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

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

[Out]

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

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Rubi [A]  time = 0.0137537, antiderivative size = 79, normalized size of antiderivative = 1., 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 = {398} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{-b x^2-1}}\right )}{2 \sqrt{2} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 398

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b^2/a), 4]}, Simp[(b*Ar
cTan[(q*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(2*Sqrt[2]*a*d*q), x] + Simp[(b*ArcTanh[(q*x)/(Sqrt[2]*(a + b*x^2)^(1
/4))])/(2*Sqrt[2]*a*d*q), 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*}

Mathematica [C]  time = 0.142623, size = 137, normalized size = 1.73 \[ \frac{6 x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-b x^2,-\frac{b x^2}{2}\right )}{\sqrt [4]{-b x^2-1} \left (b x^2+2\right ) \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-b x^2,-\frac{b x^2}{2}\right )\right )-6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-b x^2,-\frac{b x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*x*AppellF1[1/2, 1/4, 1, 3/2, -(b*x^2), -(b*x^2)/2])/((-1 - b*x^2)^(1/4)*(2 + b*x^2)*(-6*AppellF1[1/2, 1/4,
1, 3/2, -(b*x^2), -(b*x^2)/2] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, -(b*x^2), -(b*x^2)/2] + AppellF1[3/2, 5/4,
1, 5/2, -(b*x^2), -(b*x^2)/2])))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-b{x}^{2}-2}{\frac{1}{\sqrt [4]{-b{x}^{2}-1}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (b x^{2} + 2\right )}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

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]  time = 85.1679, size = 675, normalized size = 8.54 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{b x^{2} \sqrt [4]{- b x^{2} - 1} + 2 \sqrt [4]{- b x^{2} - 1}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{2} + 2\right )}{\left (-b x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

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)