(2)^2,x,method=_RETURNVERBOSE)

[Out]

1/15*(4*x^2*exp(4)-3*exp(4)+1)*exp(-4)/x^3

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maxima [A]  time = 0.37, size = 20, normalized size = 1.11 \begin {gather*} \frac {{\left (4 \, x^{2} e^{4} - 3 \, e^{4} + 1\right )} e^{\left (-4\right )}}{15 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-4*x^2+9)*exp(2)^2-3)/x^4/exp(2)^2,x, algorithm="maxima")

[Out]

1/15*(4*x^2*e^4 - 3*e^4 + 1)*e^(-4)/x^3

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mupad [B]  time = 1.97, size = 20, normalized size = 1.11 \begin {gather*} \frac {\frac {4\,x^2}{15}-\frac {{\mathrm {e}}^{-4}\,\left (3\,{\mathrm {e}}^4-1\right )}{15}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4)*((exp(4)*(4*x^2 - 9))/15 + 1/5))/x^4,x)

[Out]

((4*x^2)/15 - (exp(-4)*(3*exp(4) - 1))/15)/x^3

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sympy [A]  time = 0.18, size = 24, normalized size = 1.33 \begin {gather*} - \frac {- 4 x^{2} e^{4} - 1 + 3 e^{4}}{15 x^{3} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*((-4*x**2+9)*exp(2)**2-3)/x**4/exp(2)**2,x)

[Out]

-(-4*x**2*exp(4) - 1 + 3*exp(4))*exp(-4)/(15*x**3)

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