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

Optimal. Leaf size=87 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \]

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Rubi [A]  time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {377, 212, 206, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-b + a*x^4)^(1/4)*(b + a*x^4)),x]

[Out]

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

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])

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 70, normalized size = 0.80 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((-b + a*x^4)^(1/4)*(b + a*x^4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.32, size = 87, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{2} \sqrt [4]{a} b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((-b + a*x^4)^(1/4)*(b + a*x^4)),x]

[Out]

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

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fricas [B]  time = 172.65, size = 431, normalized size = 4.95 \begin {gather*} -\frac {1}{2} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x^{4} - b\right )}^{\frac {3}{4}} a b^{3} x \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b x^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + {\left (2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {a x^{4} - b} a b x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (3 \, a^{2} b^{3} x^{4} - a b^{4}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} b^{2} \sqrt {\frac {1}{a b^{4}}}}\right )}}{a x^{4} + b}\right ) + \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} + 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) - \frac {1}{8} \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} - b} a b^{3} x^{2} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} - 4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {1}{a b^{4}}} - 2 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a b x^{4} - b^{2}\right )} \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + b\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{4}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (a\,x^4+b\right )\,{\left (a\,x^4-b\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{a x^{4} - b} \left (a x^{4} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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