3.3.99 \(\int \frac {1}{\sqrt {(-1+x) (1+x)^3}} \, dx\) [299]

Optimal. Leaf size=26 \[ -\frac {(1-x) (1+x)}{\sqrt {-\left ((1-x) (1+x)^3\right )}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6851, 37} \begin {gather*} -\frac {(1-x) (x+1)}{\sqrt {-\left ((1-x) (x+1)^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(-1 + x)*(1 + x)^3],x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\left (\sqrt {-1+x} (1+x)^{3/2}\right ) \int \frac {1}{\sqrt {-1+x} (1+x)^{3/2}} \, dx}{\sqrt {(-1+x) (1+x)^3}}\\ &=-\frac {(1-x) (1+x)}{\sqrt {-\left ((1-x) (1+x)^3\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.77 \begin {gather*} \frac {(-1+x) (1+x)}{\sqrt {(-1+x) (1+x)^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.19, size = 29, normalized size = 1.12

method	result	size
gosper	\(\frac {\left (x -1\right ) \left (x +1\right )}{\sqrt {\left (x +1\right )^{3} \left (x -1\right )}}\)	\(19\)
risch	\(\frac {\left (x -1\right ) \left (x +1\right )}{\sqrt {\left (x +1\right )^{3} \left (x -1\right )}}\)	\(19\)
trager	\(\frac {\sqrt {x^{4}+2 x^{3}-2 x -1}}{\left (x +1\right )^{2}}\)	\(22\)
default	\(\frac {\sqrt {\left (x -1\right ) \left (x +1\right )}\, \sqrt {x^{2}-1}}{\sqrt {\left (x +1\right )^{3} \left (x -1\right )}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt((x + 1)^3*(x - 1)), x)

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Fricas [A]
time = 0.57, size = 34, normalized size = 1.31 \begin {gather*} \frac {x^{2} + 2 \, x + \sqrt {x^{4} + 2 \, x^{3} - 2 \, x - 1} + 1}{x^{2} + 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt((x - 1)*(x + 1)**3), x)

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Giac [A]
time = 0.43, size = 22, normalized size = 0.85 \begin {gather*} \frac {2}{{\left (x - \sqrt {x^{2} - 1} + 1\right )} \mathrm {sgn}\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((x - sqrt(x^2 - 1) + 1)*sgn(x + 1))

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Mupad [B]
time = 0.05, size = 17, normalized size = 0.65 \begin {gather*} \frac {x^2-1}{\sqrt {\left (x-1\right )\,{\left (x+1\right )}^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x - 1)*(x + 1)^3)^(1/2),x)

[Out]

(x^2 - 1)/((x - 1)*(x + 1)^3)^(1/2)

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Chatgpt [F] Failed to verify
time = 1.00, size = 29, normalized size = 1.12 \begin {gather*} 6 \sqrt {\frac {x +1}{x -1}+2}+\frac {1}{\frac {x +1}{x -1}+2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/((x+1)^3*(x-1))^(1/2),x)

[Out]

6*((x+1)/(x-1)+2)^(1/2)+1/((x+1)/(x-1)+2)

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