∫ (1−x)3∕2 ______________(1+x)5∕2 dx [1129]

Optimal. Leaf size=41 \[ -\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\sin ^{-1}(x) \]

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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {49, 41, 222} \begin {gather*} -\frac {2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {x+1}}+\sin ^{-1}(x) \end {gather*}

(-2*(1 - x)^(3/2))/(3*(1 + x)^(3/2)) + (2*Sqrt[1 - x])/Sqrt[1 + x] + ArcSin[x]

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b

, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(

m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},

x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0

] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}

\begin {align*} \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}-\int \frac {\sqrt {1-x}}{(1+x)^{3/2}} \, dx\\ &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 46, normalized size = 1.12 \begin {gather*} \frac {4 \sqrt {1-x} (1+2 x)}{3 (1+x)^{3/2}}-2 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

(4*Sqrt[1 - x]*(1 + 2*x))/(3*(1 + x)^(3/2)) - 2*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 4.15, size = 137, normalized size = 3.34 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-4 \sqrt {\frac {1-x}{1+x}}+\left (1+x\right ) \left (3 I \text {Log}\left [1+x\right ]+3 I \text {Log}\left [\frac {1}{1+x}\right ]+6 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]+8 \sqrt {\frac {1-x}{1+x}}\right )}{3 \left (1+x\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-2 I \text {Log}\left [1+\sqrt {1-\frac {2}{1+x}}\right ]-\frac {4 I \sqrt {1-\frac {2}{1+x}}}{3 \left (1+x\right )}+\frac {I 8 \sqrt {1-\frac {2}{1+x}}}{3}+I \text {Log}\left [\frac {1}{1+x}\right ]\right ] \end {gather*}

Piecewise[{{(-4 Sqrt[(1 - x) / (1 + x)] + (1 + x) (3 I Log[1 + x] + 3 I Log[1 / (1 + x)] + 6 ArcSin[Sqrt[2] Sq

rt[1 + x] / 2] + 8 Sqrt[(1 - x) / (1 + x)])) / (3 (1 + x)), 1 / Abs[1 + x] > 1 / 2}}, -2 I Log[1 + Sqrt[1 - 2

/ (1 + x)]] - 4 I Sqrt[1 - 2 / (1 + x)] / (3 (1 + x)) + I 8 Sqrt[1 - 2 / (1 + x)] / 3 + I Log[1 / (1 + x)]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(31)=62\).
time = 0.16, size = 73, normalized size = 1.78


method	result	size

risch	\(-\frac {4 \left (2 x^{2}-x -1\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(73\)

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).
time = 0.36, size = 66, normalized size = 1.61 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {7 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} + \arcsin \left (x\right ) \end {gather*}

-1/3*(-x^2 + 1)^(3/2)/(x^3 + 3*x^2 + 3*x + 1) - 2/3*sqrt(-x^2 + 1)/(x^2 + 2*x + 1) + 7/3*sqrt(-x^2 + 1)/(x + 1

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (31) = 62\).
time = 0.30, size = 71, normalized size = 1.73 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - 3 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 4 \, x + 2\right )}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

2/3*(2*x^2 + 2*(2*x + 1)*sqrt(x + 1)*sqrt(-x + 1) - 3*(x^2 + 2*x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [C] Result contains complex when optimal does not.
time = 2.06, size = 128, normalized size = 3.12 \begin {gather*} \begin {cases} \frac {8 \sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {4 \sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} + i \log {\left (\frac {1}{x + 1} \right )} + i \log {\left (x + 1 \right )} + 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\\frac {8 i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {4 i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} + i \log {\left (\frac {1}{x + 1} \right )} - 2 i \log {\left (\sqrt {1 - \frac {2}{x + 1}} + 1 \right )} & \text {otherwise} \end {cases} \end {gather*}

Piecewise((8*sqrt(-1 + 2/(x + 1))/3 - 4*sqrt(-1 + 2/(x + 1))/(3*(x + 1)) + I*log(1/(x + 1)) + I*log(x + 1) + 2

*asin(sqrt(2)*sqrt(x + 1)/2), 1/Abs(x + 1) > 1/2), (8*I*sqrt(1 - 2/(x + 1))/3 - 4*I*sqrt(1 - 2/(x + 1))/(3*(x

+ 1)) + I*log(1/(x + 1)) - 2*I*log(sqrt(1 - 2/(x + 1)) + 1), True))

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Giac [A]
time = 0.01, size = 65, normalized size = 1.59 \begin {gather*} \frac {2 \left (-\frac {4}{3} \sqrt {-x+1} \sqrt {-x+1}+2\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}}-2 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}

4/3*(2*x + 1)*sqrt(-x + 1)/(x + 1)^(3/2) - 2*arcsin(1/2*sqrt(2)*sqrt(-x + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (1-x\right )}^{3/2}}{{\left (x+1\right )}^{5/2}} \,d x \end {gather*}

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3.12.29 \(\int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx\) [1129]