3.102.70 \(\int \frac {32+e^4 x^2}{e^4 x^2} \, dx\) [10170]

Optimal. Leaf size=11 \[ -1-\frac {32}{e^4 x}+x \]

[Out]

x-32/exp(4)/x-1

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 14} \begin {gather*} x-\frac {32}{e^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32 + E^4*x^2)/(E^4*x^2),x]

[Out]

-32/(E^4*x) + 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 {32+e^4 x^2}{x^2} \, dx}{e^4}\\ &=\frac {\int \left (e^4+\frac {32}{x^2}\right ) \, dx}{e^4}\\ &=-\frac {32}{e^4 x}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 0.91 \begin {gather*} -\frac {32}{e^4 x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 + E^4*x^2)/(E^4*x^2),x]

[Out]

-32/(E^4*x) + x

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Maple [A]
time = 0.02, size = 16, normalized size = 1.45

method	result	size
risch	\(x -\frac {32 \,{\mathrm e}^{-4}}{x}\)	\(10\)
norman	\(\frac {x^{2}-32 \,{\mathrm e}^{-4}}{x}\)	\(15\)
default	\({\mathrm e}^{-4} \left (x \,{\mathrm e}^{4}-\frac {32}{x}\right )\)	\(16\)
gosper	\(\frac {\left (x^{2} {\mathrm e}^{4}-32\right ) {\mathrm e}^{-4}}{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(4)+32)/x^2/exp(4),x,method=_RETURNVERBOSE)

[Out]

1/exp(4)*(x*exp(4)-32/x)

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Maxima [A]
time = 0.26, size = 13, normalized size = 1.18 \begin {gather*} {\left (x e^{4} - \frac {32}{x}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(4)+32)/x^2/exp(4),x, algorithm="maxima")

[Out]

(x*e^4 - 32/x)*e^(-4)

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Fricas [A]
time = 0.33, size = 14, normalized size = 1.27 \begin {gather*} \frac {{\left (x^{2} e^{4} - 32\right )} e^{\left (-4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(4)+32)/x^2/exp(4),x, algorithm="fricas")

[Out]

(x^2*e^4 - 32)*e^(-4)/x

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Sympy [A]
time = 0.02, size = 10, normalized size = 0.91 \begin {gather*} \frac {x e^{4} - \frac {32}{x}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(4)+32)/x**2/exp(4),x)

[Out]

(x*exp(4) - 32/x)*exp(-4)

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Giac [A]
time = 0.41, size = 13, normalized size = 1.18 \begin {gather*} {\left (x e^{4} - \frac {32}{x}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(4)+32)/x^2/exp(4),x, algorithm="giac")

[Out]

(x*e^4 - 32/x)*e^(-4)

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Mupad [B]
time = 6.91, size = 9, normalized size = 0.82 \begin {gather*} x-\frac {32\,{\mathrm {e}}^{-4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4)*(x^2*exp(4) + 32))/x^2,x)

[Out]

x - (32*exp(-4))/x

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