\(\int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx\) [2060]

Optimal result

Integrand size = 15, antiderivative size = 43 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{2} \sqrt {a+\frac {b}{x^4}}+\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, 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, 52, 65, 214} \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )-\frac {1}{2} \sqrt {a+\frac {b}{x^4}} \]

[In]

Int[Sqrt[a + b/x^4]/x,x]

[Out]

-1/2*Sqrt[a + b/x^4] + (Sqrt[a]*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/2

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 214

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

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{2} \sqrt {a+\frac {b}{x^4}}+\frac {\sqrt {a} \sqrt {a+\frac {b}{x^4}} x^2 \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{2 \sqrt {b+a x^4}} \]

[In]

Integrate[Sqrt[a + b/x^4]/x,x]

[Out]

-1/2*Sqrt[a + b/x^4] + (Sqrt[a]*Sqrt[a + b/x^4]*x^2*Log[Sqrt[a]*x^2 + Sqrt[b + a*x^4]])/(2*Sqrt[b + a*x^4])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(31)=62\).

Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.51

method	result	size
risch	\(-\frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{2}+\frac {\sqrt {a}\, \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{2 \sqrt {a \,x^{4}+b}}\)	\(65\)
default	\(\frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \left (a \,x^{4} \sqrt {a \,x^{4}+b}+\sqrt {a}\, \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) b \,x^{2}-\left (a \,x^{4}+b \right )^{\frac {3}{2}}\right )}{2 \sqrt {a \,x^{4}+b}\, b}\)	\(80\)

[In]

int((a+b/x^4)^(1/2)/x,x,method=_RETURNVERBOSE)

[Out]

-1/2*((a*x^4+b)/x^4)^(1/2)+1/2*a^(1/2)*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*((a*x^4+b)/x^4)^(1/2)*x^2/(a*x^4+b)^(1/
2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\left [\frac {1}{4} \, \sqrt {a} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right ) - \frac {1}{2} \, \sqrt {\frac {a x^{4} + b}{x^{4}}}, -\frac {1}{2} \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) - \frac {1}{2} \, \sqrt {\frac {a x^{4} + b}{x^{4}}}\right ] \]

[In]

integrate((a+b/x^4)^(1/2)/x,x, algorithm="fricas")

[Out]

[1/4*sqrt(a)*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) - 1/2*sqrt((a*x^4 + b)/x^4), -1/2*sqrt(-a
)*arctan(sqrt(-a)*x^4*sqrt((a*x^4 + b)/x^4)/(a*x^4 + b)) - 1/2*sqrt((a*x^4 + b)/x^4)]

Sympy [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\frac {\sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{2} - \frac {a x^{2}}{2 \sqrt {b} \sqrt {\frac {a x^{4}}{b} + 1}} - \frac {\sqrt {b}}{2 x^{2} \sqrt {\frac {a x^{4}}{b} + 1}} \]

[In]

integrate((a+b/x**4)**(1/2)/x,x)

[Out]

sqrt(a)*asinh(sqrt(a)*x**2/sqrt(b))/2 - a*x**2/(2*sqrt(b)*sqrt(a*x**4/b + 1)) - sqrt(b)/(2*x**2*sqrt(a*x**4/b
+ 1))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{4} \, \sqrt {a} \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right ) - \frac {1}{2} \, \sqrt {a + \frac {b}{x^{4}}} \]

[In]

integrate((a+b/x^4)^(1/2)/x,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=-\frac {1}{4} \, \sqrt {a} \log \left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2}\right ) + \frac {\sqrt {a} b}{{\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b}\right )}^{2} - b} \]

[In]

integrate((a+b/x^4)^(1/2)/x,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a+\frac {b}{x^4}}}{x} \, dx=\frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2}-\frac {\sqrt {a+\frac {b}{x^4}}}{2} \]

[In]

int((a + b/x^4)^(1/2)/x,x)

[Out]

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