3.10.45 \(\int \frac {1}{x^3 (1+x^4)^{3/2}} \, dx\) [945]

Optimal. Leaf size=31 \[ -\frac {1}{2 x^2 \sqrt {1+x^4}}-\frac {x^2}{\sqrt {1+x^4}} \]

[Out]

-1/2/x^2/(x^4+1)^(1/2)-x^2/(x^4+1)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} -\frac {x^2}{\sqrt {x^4+1}}-\frac {1}{2 \sqrt {x^4+1} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 + x^4)^(3/2)),x]

[Out]

-1/2*1/(x^2*Sqrt[1 + x^4]) - x^2/Sqrt[1 + x^4]

Rule 270

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 277

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

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Mathematica [A]
time = 0.11, size = 23, normalized size = 0.74 \begin {gather*} \frac {-1-2 x^4}{2 x^2 \sqrt {1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(1 + x^4)^(3/2)),x]

[Out]

(-1 - 2*x^4)/(2*x^2*Sqrt[1 + x^4])

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Maple [A]
time = 0.14, size = 20, normalized size = 0.65

method	result	size
gosper	\(-\frac {2 x^{4}+1}{2 x^{2} \sqrt {x^{4}+1}}\)	\(20\)
default	\(-\frac {2 x^{4}+1}{2 x^{2} \sqrt {x^{4}+1}}\)	\(20\)
trager	\(-\frac {2 x^{4}+1}{2 x^{2} \sqrt {x^{4}+1}}\)	\(20\)
meijerg	\(-\frac {2 x^{4}+1}{2 x^{2} \sqrt {x^{4}+1}}\)	\(20\)
risch	\(-\frac {2 x^{4}+1}{2 x^{2} \sqrt {x^{4}+1}}\)	\(20\)
elliptic	\(-\frac {2 x^{4}+1}{2 x^{2} \sqrt {x^{4}+1}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(2*x^4+1)/x^2/(x^4+1)^(1/2)

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Maxima [A]
time = 0.29, size = 25, normalized size = 0.81 \begin {gather*} -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} - \frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

-1/2*x^2/sqrt(x^4 + 1) - 1/2*sqrt(x^4 + 1)/x^2

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Fricas [A]
time = 0.35, size = 37, normalized size = 1.19 \begin {gather*} -\frac {2 \, x^{6} + 2 \, x^{2} + {\left (2 \, x^{4} + 1\right )} \sqrt {x^{4} + 1}}{2 \, {\left (x^{6} + x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(2*x^6 + 2*x^2 + (2*x^4 + 1)*sqrt(x^4 + 1))/(x^6 + x^2)

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Sympy [A]
time = 0.38, size = 42, normalized size = 1.35 \begin {gather*} - \frac {2 x^{4} \sqrt {x^{4} + 1}}{2 x^{6} + 2 x^{2}} - \frac {\sqrt {x^{4} + 1}}{2 x^{6} + 2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(x**4+1)**(3/2),x)

[Out]

-2*x**4*sqrt(x**4 + 1)/(2*x**6 + 2*x**2) - sqrt(x**4 + 1)/(2*x**6 + 2*x**2)

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Giac [A]
time = 1.80, size = 32, normalized size = 1.03 \begin {gather*} -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} + \frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

-1/2*x^2/sqrt(x^4 + 1) + 1/((x^2 - sqrt(x^4 + 1))^2 - 1)

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Mupad [B]
time = 1.14, size = 17, normalized size = 0.55 \begin {gather*} -\frac {x^4+\frac {1}{2}}{x^2\,\sqrt {x^4+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(x^4 + 1)^(3/2)),x)

[Out]

-(x^4 + 1/2)/(x^2*(x^4 + 1)^(1/2))

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