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

Optimal. Leaf size=24 \[ \frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right ) \]

[Out]

(c*ArcTan[x])/2 + (c*ArcTanh[x])/2 + (d*ArcTanh[x^2])/2

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Rubi [A]  time = 0.0450354, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{2} c \tan ^{-1}(x)+\frac{1}{2} c \tanh ^{-1}(x)+\frac{1}{2} d \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(1 - x^4),x]

[Out]

(c*ArcTan[x])/2 + (c*ArcTanh[x])/2 + (d*ArcTanh[x^2])/2

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Rubi in Sympy [A]  time = 7.25757, size = 20, normalized size = 0.83 \[ \frac{c \operatorname{atan}{\left (x \right )}}{2} + \frac{c \operatorname{atanh}{\left (x \right )}}{2} + \frac{d \operatorname{atanh}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*atan(x)/2 + c*atanh(x)/2 + d*atanh(x**2)/2

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Mathematica [A]  time = 0.0262536, size = 42, normalized size = 1.75 \[ \frac{1}{4} \left (-(c+d) \log (1-x)+c \log (x+1)+2 c \tan ^{-1}(x)+d \log \left (x^2+1\right )-d \log (x+1)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 44, normalized size = 1.8 \[ -{\frac{\ln \left ( -1+x \right ) c}{4}}-{\frac{\ln \left ( -1+x \right ) d}{4}}+{\frac{\ln \left ( 1+x \right ) c}{4}}-{\frac{\ln \left ( 1+x \right ) d}{4}}+{\frac{d\ln \left ({x}^{2}+1 \right ) }{4}}+{\frac{c\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(-x^4+1),x)

[Out]

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

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Maxima [A]  time = 1.51955, size = 47, normalized size = 1.96 \[ \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left (x + 1\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227966, size = 47, normalized size = 1.96 \[ \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d \log \left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )} \log \left (x + 1\right ) - \frac{1}{4} \,{\left (c + d\right )} \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.771547, size = 313, normalized size = 13.04 \[ \frac{\left (c - d\right ) \log{\left (x + \frac{c^{4} \left (c - d\right ) + 5 c^{2} d^{3} + c^{2} d \left (c - d\right )^{2} - 2 d^{4} \left (c - d\right ) + 2 d^{2} \left (c - d\right )^{3}}{c^{5} + 4 c d^{4}} \right )}}{4} - \frac{\left (c + d\right ) \log{\left (x + \frac{- c^{4} \left (c + d\right ) + 5 c^{2} d^{3} + c^{2} d \left (c + d\right )^{2} + 2 d^{4} \left (c + d\right ) - 2 d^{2} \left (c + d\right )^{3}}{c^{5} + 4 c d^{4}} \right )}}{4} - \left (- \frac{i c}{4} - \frac{d}{4}\right ) \log{\left (x + \frac{- 4 c^{4} \left (- \frac{i c}{4} - \frac{d}{4}\right ) + 5 c^{2} d^{3} + 16 c^{2} d \left (- \frac{i c}{4} - \frac{d}{4}\right )^{2} + 8 d^{4} \left (- \frac{i c}{4} - \frac{d}{4}\right ) - 128 d^{2} \left (- \frac{i c}{4} - \frac{d}{4}\right )^{3}}{c^{5} + 4 c d^{4}} \right )} - \left (\frac{i c}{4} - \frac{d}{4}\right ) \log{\left (x + \frac{- 4 c^{4} \left (\frac{i c}{4} - \frac{d}{4}\right ) + 5 c^{2} d^{3} + 16 c^{2} d \left (\frac{i c}{4} - \frac{d}{4}\right )^{2} + 8 d^{4} \left (\frac{i c}{4} - \frac{d}{4}\right ) - 128 d^{2} \left (\frac{i c}{4} - \frac{d}{4}\right )^{3}}{c^{5} + 4 c d^{4}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)/(-x**4+1),x)

[Out]

(c - d)*log(x + (c**4*(c - d) + 5*c**2*d**3 + c**2*d*(c - d)**2 - 2*d**4*(c - d)
+ 2*d**2*(c - d)**3)/(c**5 + 4*c*d**4))/4 - (c + d)*log(x + (-c**4*(c + d) + 5*
c**2*d**3 + c**2*d*(c + d)**2 + 2*d**4*(c + d) - 2*d**2*(c + d)**3)/(c**5 + 4*c*
d**4))/4 - (-I*c/4 - d/4)*log(x + (-4*c**4*(-I*c/4 - d/4) + 5*c**2*d**3 + 16*c**
2*d*(-I*c/4 - d/4)**2 + 8*d**4*(-I*c/4 - d/4) - 128*d**2*(-I*c/4 - d/4)**3)/(c**
5 + 4*c*d**4)) - (I*c/4 - d/4)*log(x + (-4*c**4*(I*c/4 - d/4) + 5*c**2*d**3 + 16
*c**2*d*(I*c/4 - d/4)**2 + 8*d**4*(I*c/4 - d/4) - 128*d**2*(I*c/4 - d/4)**3)/(c*
*5 + 4*c*d**4))

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GIAC/XCAS [A]  time = 0.208833, size = 50, normalized size = 2.08 \[ \frac{1}{2} \, c \arctan \left (x\right ) + \frac{1}{4} \, d{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{4} \,{\left (c - d\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\left (c + d\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*arctan(x) + 1/4*d*ln(x^2 + 1) + 1/4*(c - d)*ln(abs(x + 1)) - 1/4*(c + d)*l
n(abs(x - 1))