3.12.80 \(\int \frac {\sqrt [4]{a-b x^4}}{x} \, dx\) [1180]

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]

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

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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {272, 52, 65, 218, 212, 209} \begin {gather*} -\frac {1}{2} \sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )+\sqrt [4]{a-b x^4}-\frac {1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \end {gather*}

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 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 209

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

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 218

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] &&  !Gt
Q[a/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 {\sqrt [4]{a-b x^4}}{x} \, dx &=\frac {1}{4} \text {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 \text {Subst}\left (\int \frac {1}{x (a-b x)^{3/4}} \, dx,x,x^4\right )\\ &=\sqrt [4]{a-b x^4}-\frac {a \text {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} \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )-\frac {1}{2} \sqrt {a} \text {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*}

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Mathematica [A]
time = 0.05, size = 69, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{x}\, dx\]

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 [A]
time = 0.50, size = 70, normalized size = 1.01 \begin {gather*} -\frac {1}{2} \, a^{\frac {1}{4}} \arctan \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) + \frac {1}{4} \, a^{\frac {1}{4}} \log \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right ) + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 98, normalized size = 1.42 \begin {gather*} 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 {gather*}

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] Result contains complex when optimal does not.
time = 0.54, size = 44, normalized size = 0.64 \begin {gather*} - \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (51) = 102\).
time = 1.04, size = 190, normalized size = 2.75 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 1.21, size = 51, normalized size = 0.74 \begin {gather*} {\left (a-b\,x^4\right )}^{1/4}-\frac {a^{1/4}\,\mathrm {atanh}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{2}-\frac {a^{1/4}\,\mathrm {atan}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^4)^(1/4)/x,x)

[Out]

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

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