Optimal. Leaf size=101 \[ \frac{8 \sqrt{x+1}}{315 \sqrt{1-x}}+\frac{8 \sqrt{x+1}}{315 (1-x)^{3/2}}+\frac{4 \sqrt{x+1}}{105 (1-x)^{5/2}}+\frac{4 \sqrt{x+1}}{63 (1-x)^{7/2}}+\frac{\sqrt{x+1}}{9 (1-x)^{9/2}} \]
[Out]
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Rubi [A] time = 0.0673135, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{8 \sqrt{x+1}}{315 \sqrt{1-x}}+\frac{8 \sqrt{x+1}}{315 (1-x)^{3/2}}+\frac{4 \sqrt{x+1}}{105 (1-x)^{5/2}}+\frac{4 \sqrt{x+1}}{63 (1-x)^{7/2}}+\frac{\sqrt{x+1}}{9 (1-x)^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - x)^(11/2)*Sqrt[1 + x]),x]
[Out]
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Rubi in Sympy [A] time = 7.8595, size = 82, normalized size = 0.81 \[ \frac{8 \sqrt{x + 1}}{315 \sqrt{- x + 1}} + \frac{8 \sqrt{x + 1}}{315 \left (- x + 1\right )^{\frac{3}{2}}} + \frac{4 \sqrt{x + 1}}{105 \left (- x + 1\right )^{\frac{5}{2}}} + \frac{4 \sqrt{x + 1}}{63 \left (- x + 1\right )^{\frac{7}{2}}} + \frac{\sqrt{x + 1}}{9 \left (- x + 1\right )^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-x)**(11/2)/(1+x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0250067, size = 40, normalized size = 0.4 \[ -\frac{\sqrt{1-x^2} \left (8 x^4-40 x^3+84 x^2-100 x+83\right )}{315 (x-1)^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - x)^(11/2)*Sqrt[1 + x]),x]
[Out]
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Maple [A] time = 0.006, size = 35, normalized size = 0.4 \[{\frac{8\,{x}^{4}-40\,{x}^{3}+84\,{x}^{2}-100\,x+83}{315}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-x)^(11/2)/(1+x)^(1/2),x)
[Out]
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Maxima [A] time = 1.51844, size = 177, normalized size = 1.75 \[ -\frac{\sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{63 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{4 \, \sqrt{-x^{2} + 1}}{105 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 1)*(-x + 1)^(11/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20443, size = 257, normalized size = 2.54 \[ \frac{91 \, x^{9} - 747 \, x^{8} + 1314 \, x^{7} + 1974 \, x^{6} - 8442 \, x^{5} + 7560 \, x^{4} + 3360 \, x^{3} - 10080 \, x^{2} + 3 \,{\left (25 \, x^{8} + 24 \, x^{7} - 658 \, x^{6} + 1624 \, x^{5} - 840 \, x^{4} - 1960 \, x^{3} + 3360 \, x^{2} - 1680 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 5040 \, x}{315 \,{\left (x^{9} - 9 \, x^{8} + 18 \, x^{7} + 18 \, x^{6} - 99 \, x^{5} + 99 \, x^{4} + 24 \, x^{3} - 108 \, x^{2} +{\left (x^{8} - 22 \, x^{6} + 60 \, x^{5} - 39 \, x^{4} - 60 \, x^{3} + 116 \, x^{2} - 72 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 72 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 1)*(-x + 1)^(11/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-x)**(11/2)/(1+x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212374, size = 57, normalized size = 0.56 \[ -\frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 8\right )} + 63\right )}{\left (x + 1\right )} - 105\right )}{\left (x + 1\right )} + 315\right )} \sqrt{x + 1} \sqrt{-x + 1}}{315 \,{\left (x - 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 1)*(-x + 1)^(11/2)),x, algorithm="giac")
[Out]