3.3.16 \(\int \frac {1}{x^6 (1+x^4)^{3/4}} \, dx\)

Optimal. Leaf size=23 \[ \frac {\sqrt [4]{x^4+1} \left (4 x^4-1\right )}{5 x^5} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {271, 264} \begin {gather*} \frac {4 \sqrt [4]{x^4+1}}{5 x}-\frac {\sqrt [4]{x^4+1}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(1 + x^4)^(3/4)),x]

[Out]

-1/5*(1 + x^4)^(1/4)/x^5 + (4*(1 + x^4)^(1/4))/(5*x)

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]

Rule 271

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

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (1+x^4\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{1+x^4}}{5 x^5}-\frac {4}{5} \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1+x^4}}{5 x^5}+\frac {4 \sqrt [4]{1+x^4}}{5 x}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{x^4+1} \left (4 x^4-1\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(1 + x^4)^(3/4)),x]

[Out]

((1 + x^4)^(1/4)*(-1 + 4*x^4))/(5*x^5)

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IntegrateAlgebraic [A]  time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{1+x^4} \left (-1+4 x^4\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^6*(1 + x^4)^(3/4)),x]

[Out]

((1 + x^4)^(1/4)*(-1 + 4*x^4))/(5*x^5)

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fricas [A]  time = 0.45, size = 19, normalized size = 0.83 \begin {gather*} \frac {{\left (4 \, x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^4+1)^(3/4),x, algorithm="fricas")

[Out]

1/5*(4*x^4 - 1)*(x^4 + 1)^(1/4)/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^4+1)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((x^4 + 1)^(3/4)*x^6), x)

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maple [A]  time = 0.06, size = 20, normalized size = 0.87

method	result	size
gosper	\(\frac {\left (x^{4}+1\right )^{\frac {1}{4}} \left (4 x^{4}-1\right )}{5 x^{5}}\)	\(20\)
trager	\(\frac {\left (x^{4}+1\right )^{\frac {1}{4}} \left (4 x^{4}-1\right )}{5 x^{5}}\)	\(20\)
meijerg	\(-\frac {\left (-4 x^{4}+1\right ) \left (x^{4}+1\right )^{\frac {1}{4}}}{5 x^{5}}\)	\(20\)
risch	\(\frac {4 x^{8}+3 x^{4}-1}{5 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(x^4+1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

1/5*(x^4+1)^(1/4)*(4*x^4-1)/x^5

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maxima [A]  time = 0.38, size = 24, normalized size = 1.04 \begin {gather*} \frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} - \frac {{\left (x^{4} + 1\right )}^{\frac {5}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^4+1)^(3/4),x, algorithm="maxima")

[Out]

(x^4 + 1)^(1/4)/x - 1/5*(x^4 + 1)^(5/4)/x^5

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mupad [B]  time = 0.20, size = 25, normalized size = 1.09 \begin {gather*} -\frac {{\left (x^4+1\right )}^{1/4}-4\,x^4\,{\left (x^4+1\right )}^{1/4}}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(x^4 + 1)^(3/4)),x)

[Out]

-((x^4 + 1)^(1/4) - 4*x^4*(x^4 + 1)^(1/4))/(5*x^5)

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sympy [B]  time = 0.77, size = 48, normalized size = 2.09 \begin {gather*} \frac {\sqrt [4]{x^{4} + 1} \Gamma \left (- \frac {5}{4}\right )}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {\sqrt [4]{x^{4} + 1} \Gamma \left (- \frac {5}{4}\right )}{16 x^{5} \Gamma \left (\frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(x**4+1)**(3/4),x)

[Out]

(x**4 + 1)**(1/4)*gamma(-5/4)/(4*x*gamma(3/4)) - (x**4 + 1)**(1/4)*gamma(-5/4)/(16*x**5*gamma(3/4))

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