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

Optimal. Leaf size=10 \[ \frac{\tanh ^{-1}(c+d x)}{d} \]

[Out]

ArcTanh[c + d*x]/d

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Rubi [A]  time = 0.00909456, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tanh ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]  Int[(1 - (c + d*x)^2)^(-1),x]

[Out]

ArcTanh[c + d*x]/d

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Rubi in Sympy [A]  time = 1.68552, size = 7, normalized size = 0.7 \[ \frac{\operatorname{atanh}{\left (c + d x \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(c + d*x)/d

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Mathematica [B]  time = 0.00802069, size = 32, normalized size = 3.2 \[ \frac{\log (c+d x+1)}{2 d}-\frac{\log (-c-d x+1)}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 26, normalized size = 2.6 \[ -{\frac{\ln \left ( dx+c-1 \right ) }{2\,d}}+{\frac{\ln \left ( dx+c+1 \right ) }{2\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 0.816423, size = 34, normalized size = 3.4 \[ \frac{\log \left (d x + c + 1\right )}{2 \, d} - \frac{\log \left (d x + c - 1\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272084, size = 30, normalized size = 3. \[ \frac{\log \left (d x + c + 1\right ) - \log \left (d x + c - 1\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.399521, size = 22, normalized size = 2.2 \[ - \frac{\frac{\log{\left (x + \frac{c - 1}{d} \right )}}{2} - \frac{\log{\left (x + \frac{c + 1}{d} \right )}}{2}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-(d*x+c)**2),x)

[Out]

-(log(x + (c - 1)/d)/2 - log(x + (c + 1)/d)/2)/d

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GIAC/XCAS [A]  time = 0.2624, size = 36, normalized size = 3.6 \[ \frac{{\rm ln}\left ({\left | d x + c + 1 \right |}\right )}{2 \, d} - \frac{{\rm ln}\left ({\left | d x + c - 1 \right |}\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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