3.118 \(\int \frac{1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=57 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]

[Out]

ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4)) + ArcTanh[(a^(1/4)*x)/(a + b*x
^4)^(1/4)]/(2*a^(5/4))

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Rubi [A]  time = 0.0694466, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4)) + ArcTanh[(a^(1/4)*x)/(a + b*x
^4)^(1/4)]/(2*a^(5/4))

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Rubi in Sympy [A]  time = 9.03974, size = 49, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 a^{\frac{5}{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(a**(1/4)*x/(a + b*x**4)**(1/4))/(2*a**(5/4)) + atanh(a**(1/4)*x/(a + b*x**4
)**(1/4))/(2*a**(5/4))

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Mathematica [A]  time = 0.109993, size = 76, normalized size = 1.33 \[ \frac{-\log \left (1-\frac{\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{4 a^{5/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{a- \left ( a-b \right ){x}^{4}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(1/(a*x**4*(a + b*x**4)**(1/4) - a*(a + b*x**4)**(1/4) - b*x**4*(a + b*
x**4)**(1/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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