\(\int \frac {1}{x^5 \sqrt {1-x^4}} \, dx\) [878]

Optimal result

Integrand size = 15, antiderivative size = 35 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^4}\right ) \]

[Out]

-1/4*arctanh((-x^4+1)^(1/2))-1/4*(-x^4+1)^(1/2)/x^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 44, 65, 212} \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^4}\right )-\frac {\sqrt {1-x^4}}{4 x^4} \]

[In]

Int[1/(x^5*Sqrt[1 - x^4]),x]

[Out]

-1/4*Sqrt[1 - x^4]/x^4 - ArcTanh[Sqrt[1 - x^4]]/4

Rule 44

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

Rule 65

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 212

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

Rule 272

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*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^2} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt {1-x^4}}{4 x^4}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^4}\right ) \\ & = -\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {1-x^4}}{4 x^4}-\frac {1}{4} \text {arctanh}\left (\sqrt {1-x^4}\right ) \]

[In]

Integrate[1/(x^5*Sqrt[1 - x^4]),x]

[Out]

-1/4*Sqrt[1 - x^4]/x^4 - ArcTanh[Sqrt[1 - x^4]]/4

Maple [A] (verified)

Time = 4.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {\sqrt {-x^{4}+1}}{4 x^{4}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{4}\)	\(28\)
elliptic	\(-\frac {\sqrt {-x^{4}+1}}{4 x^{4}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{4}\)	\(28\)
risch	\(\frac {x^{4}-1}{4 x^{4} \sqrt {-x^{4}+1}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{4}\)	\(33\)
pseudoelliptic	\(\frac {-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{4}+1}}\right ) x^{4}-\sqrt {-x^{4}+1}}{4 x^{4}}\)	\(33\)
trager	\(-\frac {\sqrt {-x^{4}+1}}{4 x^{4}}+\frac {\ln \left (\frac {-1+\sqrt {-x^{4}+1}}{x^{2}}\right )}{4}\)	\(34\)
meijerg	\(-\frac {\frac {\sqrt {\pi }}{x^{4}}-\frac {\left (1-2 \ln \left (2\right )+4 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 x^{4}+8\right )}{8 x^{4}}+\frac {\sqrt {\pi }\, \sqrt {-x^{4}+1}}{x^{4}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{4}+1}}{2}\right )}{4 \sqrt {\pi }}\)	\(82\)

[In]

int(1/x^5/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*(-x^4+1)^(1/2)/x^4-1/4*arctanh(1/(-x^4+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {x^{4} \log \left (\sqrt {-x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt {-x^{4} + 1} - 1\right ) + 2 \, \sqrt {-x^{4} + 1}}{8 \, x^{4}} \]

[In]

integrate(1/x^5/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/8*(x^4*log(sqrt(-x^4 + 1) + 1) - x^4*log(sqrt(-x^4 + 1) - 1) + 2*sqrt(-x^4 + 1))/x^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.09 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x^{2}} \right )}}{4} + \frac {1}{4 x^{2} \sqrt {-1 + \frac {1}{x^{4}}}} - \frac {1}{4 x^{6} \sqrt {-1 + \frac {1}{x^{4}}}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{4} - \frac {i \sqrt {1 - \frac {1}{x^{4}}}}{4 x^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**5/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-acosh(x**(-2))/4 + 1/(4*x**2*sqrt(-1 + x**(-4))) - 1/(4*x**6*sqrt(-1 + x**(-4))), 1/Abs(x**4) > 1)
, (I*asin(x**(-2))/4 - I*sqrt(1 - 1/x**4)/(4*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {-x^{4} + 1}}{4 \, x^{4}} - \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \]

[In]

integrate(1/x^5/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(-x^4 + 1)/x^4 - 1/8*log(sqrt(-x^4 + 1) + 1) + 1/8*log(sqrt(-x^4 + 1) - 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\sqrt {-x^{4} + 1}}{4 \, x^{4}} - \frac {1}{8} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{8} \, \log \left (-\sqrt {-x^{4} + 1} + 1\right ) \]

[In]

integrate(1/x^5/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

-1/4*sqrt(-x^4 + 1)/x^4 - 1/8*log(sqrt(-x^4 + 1) + 1) + 1/8*log(-sqrt(-x^4 + 1) + 1)

Mupad [B] (verification not implemented)

Time = 5.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^5 \sqrt {1-x^4}} \, dx=-\frac {\mathrm {atanh}\left (\sqrt {1-x^4}\right )}{4}-\frac {\sqrt {1-x^4}}{4\,x^4} \]

[In]

int(1/(x^5*(1 - x^4)^(1/2)),x)

[Out]

- atanh((1 - x^4)^(1/2))/4 - (1 - x^4)^(1/2)/(4*x^4)