3.90.73 \(\int \frac {-196 e-196 e^3}{e^2 x^2} \, dx\) [8973]

Optimal. Leaf size=15 \[ \frac {196 e \left (1+\frac {1+x}{e^2}\right )}{x} \]

[Out]

196/x*exp(1)*(1+exp(-2)*(1+x))

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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 30} \begin {gather*} \frac {196 \left (1+e^2\right )}{e x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-196*E - 196*E^3)/(E^2*x^2),x]

[Out]

(196*(1 + E^2))/(E*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\left (196 \left (1+e^2\right )\right ) \int \frac {1}{x^2} \, dx}{e}\\ &=\frac {196 \left (1+e^2\right )}{e x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 0.87 \begin {gather*} \frac {196 \left (1+e^2\right )}{e x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-196*E - 196*E^3)/(E^2*x^2),x]

[Out]

(196*(1 + E^2))/(E*x)

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Maple [A]
time = 0.41, size = 21, normalized size = 1.40

method	result	size
gosper	\(\frac {196 \,{\mathrm e} \left ({\mathrm e}^{2}+1\right ) {\mathrm e}^{-2}}{x}\)	\(16\)
norman	\(\frac {196 \,{\mathrm e} \left ({\mathrm e}^{2}+1\right ) {\mathrm e}^{-2}}{x}\)	\(16\)
risch	\(\frac {196 \,{\mathrm e}^{-2} {\mathrm e}^{3}}{x}+\frac {196 \,{\mathrm e}^{-2} {\mathrm e}}{x}\)	\(20\)
default	\(-\frac {196 \left (-{\mathrm e} \,{\mathrm e}^{2}-{\mathrm e}\right ) {\mathrm e}^{-2}}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-196*exp(1)*exp(2)-196*exp(1))/x^2/exp(2),x,method=_RETURNVERBOSE)

[Out]

-196*(-exp(1)*exp(2)-exp(1))/exp(2)/x

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.80 \begin {gather*} \frac {196 \, {\left (e^{3} + e\right )} e^{\left (-2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-196*exp(1)*exp(2)-196*exp(1))/x^2/exp(2),x, algorithm="maxima")

[Out]

196*(e^3 + e)*e^(-2)/x

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Fricas [A]
time = 0.35, size = 11, normalized size = 0.73 \begin {gather*} \frac {196 \, {\left (e^{2} + 1\right )} e^{\left (-1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-196*exp(1)*exp(2)-196*exp(1))/x^2/exp(2),x, algorithm="fricas")

[Out]

196*(e^2 + 1)*e^(-1)/x

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Sympy [A]
time = 0.01, size = 14, normalized size = 0.93 \begin {gather*} - \frac {- 196 e^{2} - 196}{e x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-196*exp(1)*exp(2)-196*exp(1))/x**2/exp(2),x)

[Out]

-(-196*exp(2) - 196)*exp(-1)/x

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Giac [A]
time = 0.40, size = 12, normalized size = 0.80 \begin {gather*} \frac {196 \, {\left (e^{3} + e\right )} e^{\left (-2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-196*exp(1)*exp(2)-196*exp(1))/x^2/exp(2),x, algorithm="giac")

[Out]

196*(e^3 + e)*e^(-2)/x

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Mupad [B]
time = 0.03, size = 11, normalized size = 0.73 \begin {gather*} \frac {196\,{\mathrm {e}}^{-1}\,\left ({\mathrm {e}}^2+1\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2)*(196*exp(1) + 196*exp(3)))/x^2,x)

[Out]

(196*exp(-1)*(exp(2) + 1))/x

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