3.31 \(\int (-1-\coth ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=67 \[ \frac{1}{2} \coth (x) \sqrt{-\coth ^2(x)-1}-\frac{5}{2} \tan ^{-1}\left (\frac{\coth (x)}{\sqrt{-\coth ^2(x)-1}}\right )+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right ) \]

[Out]

(-5*ArcTan[Coth[x]/Sqrt[-1 - Coth[x]^2]])/2 + 2*Sqrt[2]*ArcTan[(Sqrt[2]*Coth[x])/Sqrt[-1 - Coth[x]^2]] + (Coth
[x]*Sqrt[-1 - Coth[x]^2])/2

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Rubi [A]  time = 0.0478565, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3661, 416, 523, 217, 203, 377} \[ \frac{1}{2} \coth (x) \sqrt{-\coth ^2(x)-1}-\frac{5}{2} \tan ^{-1}\left (\frac{\coth (x)}{\sqrt{-\coth ^2(x)-1}}\right )+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-\coth ^2(x)-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 - Coth[x]^2)^(3/2),x]

[Out]

(-5*ArcTan[Coth[x]/Sqrt[-1 - Coth[x]^2]])/2 + 2*Sqrt[2]*ArcTan[(Sqrt[2]*Coth[x])/Sqrt[-1 - Coth[x]^2]] + (Coth
[x]*Sqrt[-1 - Coth[x]^2])/2

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 217

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

Rule 203

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

Rule 377

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

Rubi steps

\begin{align*} \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (-1-x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \coth (x) \sqrt{-1-\coth ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-3-5 x^2}{\sqrt{-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \coth (x) \sqrt{-1-\coth ^2(x)}-\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2}} \, dx,x,\coth (x)\right )+4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \coth (x) \sqrt{-1-\coth ^2(x)}-\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )+4 \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{\coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )\\ &=-\frac{5}{2} \tan ^{-1}\left (\frac{\coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \coth (x)}{\sqrt{-1-\coth ^2(x)}}\right )+\frac{1}{2} \coth (x) \sqrt{-1-\coth ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.166444, size = 118, normalized size = 1.76 \[ -\frac{1}{8} \left (-\coth ^2(x)-1\right )^{3/2} \text{sech}^2(2 x) \left (\sinh (4 x)+16 \sinh ^3(x) \sqrt{\cosh (2 x)} \tanh ^{-1}\left (\frac{\cosh (x)}{\sqrt{\cosh (2 x)}}\right )+4 \sinh ^3(x) \left (\sqrt{-\cosh (2 x)} \tan ^{-1}\left (\frac{\cosh (x)}{\sqrt{-\cosh (2 x)}}\right )-4 \sqrt{2} \sqrt{\cosh (2 x)} \log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Coth[x]^2)^(3/2),x]

[Out]

-((-1 - Coth[x]^2)^(3/2)*Sech[2*x]^2*(16*ArcTanh[Cosh[x]/Sqrt[Cosh[2*x]]]*Sqrt[Cosh[2*x]]*Sinh[x]^3 + 4*(ArcTa
n[Cosh[x]/Sqrt[-Cosh[2*x]]]*Sqrt[-Cosh[2*x]] - 4*Sqrt[2]*Sqrt[Cosh[2*x]]*Log[Sqrt[2]*Cosh[x] + Sqrt[Cosh[2*x]]
])*Sinh[x]^3 + Sinh[4*x]))/8

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Maple [B]  time = 0.018, size = 211, normalized size = 3.2 \begin{align*}{\frac{1}{6} \left ( - \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right ) \right ) ^{{\frac{3}{2}}}}+{\frac{{\rm coth} \left (x\right )}{4}\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}}-{\frac{5}{4}\arctan \left ({{\rm coth} \left (x\right ){\frac{1}{\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}}}} \right ) }-\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}+\sqrt{2}\arctan \left ({\frac{ \left ( 2\,{\rm coth} \left (x\right )-2 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}+2\,{\rm coth} \left (x\right )}}}} \right ) -{\frac{1}{6} \left ( - \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right ) \right ) ^{{\frac{3}{2}}}}+{\frac{{\rm coth} \left (x\right )}{4}\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}}-{\frac{5}{4}\arctan \left ({{\rm coth} \left (x\right ){\frac{1}{\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}}}} \right ) }+\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}-\sqrt{2}\arctan \left ({\frac{ \left ( -2-2\,{\rm coth} \left (x\right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ({\rm coth} \left (x\right )-1 \right ) ^{2}-2\,{\rm coth} \left (x\right )}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-(1+coth(x))^2+2*coth(x))^(3/2)+1/4*coth(x)*(-(1+coth(x))^2+2*coth(x))^(1/2)-5/4*arctan(coth(x)/(-(1+coth
(x))^2+2*coth(x))^(1/2))-(-(1+coth(x))^2+2*coth(x))^(1/2)+2^(1/2)*arctan(1/4*(2*coth(x)-2)*2^(1/2)/(-(1+coth(x
))^2+2*coth(x))^(1/2))-1/6*(-(coth(x)-1)^2-2*coth(x))^(3/2)+1/4*coth(x)*(-(coth(x)-1)^2-2*coth(x))^(1/2)-5/4*a
rctan(coth(x)/(-(coth(x)-1)^2-2*coth(x))^(1/2))+(-(coth(x)-1)^2-2*coth(x))^(1/2)-2^(1/2)*arctan(1/4*(-2-2*coth
(x))*2^(1/2)/(-(coth(x)-1)^2-2*coth(x))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\coth \left (x\right )^{2} - 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-coth(x)^2 - 1)^(3/2), x)

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Fricas [C]  time = 2.14427, size = 1103, normalized size = 16.46 \begin{align*} \frac{2 \,{\left (\sqrt{-2} e^{\left (4 \, x\right )} - 2 \, \sqrt{-2} e^{\left (2 \, x\right )} + \sqrt{-2}\right )} \log \left (2 \,{\left (\sqrt{-2} \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - 2 \,{\left (\sqrt{-2} e^{\left (4 \, x\right )} - 2 \, \sqrt{-2} e^{\left (2 \, x\right )} + \sqrt{-2}\right )} \log \left (-2 \,{\left (\sqrt{-2} \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} - 2 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )}\right ) +{\left (5 i \, e^{\left (4 \, x\right )} - 10 i \, e^{\left (2 \, x\right )} + 5 i\right )} \log \left ({\left (4 i \, \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} - 4 \, e^{\left (2 \, x\right )} - 4\right )} e^{\left (-2 \, x\right )}\right ) +{\left (-5 i \, e^{\left (4 \, x\right )} + 10 i \, e^{\left (2 \, x\right )} - 5 i\right )} \log \left ({\left (-4 i \, \sqrt{-2 \, e^{\left (4 \, x\right )} - 2} - 4 \, e^{\left (2 \, x\right )} - 4\right )} e^{\left (-2 \, x\right )}\right ) - 2 \,{\left (\sqrt{-2} e^{\left (4 \, x\right )} - 2 \, \sqrt{-2} e^{\left (2 \, x\right )} + \sqrt{-2}\right )} \log \left (4 \,{\left (\sqrt{-2 \, e^{\left (4 \, x\right )} - 2}{\left (e^{\left (2 \, x\right )} + 2\right )} + \sqrt{-2} e^{\left (4 \, x\right )} + \sqrt{-2} e^{\left (2 \, x\right )} + 2 \, \sqrt{-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \,{\left (\sqrt{-2} e^{\left (4 \, x\right )} - 2 \, \sqrt{-2} e^{\left (2 \, x\right )} + \sqrt{-2}\right )} \log \left (4 \,{\left (\sqrt{-2 \, e^{\left (4 \, x\right )} - 2}{\left (e^{\left (2 \, x\right )} + 2\right )} - \sqrt{-2} e^{\left (4 \, x\right )} - \sqrt{-2} e^{\left (2 \, x\right )} - 2 \, \sqrt{-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \, \sqrt{-2 \, e^{\left (4 \, x\right )} - 2}{\left (e^{\left (2 \, x\right )} + 1\right )}}{4 \,{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(sqrt(-2)*e^(4*x) - 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(2*(sqrt(-2)*sqrt(-2*e^(4*x) - 2) + 2*e^(2*x) - 2
)*e^(-2*x)) - 2*(sqrt(-2)*e^(4*x) - 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(-2*(sqrt(-2)*sqrt(-2*e^(4*x) - 2) - 2*e
^(2*x) + 2)*e^(-2*x)) + (5*I*e^(4*x) - 10*I*e^(2*x) + 5*I)*log((4*I*sqrt(-2*e^(4*x) - 2) - 4*e^(2*x) - 4)*e^(-
2*x)) + (-5*I*e^(4*x) + 10*I*e^(2*x) - 5*I)*log((-4*I*sqrt(-2*e^(4*x) - 2) - 4*e^(2*x) - 4)*e^(-2*x)) - 2*(sqr
t(-2)*e^(4*x) - 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(4*(sqrt(-2*e^(4*x) - 2)*(e^(2*x) + 2) + sqrt(-2)*e^(4*x) +
sqrt(-2)*e^(2*x) + 2*sqrt(-2))*e^(-4*x)) + 2*(sqrt(-2)*e^(4*x) - 2*sqrt(-2)*e^(2*x) + sqrt(-2))*log(4*(sqrt(-2
*e^(4*x) - 2)*(e^(2*x) + 2) - sqrt(-2)*e^(4*x) - sqrt(-2)*e^(2*x) - 2*sqrt(-2))*e^(-4*x)) + 2*sqrt(-2*e^(4*x)
- 2)*(e^(2*x) + 1))/(e^(4*x) - 2*e^(2*x) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-coth(x)**2 - 1)**(3/2), x)

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Giac [C]  time = 1.24887, size = 385, normalized size = 5.75 \begin{align*} -\frac{1}{4} \, \sqrt{2}{\left (-5 i \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \,{\left (\sqrt{2} + \sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) - 4 i \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + 4 i \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + 4 i \, \log \left (-\sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + \frac{4 \,{\left (3 i \,{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + i \,{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) +{\left (-i \, \sqrt{e^{\left (4 \, x\right )} + 1} + i \, e^{\left (2 \, x\right )}\right )} \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + i \, \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right )\right )}}{{\left ({\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} + 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*(-5*I*sqrt(2)*log(1/2*abs(-2*sqrt(2) + 2*sqrt(e^(4*x) + 1) - 2*e^(2*x) + 2)/(sqrt(2) + sqrt(e^(4*
x) + 1) - e^(2*x) + 1))*sgn(-e^(2*x) + 1) - 4*I*log(sqrt(e^(4*x) + 1) - e^(2*x) + 1)*sgn(-e^(2*x) + 1) + 4*I*l
og(sqrt(e^(4*x) + 1) - e^(2*x))*sgn(-e^(2*x) + 1) + 4*I*log(-sqrt(e^(4*x) + 1) + e^(2*x) + 1)*sgn(-e^(2*x) + 1
) + 4*(3*I*(sqrt(e^(4*x) + 1) - e^(2*x))^3*sgn(-e^(2*x) + 1) + I*(sqrt(e^(4*x) + 1) - e^(2*x))^2*sgn(-e^(2*x)
+ 1) + (-I*sqrt(e^(4*x) + 1) + I*e^(2*x))*sgn(-e^(2*x) + 1) + I*sgn(-e^(2*x) + 1))/((sqrt(e^(4*x) + 1) - e^(2*
x))^2 + 2*sqrt(e^(4*x) + 1) - 2*e^(2*x) - 1)^2)