3.12.18 \(\int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx\) [1118]

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

[Out]

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

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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 = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 52, 41, 222} \begin {gather*} -\frac {2 (1-x)^{3/2}}{\sqrt {x+1}}-3 \sqrt {x+1} \sqrt {1-x}-3 \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(3/2)/(1 + x)^(3/2),x]

[Out]

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

Rule 41

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]))

Rule 49

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]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 43, normalized size = 1.05 \begin {gather*} \frac {(-5-x) \sqrt {1-x}}{\sqrt {1+x}}-6 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(3/2)/(1 + x)^(3/2),x]

[Out]

((-5 - x)*Sqrt[1 - x])/Sqrt[1 + x] - 6*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 = 3.42, size = 105, normalized size = 2.56 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-x \sqrt {-1+x}-5 \sqrt {-1+x}+6 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {1+x}\right )}{\sqrt {1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-8}{\sqrt {1+x} \sqrt {1-x}}-6 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]+\frac {2 \sqrt {1+x}}{\sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {3}{2}}}{\sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x)^(3/2)/(1 + x)^(3/2),x]')

[Out]

Piecewise[{{I (-x Sqrt[-1 + x] - 5 Sqrt[-1 + x] + 6 ArcCosh[Sqrt[2] Sqrt[1 + x] / 2] Sqrt[1 + x]) / Sqrt[1 + x
], Abs[1 + x] > 2}}, -8 / (Sqrt[1 + x] Sqrt[1 - x]) - 6 ArcSin[Sqrt[2] Sqrt[1 + x] / 2] + 2 Sqrt[1 + x] / Sqrt
[1 - x] + (1 + x) ^ (3 / 2) / Sqrt[1 - x]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
time = 0.16, size = 71, normalized size = 1.73

method	result	size
risch	\(\frac {\left (x^{2}+4 x -5\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(3/2)/(1+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

(x^2+4*x-5)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-3*((1+x)*(1-x))^(1/2)/(1+x)^(1/2
)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]
time = 0.36, size = 41, normalized size = 1.00 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2} + 2 \, x + 1} - \frac {6 \, \sqrt {-x^{2} + 1}}{x + 1} - 3 \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(3/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

(-x^2 + 1)^(3/2)/(x^2 + 2*x + 1) - 6*sqrt(-x^2 + 1)/(x + 1) - 3*arcsin(x)

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Fricas [A]
time = 0.30, size = 53, normalized size = 1.29 \begin {gather*} -\frac {{\left (x + 5\right )} \sqrt {x + 1} \sqrt {-x + 1} - 6 \, {\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 5 \, x + 5}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(3/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

-((x + 5)*sqrt(x + 1)*sqrt(-x + 1) - 6*(x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 5*x + 5)/(x + 1)

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Sympy [A]
time = 1.38, size = 131, normalized size = 3.20 \begin {gather*} \begin {cases} 6 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {8 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 6 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {8}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(3/2)/(1+x)**(3/2),x)

[Out]

Piecewise((6*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(3/2)/sqrt(x - 1) - 2*I*sqrt(x + 1)/sqrt(x - 1) + 8*I
/(sqrt(x - 1)*sqrt(x + 1)), Abs(x + 1) > 2), (-6*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(3/2)/sqrt(1 - x) + 2*
sqrt(x + 1)/sqrt(1 - x) - 8/(sqrt(1 - x)*sqrt(x + 1)), True))

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Giac [A]
time = 0.01, size = 61, normalized size = 1.49 \begin {gather*} \frac {2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {-x+1}-3\right ) \sqrt {-x+1} \sqrt {x+1}}{x+1}+6 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(3/2)/(1+x)^(3/2),x)

[Out]

-(x + 5)*sqrt(-x + 1)/sqrt(x + 1) + 6*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 )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(3/2)/(x + 1)^(3/2),x)

[Out]

int((1 - x)^(3/2)/(x + 1)^(3/2), x)

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