3.5.44 \(\int \frac {\sqrt {1+x^6}}{x^4} \, dx\)

Optimal. Leaf size=35 \[ \frac {1}{3} \log \left (\sqrt {x^6+1}+x^3\right )-\frac {\sqrt {x^6+1}}{3 x^3} \]

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 277, 215} \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right )-\frac {\sqrt {x^6+1}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^6]/x^4,x]

[Out]

-1/3*Sqrt[1 + x^6]/x^3 + ArcSinh[x^3]/3

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 277

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x^6}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \sinh ^{-1}\left (x^3\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.71 \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right )-\frac {\sqrt {x^6+1}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^6]/x^4,x]

[Out]

-1/3*Sqrt[1 + x^6]/x^3 + ArcSinh[x^3]/3

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IntegrateAlgebraic [A]  time = 0.12, size = 35, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^6}}{3 x^3}+\frac {1}{3} \log \left (x^3+\sqrt {1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 + x^6]/x^4,x]

[Out]

-1/3*Sqrt[1 + x^6]/x^3 + Log[x^3 + Sqrt[1 + x^6]]/3

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fricas [A]  time = 0.57, size = 34, normalized size = 0.97 \begin {gather*} -\frac {x^{3} \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) + x^{3} + \sqrt {x^{6} + 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-1/3*(x^3*log(-x^3 + sqrt(x^6 + 1)) + x^3 + sqrt(x^6 + 1))/x^3

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giac [A]  time = 0.31, size = 38, normalized size = 1.09 \begin {gather*} -\frac {2 \, \sqrt {\frac {1}{x^{6}} + 1} - \log \left (\sqrt {\frac {1}{x^{6}} + 1} + 1\right ) + \log \left (\sqrt {\frac {1}{x^{6}} + 1} - 1\right )}{6 \, \mathrm {sgn}\relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^(1/2)/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.20, size = 20, normalized size = 0.57

method	result	size
risch	\(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}+\frac {\arcsinh \left (x^{3}\right )}{3}\)	\(20\)
trager	\(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}+1}\right )}{3}\)	\(30\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{3}}-4 \sqrt {\pi }\, \arcsinh \left (x^{3}\right )}{12 \sqrt {\pi }}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*(x^6+1)^(1/2)/x^3+1/3*arcsinh(x^3)

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maxima [A]  time = 0.37, size = 45, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {x^{6} + 1}}{3 \, x^{3}} + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^(1/2)/x^4,x, algorithm="maxima")

[Out]

-1/3*sqrt(x^6 + 1)/x^3 + 1/6*log(sqrt(x^6 + 1)/x^3 + 1) - 1/6*log(sqrt(x^6 + 1)/x^3 - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {x^6+1}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 + 1)^(1/2)/x^4,x)

[Out]

int((x^6 + 1)^(1/2)/x^4, x)

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sympy [A]  time = 0.98, size = 34, normalized size = 0.97 \begin {gather*} - \frac {x^{3}}{3 \sqrt {x^{6} + 1}} + \frac {\operatorname {asinh}{\left (x^{3} \right )}}{3} - \frac {1}{3 x^{3} \sqrt {x^{6} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+1)**(1/2)/x**4,x)

[Out]

-x**3/(3*sqrt(x**6 + 1)) + asinh(x**3)/3 - 1/(3*x**3*sqrt(x**6 + 1))

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