3.651 \(\int \frac{x^4}{a+c x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.279137, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} c^{5/4}}+\frac{\sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} c^{5/4}}-\frac{\sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} c^{5/4}}+\frac{x}{c} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 53.0266, size = 175, normalized size = 0.92 \[ \frac{\sqrt{2} \sqrt [4]{a} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 c^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 c^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 c^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 c^{\frac{5}{4}}} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(c*x**4+a),x)

[Out]

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

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Mathematica [A]  time = 0.0620553, size = 173, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+8 \sqrt [4]{c} x}{8 c^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.013, size = 133, normalized size = 0.7 \[{\frac{x}{c}}-{\frac{\sqrt{2}}{8\,c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^4+a),x)

[Out]

x/c-1/8/c*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a
/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))-1/4/c*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)
^(1/4)*x+1)-1/4/c*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(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(x^4/(c*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253049, size = 136, normalized size = 0.72 \[ \frac{4 \, c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}}}{x + \sqrt{c^{2} \sqrt{-\frac{a}{c^{5}}} + x^{2}}}\right ) - c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} \log \left (c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} + x\right ) + c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} \log \left (-c \left (-\frac{a}{c^{5}}\right )^{\frac{1}{4}} + x\right ) + 4 \, x}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.25531, size = 22, normalized size = 0.12 \[ \operatorname{RootSum}{\left (256 t^{4} c^{5} + a, \left ( t \mapsto t \log{\left (- 4 t c + x \right )} \right )\right )} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**4+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222944, size = 232, normalized size = 1.22 \[ \frac{x}{c} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, c^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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