Optimal. Leaf size=64 \[ \frac{3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac{c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac{3 \tanh ^{-1}(c+d x)}{8 d} \]
[Out]
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Rubi [A] time = 0.0485389, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac{c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac{3 \tanh ^{-1}(c+d x)}{8 d} \]
Antiderivative was successfully verified.
[In] Int[(1 - (c + d*x)^2)^(-3),x]
[Out]
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Rubi in Sympy [A] time = 3.41297, size = 49, normalized size = 0.77 \[ \frac{3 \left (c + d x\right )}{8 d \left (- \left (c + d x\right )^{2} + 1\right )} + \frac{c + d x}{4 d \left (- \left (c + d x\right )^{2} + 1\right )^{2}} + \frac{3 \operatorname{atanh}{\left (c + d x \right )}}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-(d*x+c)**2)**3,x)
[Out]
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Mathematica [A] time = 0.0449493, size = 65, normalized size = 1.02 \[ \frac{-\frac{6 (c+d x)}{(c+d x)^2-1}+\frac{4 (c+d x)}{\left ((c+d x)^2-1\right )^2}-3 \log (-c-d x+1)+3 \log (c+d x+1)}{16 d} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - (c + d*x)^2)^(-3),x]
[Out]
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Maple [A] time = 0.015, size = 78, normalized size = 1.2 \[{\frac{1}{16\,d \left ( dx+c-1 \right ) ^{2}}}-{\frac{3}{16\,d \left ( dx+c-1 \right ) }}-{\frac{3\,\ln \left ( dx+c-1 \right ) }{16\,d}}-{\frac{1}{16\,d \left ( dx+c+1 \right ) ^{2}}}-{\frac{3}{16\,d \left ( dx+c+1 \right ) }}+{\frac{3\,\ln \left ( dx+c+1 \right ) }{16\,d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-(d*x+c)^2)^3,x)
[Out]
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Maxima [A] time = 0.826349, size = 165, normalized size = 2.58 \[ -\frac{3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} +{\left (9 \, c^{2} - 5\right )} d x - 5 \, c}{8 \,{\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \,{\left (3 \, c^{2} - 1\right )} d^{3} x^{2} + 4 \,{\left (c^{3} - c\right )} d^{2} x +{\left (c^{4} - 2 \, c^{2} + 1\right )} d\right )}} + \frac{3 \, \log \left (d x + c + 1\right )}{16 \, d} - \frac{3 \, \log \left (d x + c - 1\right )}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((d*x + c)^2 - 1)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.268806, size = 297, normalized size = 4.64 \[ -\frac{6 \, d^{3} x^{3} + 18 \, c d^{2} x^{2} + 6 \, c^{3} + 2 \,{\left (9 \, c^{2} - 5\right )} d x - 3 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \,{\left (3 \, c^{2} - 1\right )} d^{2} x^{2} + c^{4} + 4 \,{\left (c^{3} - c\right )} d x - 2 \, c^{2} + 1\right )} \log \left (d x + c + 1\right ) + 3 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \,{\left (3 \, c^{2} - 1\right )} d^{2} x^{2} + c^{4} + 4 \,{\left (c^{3} - c\right )} d x - 2 \, c^{2} + 1\right )} \log \left (d x + c - 1\right ) - 10 \, c}{16 \,{\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \,{\left (3 \, c^{2} - 1\right )} d^{3} x^{2} + 4 \,{\left (c^{3} - c\right )} d^{2} x +{\left (c^{4} - 2 \, c^{2} + 1\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((d*x + c)^2 - 1)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.88074, size = 141, normalized size = 2.2 \[ - \frac{3 c^{3} + 9 c d^{2} x^{2} - 5 c + 3 d^{3} x^{3} + x \left (9 c^{2} d - 5 d\right )}{8 c^{4} d - 16 c^{2} d + 32 c d^{4} x^{3} + 8 d^{5} x^{4} + 8 d + x^{2} \left (48 c^{2} d^{3} - 16 d^{3}\right ) + x \left (32 c^{3} d^{2} - 32 c d^{2}\right )} - \frac{\frac{3 \log{\left (x + \frac{3 c - 3}{3 d} \right )}}{16} - \frac{3 \log{\left (x + \frac{3 c + 3}{3 d} \right )}}{16}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-(d*x+c)**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.263148, size = 119, normalized size = 1.86 \[ \frac{3 \,{\rm ln}\left ({\left | d x + c + 1 \right |}\right )}{16 \, d} - \frac{3 \,{\rm ln}\left ({\left | d x + c - 1 \right |}\right )}{16 \, d} - \frac{3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 9 \, c^{2} d x + 3 \, c^{3} - 5 \, d x - 5 \, c}{8 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((d*x + c)^2 - 1)^3,x, algorithm="giac")
[Out]