3.4.14 \(\int \frac {1}{(-2+b x^2) \sqrt [4]{-1+b x^2}} \, dx\) [314]

Optimal. Leaf size=77 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1+b x^2}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1+b x^2}}\right )}{2 \sqrt {2} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {407} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{b x^2-1}}\right )}{2 \sqrt {2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-2 + b*x^2)*(-1 + b*x^2)^(1/4)),x]

[Out]

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

Rule 407

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*S
qrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2
]*(a + b*x^2)^(1/4)))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 67, normalized size = 0.87 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+b x^2}}{\sqrt {b} x}\right )-\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1+b x^2}}\right )}{2 \sqrt {2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((-2 + b*x^2)*(-1 + b*x^2)^(1/4)),x]

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{2}-2\right ) \left (b \,x^{2}-1\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2-2)/(b*x^2-1)^(1/4),x)

[Out]

int(1/(b*x^2-2)/(b*x^2-1)^(1/4),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2-2)/(b*x^2-1)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 - 1)^(1/4)*(b*x^2 - 2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (55) = 110\).
time = 7.38, size = 274, normalized size = 3.56 \begin {gather*} \left [\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}}}{\sqrt {b} x}\right ) + \sqrt {2} \sqrt {b} \log \left (-\frac {b^{2} x^{4} - 2 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} b^{\frac {3}{2}} x^{3} + 4 \, \sqrt {b x^{2} - 1} b x^{2} + 4 \, b x^{2} - 4 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{8 \, b}, \frac {2 \, \sqrt {2} \sqrt {-b} \arctan \left (\frac {\sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b}}{b x}\right ) - \sqrt {2} \sqrt {-b} \log \left (-\frac {b^{2} x^{4} + 2 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b} b x^{3} - 4 \, \sqrt {b x^{2} - 1} b x^{2} + 4 \, b x^{2} - 4 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {-b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{8 \, b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2-2)/(b*x^2-1)^(1/4),x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(2)*sqrt(b)*arctan(sqrt(2)*(b*x^2 - 1)^(1/4)/(sqrt(b)*x)) + sqrt(2)*sqrt(b)*log(-(b^2*x^4 - 2*sqrt
(2)*(b*x^2 - 1)^(1/4)*b^(3/2)*x^3 + 4*sqrt(b*x^2 - 1)*b*x^2 + 4*b*x^2 - 4*sqrt(2)*(b*x^2 - 1)^(3/4)*sqrt(b)*x
- 4)/(b^2*x^4 - 4*b*x^2 + 4)))/b, 1/8*(2*sqrt(2)*sqrt(-b)*arctan(sqrt(2)*(b*x^2 - 1)^(1/4)*sqrt(-b)/(b*x)) - s
qrt(2)*sqrt(-b)*log(-(b^2*x^4 + 2*sqrt(2)*(b*x^2 - 1)^(1/4)*sqrt(-b)*b*x^3 - 4*sqrt(b*x^2 - 1)*b*x^2 + 4*b*x^2
- 4*sqrt(2)*(b*x^2 - 1)^(3/4)*sqrt(-b)*x - 4)/(b^2*x^4 - 4*b*x^2 + 4)))/b]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b x^{2} - 2\right ) \sqrt [4]{b x^{2} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2-2)/(b*x**2-1)**(1/4),x)

[Out]

Integral(1/((b*x**2 - 2)*(b*x**2 - 1)**(1/4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2-2)/(b*x^2-1)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 - 1)^(1/4)*(b*x^2 - 2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,x^2-1\right )}^{1/4}\,\left (b\,x^2-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x^2 - 1)^(1/4)*(b*x^2 - 2)),x)

[Out]

int(1/((b*x^2 - 1)^(1/4)*(b*x^2 - 2)), x)

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