3.33.12 $\int \frac{64+e^{\frac{x^4}{256}}{x}^4}{32x}dx$

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

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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} $$\begin{array}{c}2e^{\frac{x^4}{256}}+2\log (x)\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{32}\int \frac{64+e^{\frac{x^4}{256}}{x}^4}{x}dx\\ & =\frac{1}{32}\int (\frac{64}{x}+e^{\frac{x^4}{256}}{x}^3)dx\\ & =2\log (x)+\frac{1}{32}\int e^{\frac{x^4}{256}}{x}^{3}dx\\ & =2e^{\frac{x^4}{256}}+2\log (x)\end{array}\end{array}$$

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

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 $$\begin{array}{c}2e^{\left(\frac{1}{256}{x}^4\right)}+2\log (x)\end{array}$$

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 $$\begin{array}{c}2e^{\left(\frac{1}{256}{x}^4\right)}+\frac{1}{2}\log \left(\frac{1}{256}{x}^4\right)\end{array}$$

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	$2{\mathrm{e}}^{\frac{x^4}{256}}+2\ln (x)$	$14$
norman	$2{\mathrm{e}}^{\frac{x^4}{256}}+2\ln (x)$	$14$
risch	$2{\mathrm{e}}^{\frac{x^4}{256}}+2\ln (x)$	$14$

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 $$\begin{array}{c}2e^{\left(\frac{1}{256}{x}^4\right)}+2\log (x)\end{array}$$

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 $$\begin{array}{c}2{\mathrm{e}}^{\frac{x^4}{256}}+2\ln (x)\end{array}$$

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 $$\begin{array}{c}2e^{\frac{x^4}{256}}+2\log (x)\end{array}$$

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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