3.4.15 \(\int \frac {1}{(-2-b x^2) \sqrt [4]{-1-b x^2}} \, dx\) [315]

Optimal. Leaf size=79 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{2 \sqrt {2} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{2 \sqrt {2} \sqrt {b}} \]

[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 = 79, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, 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 = 69, 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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (57) = 114\).
time = 6.82, size = 273, normalized size = 3.46 \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} + 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} + 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {b} - 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} - 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} + 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} - 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {-b} - 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 + 4*sqr
t(-b*x^2 - 1)*b*x^2 - 4*b*x^2 - 2*sqrt(2)*((-b*x^2 - 1)^(1/4)*b*x^3 + 2*(-b*x^2 - 1)^(3/4)*x)*sqrt(b) - 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)) - sqrt(2)*
sqrt(-b)*log(-(b^2*x^4 - 4*sqrt(-b*x^2 - 1)*b*x^2 - 4*b*x^2 + 2*sqrt(2)*((-b*x^2 - 1)^(1/4)*b*x^3 - 2*(-b*x^2
- 1)^(3/4)*x)*sqrt(-b) - 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}{b x^{2} \sqrt [4]{- b x^{2} - 1} + 2 \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*(-b*x**2 - 1)**(1/4) + 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 + 2)*(-b*x^2 - 1)^(1/4)), 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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