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

Optimal. Leaf size=31 \[ \frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {446, 78, 63, 207} \begin {gather*} \frac {\sqrt {x^6+1}}{6 x^6}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^6)/(x^7*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/(6*x^6) - ArcTanh[Sqrt[1 + x^6]]/2

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {-1+x^6}{x^7 \sqrt {1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+x}{x^2 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{6 x^6}+\frac {1}{4} \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}{2} \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}{2} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {x^6+1}}{x^6}-3 \tanh ^{-1}\left (\sqrt {x^6+1}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^6)/(x^7*Sqrt[1 + x^6]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 31, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^6}}{6 x^6}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^6)/(x^7*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/(6*x^6) - ArcTanh[Sqrt[1 + x^6]]/2

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fricas [A]  time = 0.48, size = 45, normalized size = 1.45 \begin {gather*} -\frac {3 \, x^{6} \log \left (\sqrt {x^{6} + 1} + 1\right ) - 3 \, 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)/x^7/(x^6+1)^(1/2),x, algorithm="fricas")

[Out]

-1/12*(3*x^6*log(sqrt(x^6 + 1) + 1) - 3*x^6*log(sqrt(x^6 + 1) - 1) - 2*sqrt(x^6 + 1))/x^6

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giac [A]  time = 0.32, size = 37, normalized size = 1.19 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{4} \, \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)/x^7/(x^6+1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.24, size = 30, normalized size = 0.97

method	result	size
trager	\(\frac {\sqrt {x^{6}+1}}{6 x^{6}}-\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{2}\)	\(30\)
risch	\(\frac {\sqrt {x^{6}+1}}{6 x^{6}}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{2}\)	\(32\)
meijerg	\(\frac {\left (-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }-2 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{6 \sqrt {\pi }}-\frac {-\frac {\sqrt {\pi }}{x^{6}}-\frac {\left (1-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right )}{8 x^{6}}-\frac {\sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{6}}+\ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{6 \sqrt {\pi }}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 37, normalized size = 1.19 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} - \frac {1}{4} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {1}{4} \, \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)/x^7/(x^6+1)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.52, size = 23, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x^6+1}}{6\,x^6}-\frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)/(x^7*(x^6 + 1)^(1/2)),x)

[Out]

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

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sympy [B]  time = 98.97, size = 56, normalized size = 1.81 \begin {gather*} \frac {\log {\left (-1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{4} - \frac {\log {\left (1 + \frac {1}{\sqrt {x^{6} + 1}} \right )}}{4} - \frac {1}{12 \left (1 + \frac {1}{\sqrt {x^{6} + 1}}\right )} - \frac {1}{12 \left (-1 + \frac {1}{\sqrt {x^{6} + 1}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-1 + 1/sqrt(x**6 + 1))/4 - log(1 + 1/sqrt(x**6 + 1))/4 - 1/(12*(1 + 1/sqrt(x**6 + 1))) - 1/(12*(-1 + 1/sqr
t(x**6 + 1)))

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