3.1.12 \(\int (1-\coth ^2(x))^{3/2} \, dx\) [12]

Optimal. Leaf size=24 \[ \frac {1}{2} \text {ArcSin}(\coth (x))+\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)} \]

[Out]

1/2*arcsin(coth(x))+1/2*coth(x)*(-csch(x)^2)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3738, 4207, 201, 222} \begin {gather*} \frac {1}{2} \text {ArcSin}(\coth (x))+\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Coth[x]^2)^(3/2),x]

[Out]

ArcSin[Coth[x]]/2 + (Coth[x]*Sqrt[-Csch[x]^2])/2

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rule 3738

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 41, normalized size = 1.71 \begin {gather*} \frac {1}{4} \text {csch}\left (\frac {x}{2}\right ) \sqrt {-\text {csch}^2(x)} \text {sech}\left (\frac {x}{2}\right ) \left (\cosh (x)+\log \left (\tanh \left (\frac {x}{2}\right )\right ) \sinh ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - Coth[x]^2)^(3/2),x]

[Out]

(Csch[x/2]*Sqrt[-Csch[x]^2]*Sech[x/2]*(Cosh[x] + Log[Tanh[x/2]]*Sinh[x]^2))/4

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Maple [A]
time = 0.40, size = 21, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\coth \left (x \right ) \sqrt {1-\left (\coth ^{2}\left (x \right )\right )}}{2}+\frac {\arcsin \left (\coth \left (x \right )\right )}{2}\)	\(21\)
default	\(\frac {\coth \left (x \right ) \sqrt {1-\left (\coth ^{2}\left (x \right )\right )}}{2}+\frac {\arcsin \left (\coth \left (x \right )\right )}{2}\)	\(21\)
risch	\(\frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}+\frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}+1\right )}{2}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 49, normalized size = 2.04 \begin {gather*} \frac {i \, e^{\left (-x\right )} + i \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (1 - \coth ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((1 - coth(x)**2)**(3/2), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.39, size = 60, normalized size = 2.50 \begin {gather*} -\frac {1}{4} \, {\left (\frac {4 \, {\left (i \, e^{\left (-x\right )} + i \, e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - i \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + i \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(I*e^(-x) + I*e^x)/((e^(-x) + e^x)^2 - 4) - I*log(e^(-x) + e^x + 2) + I*log(e^(-x) + e^x - 2))*sgn(-e^
(2*x) + 1)

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Mupad [B]
time = 1.21, size = 20, normalized size = 0.83 \begin {gather*} \frac {\mathrm {asin}\left (\mathrm {coth}\left (x\right )\right )}{2}+\frac {\mathrm {coth}\left (x\right )\,\sqrt {1-{\mathrm {coth}\left (x\right )}^2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - coth(x)^2)^(3/2),x)

[Out]

asin(coth(x))/2 + (coth(x)*(1 - coth(x)^2)^(1/2))/2

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