3.5.32 \(\int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx\)

Optimal. Leaf size=35 \[ \frac {\sqrt {x^4+1}}{2 x^2}+\frac {1}{2} \log \left (\sqrt {x^4+1}+x^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {451, 275, 215} \begin {gather*} \frac {1}{2} \sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^4)/(x^3*Sqrt[1 + x^4]),x]

[Out]

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 451

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.63 \begin {gather*} \frac {1}{2} \left (\sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1}}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^4)/(x^3*Sqrt[1 + x^4]),x]

[Out]

(Sqrt[1 + x^4]/x^2 + ArcSinh[x^2])/2

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IntegrateAlgebraic [A]  time = 0.06, size = 37, normalized size = 1.06 \begin {gather*} \frac {\sqrt {1+x^4}}{2 x^2}-\frac {1}{2} \log \left (-x^2+\sqrt {1+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 38, normalized size = 1.09 \begin {gather*} -\frac {x^{2} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) - x^{2} - \sqrt {x^{4} + 1}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.31, size = 38, normalized size = 1.09 \begin {gather*} -\frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.18, size = 20, normalized size = 0.57

method	result	size
default	\(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
meijerg	\(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
risch	\(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
elliptic	\(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\)	\(20\)
trager	\(\frac {\sqrt {x^{4}+1}}{2 x^{2}}-\frac {\ln \left (x^{2}-\sqrt {x^{4}+1}\right )}{2}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 45, normalized size = 1.29 \begin {gather*} \frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.39, size = 19, normalized size = 0.54 \begin {gather*} \frac {\mathrm {asinh}\left (x^2\right )}{2}+\frac {\sqrt {x^4+1}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - 1)/(x^3*(x^4 + 1)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.53, size = 19, normalized size = 0.54 \begin {gather*} \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} + \frac {\sqrt {x^{4} + 1}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x**2)/2 + sqrt(x**4 + 1)/(2*x**2)

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