3.64.36 \(\int \frac {-1+e^x x-16 x^4-6 x^6}{x} \, dx\)

Optimal. Leaf size=20 \[ e^x-x^4 \left (4+x^2\right )+\log \left (\frac {3}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2194} \begin {gather*} -x^6-4 x^4+e^x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + E^x*x - 16*x^4 - 6*x^6)/x,x]

[Out]

E^x - 4*x^4 - x^6 - Log[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} &=\int \left (e^x+\frac {-1-16 x^4-6 x^6}{x}\right ) \, dx\\ &=\int e^x \, dx+\int \frac {-1-16 x^4-6 x^6}{x} \, dx\\ &=e^x+\int \left (-\frac {1}{x}-16 x^3-6 x^5\right ) \, dx\\ &=e^x-4 x^4-x^6-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} e^x-4 x^4-x^6-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^x*x - 16*x^4 - 6*x^6)/x,x]

[Out]

E^x - 4*x^4 - x^6 - Log[x]

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fricas [A]  time = 0.62, size = 17, normalized size = 0.85 \begin {gather*} -x^{6} - 4 \, x^{4} + e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-6*x^6-16*x^4-1)/x,x, algorithm="fricas")

[Out]

-x^6 - 4*x^4 + e^x - log(x)

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giac [A]  time = 0.13, size = 17, normalized size = 0.85 \begin {gather*} -x^{6} - 4 \, x^{4} + e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-6*x^6-16*x^4-1)/x,x, algorithm="giac")

[Out]

-x^6 - 4*x^4 + e^x - log(x)

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maple [A]  time = 0.02, size = 18, normalized size = 0.90

method	result	size
default	\(-\ln \relax (x )-4 x^{4}-x^{6}+{\mathrm e}^{x}\)	\(18\)
norman	\(-\ln \relax (x )-4 x^{4}-x^{6}+{\mathrm e}^{x}\)	\(18\)
risch	\(-\ln \relax (x )-4 x^{4}-x^{6}+{\mathrm e}^{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*x-6*x^6-16*x^4-1)/x,x,method=_RETURNVERBOSE)

[Out]

-ln(x)-4*x^4-x^6+exp(x)

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maxima [A]  time = 0.37, size = 17, normalized size = 0.85 \begin {gather*} -x^{6} - 4 \, x^{4} + e^{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-6*x^6-16*x^4-1)/x,x, algorithm="maxima")

[Out]

-x^6 - 4*x^4 + e^x - log(x)

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mupad [B]  time = 4.11, size = 17, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^x-\ln \relax (x)-4\,x^4-x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x^4 - x*exp(x) + 6*x^6 + 1)/x,x)

[Out]

exp(x) - log(x) - 4*x^4 - x^6

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sympy [A]  time = 0.10, size = 14, normalized size = 0.70 \begin {gather*} - x^{6} - 4 x^{4} + e^{x} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-6*x**6-16*x**4-1)/x,x)

[Out]

-x**6 - 4*x**4 + exp(x) - log(x)

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