3.5.43 \(\int \frac {x^8}{\sqrt {1+x^6}} \, dx\)

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

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 321, 215} \begin {gather*} \frac {1}{6} x^3 \sqrt {x^6+1}-\frac {1}{6} \sinh ^{-1}\left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/Sqrt[1 + x^6],x]

[Out]

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

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 321

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

Rubi steps

\begin {align*} \int \frac {x^8}{\sqrt {1+x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {1+x^6}-\frac {1}{6} \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}{6} x^3 \sqrt {x^6+1}-\frac {1}{6} \sinh ^{-1}\left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/Sqrt[1 + x^6],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.14, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{6} x^3 \sqrt {1+x^6}-\frac {1}{6} \log \left (x^3+\sqrt {1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^8/Sqrt[1 + x^6],x]

[Out]

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

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fricas [A]  time = 0.55, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{6} \, \sqrt {x^{6} + 1} x^{3} + \frac {1}{6} \, \log \left (-x^{3} + \sqrt {x^{6} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^6+1)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\sqrt {x^{6} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^6+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(x^6 + 1), x)

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maple [A]  time = 0.16, size = 20, normalized size = 0.57

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^6+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.34, size = 58, normalized size = 1.66 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{6 \, x^{3} {\left (\frac {x^{6} + 1}{x^{6}} - 1\right )}} - \frac {1}{12} \, \log \left (\frac {\sqrt {x^{6} + 1}}{x^{3}} + 1\right ) + \frac {1}{12} \, \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^8/(x^6+1)^(1/2),x, algorithm="maxima")

[Out]

1/6*sqrt(x^6 + 1)/(x^3*((x^6 + 1)/x^6 - 1)) - 1/12*log(sqrt(x^6 + 1)/x^3 + 1) + 1/12*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 {x^8}{\sqrt {x^6+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^6 + 1)^(1/2),x)

[Out]

int(x^8/(x^6 + 1)^(1/2), x)

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sympy [A]  time = 1.52, size = 19, normalized size = 0.54 \begin {gather*} \frac {x^{3} \sqrt {x^{6} + 1}}{6} - \frac {\operatorname {asinh}{\left (x^{3} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**6+1)**(1/2),x)

[Out]

x**3*sqrt(x**6 + 1)/6 - asinh(x**3)/6

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