3.55.45 \(\int \frac {-6+x^4+e^{x^4} x^7}{x^4} \, dx\)

Optimal. Leaf size=16 \[ \frac {e^{x^4}}{4}+\frac {2}{x^3}+x \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14, 2209} \begin {gather*} \frac {e^{x^4}}{4}+\frac {2}{x^3}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 + x^4 + E^x^4*x^7)/x^4,x]

[Out]

E^x^4/4 + 2/x^3 + 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} &=\int \left (e^{x^4} x^3+\frac {-6+x^4}{x^4}\right ) \, dx\\ &=\int e^{x^4} x^3 \, dx+\int \frac {-6+x^4}{x^4} \, dx\\ &=\frac {e^{x^4}}{4}+\int \left (1-\frac {6}{x^4}\right ) \, dx\\ &=\frac {e^{x^4}}{4}+\frac {2}{x^3}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {e^{x^4}}{4}+\frac {2}{x^3}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + x^4 + E^x^4*x^7)/x^4,x]

[Out]

E^x^4/4 + 2/x^3 + x

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fricas [A]  time = 0.86, size = 20, normalized size = 1.25 \begin {gather*} \frac {4 \, x^{4} + x^{3} e^{\left (x^{4}\right )} + 8}{4 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7*exp(x^4)+x^4-6)/x^4,x, algorithm="fricas")

[Out]

1/4*(4*x^4 + x^3*e^(x^4) + 8)/x^3

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giac [A]  time = 0.13, size = 20, normalized size = 1.25 \begin {gather*} \frac {4 \, x^{4} + x^{3} e^{\left (x^{4}\right )} + 8}{4 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7*exp(x^4)+x^4-6)/x^4,x, algorithm="giac")

[Out]

1/4*(4*x^4 + x^3*e^(x^4) + 8)/x^3

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maple [A]  time = 0.02, size = 14, normalized size = 0.88

method	result	size
default	\(\frac {2}{x^{3}}+x +\frac {{\mathrm e}^{x^{4}}}{4}\)	\(14\)
risch	\(\frac {2}{x^{3}}+x +\frac {{\mathrm e}^{x^{4}}}{4}\)	\(14\)
norman	\(\frac {2+x^{4}+\frac {x^{3} {\mathrm e}^{x^{4}}}{4}}{x^{3}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*exp(x^4)+x^4-6)/x^4,x,method=_RETURNVERBOSE)

[Out]

2/x^3+x+1/4*exp(x^4)

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maxima [A]  time = 0.37, size = 13, normalized size = 0.81 \begin {gather*} x + \frac {2}{x^{3}} + \frac {1}{4} \, e^{\left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7*exp(x^4)+x^4-6)/x^4,x, algorithm="maxima")

[Out]

x + 2/x^3 + 1/4*e^(x^4)

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mupad [B]  time = 3.39, size = 13, normalized size = 0.81 \begin {gather*} x+\frac {{\mathrm {e}}^{x^4}}{4}+\frac {2}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*exp(x^4) + x^4 - 6)/x^4,x)

[Out]

x + exp(x^4)/4 + 2/x^3

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sympy [A]  time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} x + \frac {e^{x^{4}}}{4} + \frac {2}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**7*exp(x**4)+x**4-6)/x**4,x)

[Out]

x + exp(x**4)/4 + 2/x**3

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