3.1209 \(\int \frac{1}{x \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=57 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0347115, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {266, 63, 298, 203, 206} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a - b*x^4)^(1/4)),x]

[Out]

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

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]]

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 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.009278, size = 50, normalized size = 0.88 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a - b*x^4)^(1/4)),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x/(-b*x^4+a)^(1/4),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.58053, size = 248, normalized size = 4.35 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{\sqrt{-b x^{4} + a} + \sqrt{a}}}{a^{\frac{1}{4}}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{a^{\frac{1}{4}}}\right )}{a^{\frac{1}{4}}} - \frac{\log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} + a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} + \frac{\log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} - a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 1.41023, size = 39, normalized size = 0.68 \begin{align*} - \frac{e^{- \frac{i \pi }{4}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(-I*pi/4)*gamma(1/4)*hyper((1/4, 1/4), (5/4,), a/(b*x**4))/(4*b**(1/4)*x*gamma(5/4))

________________________________________________________________________________________

Giac [B]  time = 1.15139, size = 259, normalized size = 4.54 \begin{align*} -\frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} + \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a - 1/4*sqr
t(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4))/a + 1/8*sqrt(2)*(-
a)^(3/4)*log(sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a - 1/8*sqrt(2)*(-a)^(3/4)*l
og(-sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a