3.124 \(\int \frac{c+d x}{1+x^4} \, dx\)

Optimal. Leaf size=98 \[ -\frac{c \log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{c \log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{c \tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{c \tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}}+\frac{1}{2} d \tan ^{-1}\left (x^2\right ) \]

[Out]

(d*ArcTan[x^2])/2 - (c*ArcTan[1 - Sqrt[2]*x])/(2*Sqrt[2]) + (c*ArcTan[1 + Sqrt[2
]*x])/(2*Sqrt[2]) - (c*Log[1 - Sqrt[2]*x + x^2])/(4*Sqrt[2]) + (c*Log[1 + Sqrt[2
]*x + x^2])/(4*Sqrt[2])

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Rubi [A]  time = 0.151425, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ -\frac{c \log \left (x^2-\sqrt{2} x+1\right )}{4 \sqrt{2}}+\frac{c \log \left (x^2+\sqrt{2} x+1\right )}{4 \sqrt{2}}-\frac{c \tan ^{-1}\left (1-\sqrt{2} x\right )}{2 \sqrt{2}}+\frac{c \tan ^{-1}\left (\sqrt{2} x+1\right )}{2 \sqrt{2}}+\frac{1}{2} d \tan ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(1 + x^4),x]

[Out]

(d*ArcTan[x^2])/2 - (c*ArcTan[1 - Sqrt[2]*x])/(2*Sqrt[2]) + (c*ArcTan[1 + Sqrt[2
]*x])/(2*Sqrt[2]) - (c*Log[1 - Sqrt[2]*x + x^2])/(4*Sqrt[2]) + (c*Log[1 + Sqrt[2
]*x + x^2])/(4*Sqrt[2])

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Rubi in Sympy [A]  time = 19.0504, size = 88, normalized size = 0.9 \[ - \frac{\sqrt{2} c \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} c \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{8} + \frac{\sqrt{2} c \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{4} + \frac{\sqrt{2} c \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{4} + \frac{d \operatorname{atan}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)/(x**4+1),x)

[Out]

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

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Mathematica [C]  time = 0.119185, size = 99, normalized size = 1.01 \[ \frac{1}{4} \left (-\left (\sqrt [4]{-1} c+i d\right ) \log \left (\sqrt [4]{-1}-x\right )+\left (-(-1)^{3/4} c+i d\right ) \log \left ((-1)^{3/4}-x\right )+\left (\sqrt [4]{-1} c-i d\right ) \log \left (x+\sqrt [4]{-1}\right )+\left ((-1)^{3/4} c+i d\right ) \log \left (x+(-1)^{3/4}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)/(1 + x^4),x]

[Out]

(-(((-1)^(1/4)*c + I*d)*Log[(-1)^(1/4) - x]) + (-((-1)^(3/4)*c) + I*d)*Log[(-1)^
(3/4) - x] + ((-1)^(1/4)*c - I*d)*Log[(-1)^(1/4) + x] + ((-1)^(3/4)*c + I*d)*Log
[(-1)^(3/4) + x])/4

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Maple [A]  time = 0.005, size = 68, normalized size = 0.7 \[{\frac{c\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{4}}+{\frac{c\arctan \left ( x\sqrt{2}-1 \right ) \sqrt{2}}{4}}+{\frac{c\sqrt{2}}{8}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }+{\frac{d\arctan \left ({x}^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(x^4+1),x)

[Out]

1/4*c*arctan(1+x*2^(1/2))*2^(1/2)+1/4*c*arctan(x*2^(1/2)-1)*2^(1/2)+1/8*c*2^(1/2
)*ln((1+x^2+x*2^(1/2))/(1+x^2-x*2^(1/2)))+1/2*d*arctan(x^2)

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Maxima [A]  time = 1.52263, size = 116, normalized size = 1.18 \[ \frac{1}{8} \, \sqrt{2} c \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2} c \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{1}{4} \,{\left (\sqrt{2} c - 2 \, d\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \,{\left (\sqrt{2} c + 2 \, d\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*c*log(x^2 + sqrt(2)*x + 1) - 1/8*sqrt(2)*c*log(x^2 - sqrt(2)*x + 1)
+ 1/4*(sqrt(2)*c - 2*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) + 1/4*(sqrt(2)*c + 2
*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.744517, size = 83, normalized size = 0.85 \[ \operatorname{RootSum}{\left (256 t^{4} + 32 t^{2} d^{2} - 16 t c^{2} d + c^{4} + d^{4}, \left ( t \mapsto t \log{\left (x + \frac{128 t^{3} d^{2} + 16 t^{2} c^{2} d + 4 t c^{4} + 8 t d^{4} - 5 c^{2} d^{3}}{c^{5} - 4 c d^{4}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)/(x**4+1),x)

[Out]

RootSum(256*_t**4 + 32*_t**2*d**2 - 16*_t*c**2*d + c**4 + d**4, Lambda(_t, _t*lo
g(x + (128*_t**3*d**2 + 16*_t**2*c**2*d + 4*_t*c**4 + 8*_t*d**4 - 5*c**2*d**3)/(
c**5 - 4*c*d**4))))

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GIAC/XCAS [A]  time = 0.213021, size = 116, normalized size = 1.18 \[ \frac{1}{8} \, \sqrt{2} c{\rm ln}\left (x^{2} + \sqrt{2} x + 1\right ) - \frac{1}{8} \, \sqrt{2} c{\rm ln}\left (x^{2} - \sqrt{2} x + 1\right ) + \frac{1}{4} \,{\left (\sqrt{2} c - 2 \, d\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{4} \,{\left (\sqrt{2} c + 2 \, d\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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