3.71.90 \(\int \frac {1}{25} (25+12 e^{2 x}-12 e^{4 x}-50 x) \, dx\)

Optimal. Leaf size=27 \[ 3-\frac {3}{25} \left (-1+e^{2 x}\right )^2+x-x^2-\log ^2(2) \]

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 2194} \begin {gather*} -x^2+x+\frac {6 e^{2 x}}{25}-\frac {3 e^{4 x}}{25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(25 + 12*E^(2*x) - 12*E^(4*x) - 50*x)/25,x]

[Out]

(6*E^(2*x))/25 - (3*E^(4*x))/25 + x - x^2

Rule 12

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \left (25+12 e^{2 x}-12 e^{4 x}-50 x\right ) \, dx\\ &=x-x^2+\frac {12}{25} \int e^{2 x} \, dx-\frac {12}{25} \int e^{4 x} \, dx\\ &=\frac {6 e^{2 x}}{25}-\frac {3 e^{4 x}}{25}+x-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 0.93 \begin {gather*} \frac {6 e^{2 x}}{25}-\frac {3 e^{4 x}}{25}+x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + 12*E^(2*x) - 12*E^(4*x) - 50*x)/25,x]

[Out]

(6*E^(2*x))/25 - (3*E^(4*x))/25 + x - x^2

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fricas [A]  time = 0.56, size = 19, normalized size = 0.70 \begin {gather*} -x^{2} + x - \frac {3}{25} \, e^{\left (4 \, x\right )} + \frac {6}{25} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/25*exp(2*x)^2+12/25*exp(2*x)-2*x+1,x, algorithm="fricas")

[Out]

-x^2 + x - 3/25*e^(4*x) + 6/25*e^(2*x)

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giac [A]  time = 0.13, size = 19, normalized size = 0.70 \begin {gather*} -x^{2} + x - \frac {3}{25} \, e^{\left (4 \, x\right )} + \frac {6}{25} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/25*exp(2*x)^2+12/25*exp(2*x)-2*x+1,x, algorithm="giac")

[Out]

-x^2 + x - 3/25*e^(4*x) + 6/25*e^(2*x)

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maple [A]  time = 0.03, size = 20, normalized size = 0.74

method	result	size
risch	\(x -x^{2}-\frac {3 \,{\mathrm e}^{4 x}}{25}+\frac {6 \,{\mathrm e}^{2 x}}{25}\)	\(20\)
derivativedivides	\(x -x^{2}-\frac {3 \,{\mathrm e}^{4 x}}{25}+\frac {6 \,{\mathrm e}^{2 x}}{25}\)	\(22\)
default	\(x -x^{2}-\frac {3 \,{\mathrm e}^{4 x}}{25}+\frac {6 \,{\mathrm e}^{2 x}}{25}\)	\(22\)
norman	\(x -x^{2}-\frac {3 \,{\mathrm e}^{4 x}}{25}+\frac {6 \,{\mathrm e}^{2 x}}{25}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-12/25*exp(2*x)^2+12/25*exp(2*x)-2*x+1,x,method=_RETURNVERBOSE)

[Out]

x-x^2-3/25*exp(4*x)+6/25*exp(2*x)

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maxima [A]  time = 0.37, size = 19, normalized size = 0.70 \begin {gather*} -x^{2} + x - \frac {3}{25} \, e^{\left (4 \, x\right )} + \frac {6}{25} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/25*exp(2*x)^2+12/25*exp(2*x)-2*x+1,x, algorithm="maxima")

[Out]

-x^2 + x - 3/25*e^(4*x) + 6/25*e^(2*x)

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mupad [B]  time = 0.06, size = 19, normalized size = 0.70 \begin {gather*} x+\frac {6\,{\mathrm {e}}^{2\,x}}{25}-\frac {3\,{\mathrm {e}}^{4\,x}}{25}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*exp(2*x))/25 - 2*x - (12*exp(4*x))/25 + 1,x)

[Out]

x + (6*exp(2*x))/25 - (3*exp(4*x))/25 - x^2

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sympy [A]  time = 0.11, size = 20, normalized size = 0.74 \begin {gather*} - x^{2} + x - \frac {3 e^{4 x}}{25} + \frac {6 e^{2 x}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/25*exp(2*x)**2+12/25*exp(2*x)-2*x+1,x)

[Out]

-x**2 + x - 3*exp(4*x)/25 + 6*exp(2*x)/25

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