3.33.12 \(\int \frac {64+e^{\frac {x^4}{256}} x^4}{32 x} \, dx\)

Optimal. Leaf size=25 \[ \log \left (\frac {e^{2 e^{\frac {x^4}{256}}} x^2}{3 \log (4)}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 14, 2209} \begin {gather*} 2 e^{\frac {x^4}{256}}+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(64 + E^(x^4/256)*x^4)/(32*x),x]

[Out]

2*E^(x^4/256) + 2*Log[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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{32} \int \frac {64+e^{\frac {x^4}{256}} x^4}{x} \, dx\\ &=\frac {1}{32} \int \left (\frac {64}{x}+e^{\frac {x^4}{256}} x^3\right ) \, dx\\ &=2 \log (x)+\frac {1}{32} \int e^{\frac {x^4}{256}} x^3 \, dx\\ &=2 e^{\frac {x^4}{256}}+2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.64 \begin {gather*} 2 e^{\frac {x^4}{256}}+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64 + E^(x^4/256)*x^4)/(32*x),x]

[Out]

2*E^(x^4/256) + 2*Log[x]

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fricas [A]  time = 0.61, size = 13, normalized size = 0.52 \begin {gather*} 2 \, e^{\left (\frac {1}{256} \, x^{4}\right )} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(x^4*exp(1/256*x^4)+64)/x,x, algorithm="fricas")

[Out]

2*e^(1/256*x^4) + 2*log(x)

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giac [A]  time = 0.16, size = 17, normalized size = 0.68 \begin {gather*} 2 \, e^{\left (\frac {1}{256} \, x^{4}\right )} + \frac {1}{2} \, \log \left (\frac {1}{256} \, x^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(x^4*exp(1/256*x^4)+64)/x,x, algorithm="giac")

[Out]

2*e^(1/256*x^4) + 1/2*log(1/256*x^4)

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maple [A]  time = 0.02, size = 14, normalized size = 0.56

method	result	size
default	\(2 \,{\mathrm e}^{\frac {x^{4}}{256}}+2 \ln \relax (x )\)	\(14\)
norman	\(2 \,{\mathrm e}^{\frac {x^{4}}{256}}+2 \ln \relax (x )\)	\(14\)
risch	\(2 \,{\mathrm e}^{\frac {x^{4}}{256}}+2 \ln \relax (x )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/32*(x^4*exp(1/256*x^4)+64)/x,x,method=_RETURNVERBOSE)

[Out]

2*exp(1/256*x^4)+2*ln(x)

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maxima [A]  time = 0.34, size = 13, normalized size = 0.52 \begin {gather*} 2 \, e^{\left (\frac {1}{256} \, x^{4}\right )} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(x^4*exp(1/256*x^4)+64)/x,x, algorithm="maxima")

[Out]

2*e^(1/256*x^4) + 2*log(x)

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mupad [B]  time = 1.90, size = 13, normalized size = 0.52 \begin {gather*} 2\,{\mathrm {e}}^{\frac {x^4}{256}}+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4*exp(x^4/256))/32 + 2)/x,x)

[Out]

2*exp(x^4/256) + 2*log(x)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.48 \begin {gather*} 2 e^{\frac {x^{4}}{256}} + 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(x**4*exp(1/256*x**4)+64)/x,x)

[Out]

2*exp(x**4/256) + 2*log(x)

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