3.1180 \(\int \frac{\sqrt [4]{a-b x^4}}{x} \, dx\)

Optimal. Leaf size=69 \[ \sqrt [4]{a-b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \]

[Out]

(a - b*x^4)^(1/4) - (a^(1/4)*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

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Rubi [A]  time = 0.0432912, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {266, 50, 63, 212, 206, 203} \[ \sqrt [4]{a-b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^4)^(1/4)/x,x]

[Out]

(a - b*x^4)^(1/4) - (a^(1/4)*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

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 50

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 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 212

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

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])

Rubi steps

\begin{align*} \int \frac{\sqrt [4]{a-b x^4}}{x} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt [4]{a-b x}}{x} \, dx,x,x^4\right )\\ &=\sqrt [4]{a-b x^4}+\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{x (a-b x)^{3/4}} \, dx,x,x^4\right )\\ &=\sqrt [4]{a-b x^4}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}-\frac{x^4}{b}} \, dx,x,\sqrt [4]{a-b x^4}\right )}{b}\\ &=\sqrt [4]{a-b x^4}-\frac{1}{2} \sqrt{a} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )-\frac{1}{2} \sqrt{a} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )\\ &=\sqrt [4]{a-b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0143432, size = 69, normalized size = 1. \[ \sqrt [4]{a-b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^4)^(1/4)/x,x]

[Out]

(a - b*x^4)^(1/4) - (a^(1/4)*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

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt [4]{-b{x}^{4}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.63366, size = 281, normalized size = 4.07 \begin{align*} a^{\frac{1}{4}} \arctan \left (\frac{a^{\frac{3}{4}} \sqrt{\sqrt{-b x^{4} + a} + \sqrt{a}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{\frac{3}{4}}}{a}\right ) - \frac{1}{4} \, a^{\frac{1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} + a^{\frac{1}{4}}\right ) + \frac{1}{4} \, a^{\frac{1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} - a^{\frac{1}{4}}\right ) +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^(1/4)*arctan((a^(3/4)*sqrt(sqrt(-b*x^4 + a) + sqrt(a)) - (-b*x^4 + a)^(1/4)*a^(3/4))/a) - 1/4*a^(1/4)*log((-
b*x^4 + a)^(1/4) + a^(1/4)) + 1/4*a^(1/4)*log((-b*x^4 + a)^(1/4) - a^(1/4)) + (-b*x^4 + a)^(1/4)

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Sympy [C]  time = 1.09989, size = 44, normalized size = 0.64 \begin{align*} - \frac{\sqrt [4]{b} x e^{\frac{i \pi }{4}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**(1/4)*x*exp(I*pi/4)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), a/(b*x**4))/(4*gamma(3/4))

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Giac [B]  time = 1.17412, size = 257, normalized size = 3.72 \begin{align*} -\frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{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 ) - \frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{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 ) - \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{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 ) + \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{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 ) +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4)) - 1/4*sqrt(
2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4)) - 1/8*sqrt(2)*(-a)^(
1/4)*log(sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a)) + 1/8*sqrt(2)*(-a)^(1/4)*log(-sq
rt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a)) + (-b*x^4 + a)^(1/4)