3.5.15 \(\int \frac {(-1+x)^{3/2}+(1+x)^{3/2}}{(-1+x)^{3/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac {2 \sqrt {x-1}}{1-x}+\frac {2 \sqrt {x+1}}{-x-1} \]

________________________________________________________________________________________

Rubi [A]  time = 0.29, antiderivative size = 19, normalized size of antiderivative = 0.58, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6688} \begin {gather*} -\frac {2}{\sqrt {x+1}}-\frac {2}{\sqrt {x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/Sqrt[-1 + x] - 2/Sqrt[1 + x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 19, normalized size = 0.58 \begin {gather*} -\frac {2}{\sqrt {x+1}}-\frac {2}{\sqrt {x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/Sqrt[-1 + x] - 2/Sqrt[1 + x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.35, size = 19, normalized size = 0.58 \begin {gather*} -\frac {2}{\sqrt {-1+x}}-\frac {2}{\sqrt {1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/Sqrt[-1 + x] - 2/Sqrt[1 + x]

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 28, normalized size = 0.85 \begin {gather*} -\frac {2 \, {\left ({\left (x + 1\right )} \sqrt {x - 1} + \sqrt {x + 1} {\left (x - 1\right )}\right )}}{x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.39, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2}{\sqrt {x + 1}} - \frac {2}{\sqrt {x - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/sqrt(x + 1) - 2/sqrt(x - 1)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 16, normalized size = 0.48

method	result	size
default	\(-\frac {2}{\sqrt {1+x}}-\frac {2}{\sqrt {-1+x}}\)	\(16\)
meijerg	\(\frac {2 \sqrt {\pi }-\frac {2 \sqrt {\pi }}{\sqrt {1+x}}}{\sqrt {\pi }}-\frac {2 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {3}{2}} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-x}}\right )}{\sqrt {\pi }\, \mathrm {signum}\left (-1+x \right )^{\frac {3}{2}}}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2}{\sqrt {x + 1}} - \frac {2}{\sqrt {x - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/sqrt(x + 1) - 2/sqrt(x - 1)

________________________________________________________________________________________

mupad [B]  time = 0.28, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2}{\sqrt {x-1}}-\frac {2}{\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 2/(x - 1)^(1/2) - 2/(x + 1)^(1/2)

________________________________________________________________________________________

sympy [B]  time = 4.91, size = 56, normalized size = 1.70 \begin {gather*} - \frac {2 x \sqrt {x - 1}}{x^{2} - 1} - \frac {2 x \sqrt {x + 1}}{x^{2} - 1} - \frac {2 \sqrt {x - 1}}{x^{2} - 1} + \frac {2 \sqrt {x + 1}}{x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________