3.1.31 \(\int (-1-\coth ^2(x))^{3/2} \, dx\) [31]

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

[Out]

-5/2*arctan(coth(x)/(-1-coth(x)^2)^(1/2))+2*arctan(coth(x)*2^(1/2)/(-1-coth(x)^2)^(1/2))*2^(1/2)+1/2*coth(x)*(
-1-coth(x)^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 537, 223, 209, 385} \begin {gather*} -\frac {5}{2} \text {ArcTan}\left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )+2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )+\frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)-1} \end {gather*}

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 209

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

Rule 223

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 385

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]

Rule 427

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 537

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 3742

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])

Rubi steps

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

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Mathematica [A]
time = 0.13, size = 118, normalized size = 1.76 \begin {gather*} -\frac {1}{8} \left (-1-\coth ^2(x)\right )^{3/2} \text {sech}^2(2 x) \left (16 \tanh ^{-1}\left (\frac {\cosh (x)}{\sqrt {\cosh (2 x)}}\right ) \sqrt {\cosh (2 x)} \sinh ^3(x)+4 \left (\text {ArcTan}\left (\frac {\cosh (x)}{\sqrt {-\cosh (2 x)}}\right ) \sqrt {-\cosh (2 x)}-4 \sqrt {2} \sqrt {\cosh (2 x)} \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )\right ) \sinh ^3(x)+\sinh (4 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Coth[x]^2)^(3/2),x]

[Out]

-1/8*((-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*(A
rcTan[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]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(210\) vs. \(2(53)=106\).
time = 0.82, size = 211, normalized size = 3.15

method	result	size
derivativedivides	\(\frac {\left (-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )-\frac {\left (-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )\)	\(211\)
default	\(\frac {\left (-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )-\frac {\left (-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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+2*coth(x))*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*
arctan(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*cot
h(x))*2^(1/2)/(-(coth(x)-1)^2-2*coth(x))^(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-coth(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((-coth(x)^2 - 1)^(3/2), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.39, size = 361, normalized size = 5.39 \begin {gather*} \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 ) - 5 \, {\left (i \, e^{\left (4 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + i\right )} \log \left (-4 \, {\left (i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) - 5 \, {\left (-i \, e^{\left (4 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (-4 \, {\left (-i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\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 {gather*}

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) - 2*I*e^(2*x) + I)*log(-4*(I*sqrt(-2*e^(4*x) - 2) + e^(2*x) + 1)*e^(-2*x)
) - 5*(-I*e^(4*x) + 2*I*e^(2*x) - I)*log(-4*(-I*sqrt(-2*e^(4*x) - 2) + e^(2*x) + 1)*e^(-2*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*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \coth ^{2}{\left (x \right )} - 1\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((-coth(x)**2 - 1)**(3/2), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 285, normalized size = 4.25 \begin {gather*} -\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 {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*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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (-{\mathrm {coth}\left (x\right )}^2-1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- coth(x)^2 - 1)^(3/2),x)

[Out]

int((- coth(x)^2 - 1)^(3/2), x)

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