3.54.43 \(\int \frac {-3+4 x^4-4 e^{-1+x} x^4}{4 x^4} \, dx\)

Optimal. Leaf size=17 \[ -2-e^{-1+x}+\frac {1}{4 x^3}+x \]

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 14, 2194} \begin {gather*} \frac {1}{4 x^3}+x-e^{x-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 4*x^4 - 4*E^(-1 + x)*x^4)/(4*x^4),x]

[Out]

-E^(-1 + x) + 1/(4*x^3) + 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]]

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} -e^{-1+x}+\frac {1}{4 x^3}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 4*x^4 - 4*E^(-1 + x)*x^4)/(4*x^4),x]

[Out]

-E^(-1 + x) + 1/(4*x^3) + x

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fricas [A]  time = 0.59, size = 21, normalized size = 1.24 \begin {gather*} \frac {4 \, x^{4} - 4 \, x^{3} e^{\left (x - 1\right )} + 1}{4 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*x^4 - 4*x^3*e^(x - 1) + 1)/x^3

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giac [A]  time = 0.20, size = 24, normalized size = 1.41 \begin {gather*} \frac {{\left (4 \, x^{4} e - 4 \, x^{3} e^{x} + e\right )} e^{\left (-1\right )}}{4 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 14, normalized size = 0.82

method	result	size
risch	\(x +\frac {1}{4 x^{3}}-{\mathrm e}^{x -1}\)	\(14\)
derivativedivides	\(\frac {1}{4 x^{3}}+x -1-{\mathrm e}^{x -1}\)	\(15\)
default	\(\frac {1}{4 x^{3}}+x -1-{\mathrm e}^{x -1}\)	\(15\)
norman	\(\frac {\frac {1}{4}+x^{4}-x^{3} {\mathrm e}^{x -1}}{x^{3}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+1/4/x^3-exp(x-1)

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maxima [A]  time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} x + \frac {1}{4 \, x^{3}} - e^{\left (x - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/4/x^3 - e^(x - 1)

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mupad [B]  time = 0.07, size = 13, normalized size = 0.76 \begin {gather*} x-{\mathrm {e}}^{x-1}+\frac {1}{4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*exp(x - 1) - x^4 + 3/4)/x^4,x)

[Out]

x - exp(x - 1) + 1/(4*x^3)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.71 \begin {gather*} x - e^{x - 1} + \frac {1}{4 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - exp(x - 1) + 1/(4*x**3)

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