3.93 \(\int \frac{1}{\left (1-(c+d x)^2\right )^2} \, dx\)

Optimal. Leaf size=39 \[ \frac{c+d x}{2 d \left (1-(c+d x)^2\right )}+\frac{\tanh ^{-1}(c+d x)}{2 d} \]

[Out]

(c + d*x)/(2*d*(1 - (c + d*x)^2)) + ArcTanh[c + d*x]/(2*d)

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Rubi [A]  time = 0.0294724, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{c+d x}{2 d \left (1-(c+d x)^2\right )}+\frac{\tanh ^{-1}(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(1 - (c + d*x)^2)^(-2),x]

[Out]

(c + d*x)/(2*d*(1 - (c + d*x)^2)) + ArcTanh[c + d*x]/(2*d)

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Rubi in Sympy [A]  time = 2.59983, size = 26, normalized size = 0.67 \[ \frac{c + d x}{2 d \left (- \left (c + d x\right )^{2} + 1\right )} + \frac{\operatorname{atanh}{\left (c + d x \right )}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-(d*x+c)**2)**2,x)

[Out]

(c + d*x)/(2*d*(-(c + d*x)**2 + 1)) + atanh(c + d*x)/(2*d)

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Mathematica [A]  time = 0.0325388, size = 45, normalized size = 1.15 \[ \frac{-\frac{2 (c+d x)}{(c+d x)^2-1}-\log (-c-d x+1)+\log (c+d x+1)}{4 d} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - (c + d*x)^2)^(-2),x]

[Out]

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

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Maple [A]  time = 0.014, size = 52, normalized size = 1.3 \[ -{\frac{1}{4\,d \left ( dx+c-1 \right ) }}-{\frac{\ln \left ( dx+c-1 \right ) }{4\,d}}-{\frac{1}{4\,d \left ( dx+c+1 \right ) }}+{\frac{\ln \left ( dx+c+1 \right ) }{4\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-(d*x+c)^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.803937, size = 76, normalized size = 1.95 \[ -\frac{d x + c}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x +{\left (c^{2} - 1\right )} d\right )}} + \frac{\log \left (d x + c + 1\right )}{4 \, d} - \frac{\log \left (d x + c - 1\right )}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2 - 1)^(-2),x, algorithm="maxima")

[Out]

-1/2*(d*x + c)/(d^3*x^2 + 2*c*d^2*x + (c^2 - 1)*d) + 1/4*log(d*x + c + 1)/d - 1/
4*log(d*x + c - 1)/d

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Fricas [A]  time = 0.25824, size = 115, normalized size = 2.95 \[ -\frac{2 \, d x -{\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \log \left (d x + c + 1\right ) +{\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \log \left (d x + c - 1\right ) + 2 \, c}{4 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x +{\left (c^{2} - 1\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2 - 1)^(-2),x, algorithm="fricas")

[Out]

-1/4*(2*d*x - (d^2*x^2 + 2*c*d*x + c^2 - 1)*log(d*x + c + 1) + (d^2*x^2 + 2*c*d*
x + c^2 - 1)*log(d*x + c - 1) + 2*c)/(d^3*x^2 + 2*c*d^2*x + (c^2 - 1)*d)

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Sympy [A]  time = 2.27859, size = 53, normalized size = 1.36 \[ - \frac{c + d x}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2} - 2 d} + \frac{- \frac{\log{\left (x + \frac{c - 1}{d} \right )}}{4} + \frac{\log{\left (x + \frac{c + 1}{d} \right )}}{4}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-(d*x+c)**2)**2,x)

[Out]

-(c + d*x)/(2*c**2*d + 4*c*d**2*x + 2*d**3*x**2 - 2*d) + (-log(x + (c - 1)/d)/4
+ log(x + (c + 1)/d)/4)/d

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GIAC/XCAS [A]  time = 0.264893, size = 76, normalized size = 1.95 \[ \frac{{\rm ln}\left ({\left | d x + c + 1 \right |}\right )}{4 \, d} - \frac{{\rm ln}\left ({\left | d x + c - 1 \right |}\right )}{4 \, d} - \frac{d x + c}{2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2 - 1)^(-2),x, algorithm="giac")

[Out]

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