3.10.41 \(\int \frac {1}{x^5 (1+x^4)^{3/2}} \, dx\) [941]

Optimal. Leaf size=44 \[ -\frac {3}{4 \sqrt {1+x^4}}-\frac {1}{4 x^4 \sqrt {1+x^4}}+\frac {3}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \]

[Out]

3/4*arctanh((x^4+1)^(1/2))-3/4/(x^4+1)^(1/2)-1/4/x^4/(x^4+1)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 44, 53, 65, 213} \begin {gather*} -\frac {1}{4 x^4 \sqrt {x^4+1}}-\frac {3}{4 \sqrt {x^4+1}}+\frac {3}{4} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 + x^4)^(3/2)),x]

[Out]

-3/(4*Sqrt[1 + x^4]) - 1/(4*x^4*Sqrt[1 + x^4]) + (3*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 53

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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*} \int \frac {1}{x^5 \left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 (1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{4 x^4}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{4 x^4}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{4 x^4}+\frac {3}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 38, normalized size = 0.86 \begin {gather*} \frac {-1-3 x^4}{4 x^4 \sqrt {1+x^4}}+\frac {3}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(1 + x^4)^(3/2)),x]

[Out]

(-1 - 3*x^4)/(4*x^4*Sqrt[1 + x^4]) + (3*ArcTanh[Sqrt[1 + x^4]])/4

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Maple [A]
time = 0.19, size = 33, normalized size = 0.75

method	result	size
risch	\(-\frac {3 x^{4}+1}{4 x^{4} \sqrt {x^{4}+1}}+\frac {3 \arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4}\)	\(31\)
default	\(-\frac {1}{4 x^{4} \sqrt {x^{4}+1}}-\frac {3}{4 \sqrt {x^{4}+1}}+\frac {3 \arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4}\)	\(33\)
elliptic	\(-\frac {1}{4 x^{4} \sqrt {x^{4}+1}}-\frac {3}{4 \sqrt {x^{4}+1}}+\frac {3 \arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4}\)	\(33\)
trager	\(-\frac {3 x^{4}+1}{4 x^{4} \sqrt {x^{4}+1}}-\frac {3 \ln \left (\frac {-1+\sqrt {x^{4}+1}}{x^{2}}\right )}{4}\)	\(37\)
meijerg	\(\frac {\frac {\sqrt {\pi }\, \left (20 x^{4}+8\right )}{16 x^{4}}-\frac {\sqrt {\pi }\, \left (24 x^{4}+8\right )}{16 x^{4} \sqrt {x^{4}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+4 \ln \left (x \right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }}{2 x^{4}}}{2 \sqrt {\pi }}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/4/x^4/(x^4+1)^(1/2)-3/4/(x^4+1)^(1/2)+3/4*arctanh(1/(x^4+1)^(1/2))

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Maxima [A]
time = 0.29, size = 53, normalized size = 1.20 \begin {gather*} -\frac {3 \, x^{4} + 1}{4 \, {\left ({\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1}\right )}} + \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) - \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

-1/4*(3*x^4 + 1)/((x^4 + 1)^(3/2) - sqrt(x^4 + 1)) + 3/8*log(sqrt(x^4 + 1) + 1) - 3/8*log(sqrt(x^4 + 1) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
time = 0.36, size = 66, normalized size = 1.50 \begin {gather*} \frac {3 \, {\left (x^{8} + x^{4}\right )} \log \left (\sqrt {x^{4} + 1} + 1\right ) - 3 \, {\left (x^{8} + x^{4}\right )} \log \left (\sqrt {x^{4} + 1} - 1\right ) - 2 \, {\left (3 \, x^{4} + 1\right )} \sqrt {x^{4} + 1}}{8 \, {\left (x^{8} + x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/8*(3*(x^8 + x^4)*log(sqrt(x^4 + 1) + 1) - 3*(x^8 + x^4)*log(sqrt(x^4 + 1) - 1) - 2*(3*x^4 + 1)*sqrt(x^4 + 1)
)/(x^8 + x^4)

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Sympy [A]
time = 1.47, size = 42, normalized size = 0.95 \begin {gather*} \frac {3 \operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{4} - \frac {3}{4 x^{2} \sqrt {1 + \frac {1}{x^{4}}}} - \frac {1}{4 x^{6} \sqrt {1 + \frac {1}{x^{4}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(x**4+1)**(3/2),x)

[Out]

3*asinh(x**(-2))/4 - 3/(4*x**2*sqrt(1 + x**(-4))) - 1/(4*x**6*sqrt(1 + x**(-4)))

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Giac [A]
time = 1.51, size = 53, normalized size = 1.20 \begin {gather*} -\frac {3 \, x^{4} + 1}{4 \, {\left ({\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1}\right )}} + \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) - \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

-1/4*(3*x^4 + 1)/((x^4 + 1)^(3/2) - sqrt(x^4 + 1)) + 3/8*log(sqrt(x^4 + 1) + 1) - 3/8*log(sqrt(x^4 + 1) - 1)

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Mupad [B]
time = 1.24, size = 32, normalized size = 0.73 \begin {gather*} \frac {3\,\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{4}-\frac {3}{4\,\sqrt {x^4+1}}-\frac {1}{4\,x^4\,\sqrt {x^4+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(x^4 + 1)^(3/2)),x)

[Out]

(3*atanh((x^4 + 1)^(1/2)))/4 - 3/(4*(x^4 + 1)^(1/2)) - 1/(4*x^4*(x^4 + 1)^(1/2))

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