3.2055 \(\int \frac{1}{a+\frac{b}{x^4}} \, dx\)

Optimal. Leaf size=190 \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a} x^2+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a} x^2+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{2 \sqrt{2} a^{5/4}}+\frac{x}{a} \]

[Out]

x/a + (b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*x)/b^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (b
^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4)*x)/b^(1/4)])/(2*Sqrt[2]*a^(5/4)) + (b^(1/4)*L
og[Sqrt[b] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2])/(4*Sqrt[2]*a^(5/4)) - (b^
(1/4)*Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2])/(4*Sqrt[2]*a^(5/4)
)

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Rubi [A]  time = 0.317155, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889 \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a} x^2+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a} x^2+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{2 \sqrt{2} a^{5/4}}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(-1),x]

[Out]

x/a + (b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*x)/b^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (b
^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4)*x)/b^(1/4)])/(2*Sqrt[2]*a^(5/4)) + (b^(1/4)*L
og[Sqrt[b] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2])/(4*Sqrt[2]*a^(5/4)) - (b^
(1/4)*Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2])/(4*Sqrt[2]*a^(5/4)
)

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Rubi in Sympy [A]  time = 52.8618, size = 175, normalized size = 0.92 \[ \frac{x}{a} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} x^{2} + \sqrt{b} \right )}}{8 a^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} x^{2} + \sqrt{b} \right )}}{8 a^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}} - 1 \right )}}{4 a^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}} + 1 \right )}}{4 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4),x)

[Out]

x/a + sqrt(2)*b**(1/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a)*x**2 + sqrt(b)
)/(8*a**(5/4)) - sqrt(2)*b**(1/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a)*x**2
+ sqrt(b))/(8*a**(5/4)) - sqrt(2)*b**(1/4)*atan(sqrt(2)*a**(1/4)*x/b**(1/4) - 1
)/(4*a**(5/4)) - sqrt(2)*b**(1/4)*atan(sqrt(2)*a**(1/4)*x/b**(1/4) + 1)/(4*a**(5
/4))

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Mathematica [A]  time = 0.0848368, size = 173, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a} x^2+\sqrt{b}\right )-\sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a} x^2+\sqrt{b}\right )+2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )-2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )+8 \sqrt [4]{a} x}{8 a^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(-1),x]

[Out]

(8*a^(1/4)*x + 2*Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*x)/b^(1/4)] - 2*Sqr
t[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4)*x)/b^(1/4)] + Sqrt[2]*b^(1/4)*Log[Sqrt[
b] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2] - Sqrt[2]*b^(1/4)*Log[Sqrt[b] + Sq
rt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2])/(8*a^(5/4))

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Maple [A]  time = 0.011, size = 133, normalized size = 0.7 \[{\frac{x}{a}}-{\frac{\sqrt{2}}{8\,a}\sqrt [4]{{\frac{b}{a}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{b}{a}}}x\sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{b}{a}}}x\sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4\,a}\sqrt [4]{{\frac{b}{a}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,a}\sqrt [4]{{\frac{b}{a}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4),x)

[Out]

x/a-1/8/a*(b/a)^(1/4)*2^(1/2)*ln((x^2+(b/a)^(1/4)*x*2^(1/2)+(b/a)^(1/2))/(x^2-(b
/a)^(1/4)*x*2^(1/2)+(b/a)^(1/2)))-1/4/a*(b/a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/a)
^(1/4)*x+1)-1/4/a*(b/a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/a)^(1/4)*x-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246337, size = 136, normalized size = 0.72 \[ \frac{4 \, a \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}}}{x + \sqrt{a^{2} \sqrt{-\frac{b}{a^{5}}} + x^{2}}}\right ) - a \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (a \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} + x\right ) + a \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} + x\right ) + 4 \, x}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^4),x, algorithm="fricas")

[Out]

1/4*(4*a*(-b/a^5)^(1/4)*arctan(a*(-b/a^5)^(1/4)/(x + sqrt(a^2*sqrt(-b/a^5) + x^2
))) - a*(-b/a^5)^(1/4)*log(a*(-b/a^5)^(1/4) + x) + a*(-b/a^5)^(1/4)*log(-a*(-b/a
^5)^(1/4) + x) + 4*x)/a

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Sympy [A]  time = 1.31298, size = 22, normalized size = 0.12 \[ \operatorname{RootSum}{\left (256 t^{4} a^{5} + b, \left ( t \mapsto t \log{\left (- 4 t a + x \right )} \right )\right )} + \frac{x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4),x)

[Out]

RootSum(256*_t**4*a**5 + b, Lambda(_t, _t*log(-4*_t*a + x))) + x/a

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GIAC/XCAS [A]  time = 0.235881, size = 232, normalized size = 1.22 \[ \frac{x}{a} - \frac{\sqrt{2} \left (a^{3} b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{b}{a}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} - \frac{\sqrt{2} \left (a^{3} b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{b}{a}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} - \frac{\sqrt{2} \left (a^{3} b\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{b}{a}\right )^{\frac{1}{4}} + \sqrt{\frac{b}{a}}\right )}{8 \, a^{2}} + \frac{\sqrt{2} \left (a^{3} b\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{b}{a}\right )^{\frac{1}{4}} + \sqrt{\frac{b}{a}}\right )}{8 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a + b/x^4),x, algorithm="giac")

[Out]

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