3.863 \(\int \frac{1}{\left (a+b x^4\right ) \sqrt{-c x^2+d \sqrt{a+b x^4}}} \, dx\)

Optimal. Leaf size=41 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}-c x^2}}\right )}{a \sqrt{c}} \]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[-(c*x^2) + d*Sqrt[a + b*x^4]]]/(a*Sqrt[c])

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Rubi [A]  time = 0.239726, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}-c x^2}}\right )}{a \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^4)*Sqrt[-(c*x^2) + d*Sqrt[a + b*x^4]]),x]

[Out]

ArcTan[(Sqrt[c]*x)/Sqrt[-(c*x^2) + d*Sqrt[a + b*x^4]]]/(a*Sqrt[c])

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Rubi in Sympy [A]  time = 6.46754, size = 34, normalized size = 0.83 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{- c x^{2} + d \sqrt{a + b x^{4}}}} \right )}}{a \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)/(-c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)

[Out]

atan(sqrt(c)*x/sqrt(-c*x**2 + d*sqrt(a + b*x**4)))/(a*sqrt(c))

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Mathematica [A]  time = 0.152236, size = 47, normalized size = 1.15 \[ \frac{\sqrt{\frac{1}{c}} \cot ^{-1}\left (\frac{\sqrt{\frac{1}{c}} \sqrt{d \sqrt{a+b x^4}-c x^2}}{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^4)*Sqrt[-(c*x^2) + d*Sqrt[a + b*x^4]]),x]

[Out]

(Sqrt[c^(-1)]*ArcCot[(Sqrt[c^(-1)]*Sqrt[-(c*x^2) + d*Sqrt[a + b*x^4]])/x])/a

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a}{\frac{1}{\sqrt{-c{x}^{2}+d\sqrt{b{x}^{4}+a}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)/(-c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)

[Out]

int(1/(b*x^4+a)/(-c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{-c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(-c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(-c*x^2 + sqrt(b*x^4 + a)*d)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(-c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{4}\right ) \sqrt{- c x^{2} + d \sqrt{a + b x^{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**4+a)/(-c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)

[Out]

Integral(1/((a + b*x**4)*sqrt(-c*x**2 + d*sqrt(a + b*x**4))), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{-c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)*sqrt(-c*x^2 + sqrt(b*x^4 + a)*d)),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)*sqrt(-c*x^2 + sqrt(b*x^4 + a)*d)), x)