3.1.86 \(\int \frac {(1+x^4)^{3/4}}{x^8} \, dx\)

Optimal. Leaf size=16 \[ -\frac {\left (x^4+1\right )^{7/4}}{7 x^7} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} -\frac {\left (x^4+1\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)^(3/4)/x^8,x]

[Out]

-1/7*(1 + x^4)^(7/4)/x^7

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (1+x^4\right )^{3/4}}{x^8} \, dx &=-\frac {\left (1+x^4\right )^{7/4}}{7 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\left (x^4+1\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)^(3/4)/x^8,x]

[Out]

-1/7*(1 + x^4)^(7/4)/x^7

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IntegrateAlgebraic [A]  time = 0.11, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^4\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^4)^(3/4)/x^8,x]

[Out]

-1/7*(1 + x^4)^(7/4)/x^7

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fricas [A]  time = 0.45, size = 12, normalized size = 0.75 \begin {gather*} -\frac {{\left (x^{4} + 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(3/4)/x^8,x, algorithm="fricas")

[Out]

-1/7*(x^4 + 1)^(7/4)/x^7

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )}^{\frac {3}{4}}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(3/4)/x^8,x, algorithm="giac")

[Out]

integrate((x^4 + 1)^(3/4)/x^8, x)

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maple [A]  time = 0.06, size = 13, normalized size = 0.81

method	result	size
gosper	\(-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\)	\(13\)
trager	\(-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\)	\(13\)
meijerg	\(-\frac {\left (x^{4}+1\right )^{\frac {7}{4}}}{7 x^{7}}\)	\(13\)
risch	\(-\frac {x^{8}+2 x^{4}+1}{7 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)^(3/4)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*(x^4+1)^(7/4)/x^7

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maxima [A]  time = 0.50, size = 12, normalized size = 0.75 \begin {gather*} -\frac {{\left (x^{4} + 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(3/4)/x^8,x, algorithm="maxima")

[Out]

-1/7*(x^4 + 1)^(7/4)/x^7

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mupad [B]  time = 0.21, size = 24, normalized size = 1.50 \begin {gather*} -\frac {{\left (x^4+1\right )}^{3/4}+x^4\,{\left (x^4+1\right )}^{3/4}}{7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 1)^(3/4)/x^8,x)

[Out]

-((x^4 + 1)^(3/4) + x^4*(x^4 + 1)^(3/4))/(7*x^7)

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sympy [B]  time = 0.90, size = 53, normalized size = 3.31 \begin {gather*} \frac {\left (1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} + \frac {\left (1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 x^{4} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)**(3/4)/x**8,x)

[Out]

(1 + x**(-4))**(3/4)*gamma(-7/4)/(4*gamma(-3/4)) + (1 + x**(-4))**(3/4)*gamma(-7/4)/(4*x**4*gamma(-3/4))

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