3.4.86 \(\int \frac {\sqrt {1+x^6}}{x^7} \, dx\)

Optimal. Leaf size=31 \[ -\frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 47, 63, 207} \begin {gather*} -\frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^6]/x^7,x]

[Out]

-1/6*Sqrt[1 + x^6]/x^6 - ArcTanh[Sqrt[1 + x^6]]/6

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 266

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x^6}}{x^7} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=-\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 1.42 \begin {gather*} -\frac {1}{6 x^6 \sqrt {x^6+1}}-\frac {1}{6 \sqrt {x^6+1}}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^6]/x^7,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{6} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 + x^6]/x^7,x]

[Out]

-1/6*Sqrt[1 + x^6]/x^6 - ArcTanh[Sqrt[1 + x^6]]/6

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fricas [A]  time = 0.46, size = 44, normalized size = 1.42 \begin {gather*} -\frac {x^{6} \log \left (\sqrt {x^{6} + 1} + 1\right ) - x^{6} \log \left (\sqrt {x^{6} + 1} - 1\right ) + 2 \, \sqrt {x^{6} + 1}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^(1/2)/x^7,x, algorithm="fricas")

[Out]

-1/12*(x^6*log(sqrt(x^6 + 1) + 1) - x^6*log(sqrt(x^6 + 1) - 1) + 2*sqrt(x^6 + 1))/x^6

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giac [A]  time = 0.23, size = 37, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^(1/2)/x^7,x, algorithm="giac")

[Out]

-1/6*sqrt(x^6 + 1)/x^6 - 1/12*log(sqrt(x^6 + 1) + 1) + 1/12*log(sqrt(x^6 + 1) - 1)

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maple [A]  time = 0.22, size = 30, normalized size = 0.97

method	result	size
trager	\(-\frac {\sqrt {x^{6}+1}}{6 x^{6}}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{x^{3}}\right )}{6}\)	\(30\)
risch	\(-\frac {\sqrt {x^{6}+1}}{6 x^{6}}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{6}\)	\(32\)
meijerg	\(-\frac {\frac {2 \sqrt {\pi }}{x^{6}}-\left (-2 \ln \relax (2)-1+6 \ln \relax (x )\right ) \sqrt {\pi }-\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right )}{4 x^{6}}+\frac {2 \sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{6}}+2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{12 \sqrt {\pi }}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.58, size = 37, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{12} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)^(1/2)/x^7,x, algorithm="maxima")

[Out]

-1/6*sqrt(x^6 + 1)/x^6 - 1/12*log(sqrt(x^6 + 1) + 1) + 1/12*log(sqrt(x^6 + 1) - 1)

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mupad [B]  time = 0.33, size = 23, normalized size = 0.74 \begin {gather*} -\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{6}-\frac {\sqrt {x^6+1}}{6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 + 1)^(1/2)/x^7,x)

[Out]

- atanh((x^6 + 1)^(1/2))/6 - (x^6 + 1)^(1/2)/(6*x^6)

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sympy [A]  time = 1.35, size = 24, normalized size = 0.77 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{3}} \right )}}{6} - \frac {\sqrt {1 + \frac {1}{x^{6}}}}{6 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+1)**(1/2)/x**7,x)

[Out]

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

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