3.22 \(\int (-1+\text{csch}^2(x))^{3/2} \, dx\)

Optimal. Leaf size=47 \[ -\frac{1}{2} \coth (x) \sqrt{\coth ^2(x)-2}+\tan ^{-1}\left (\frac{\coth (x)}{\sqrt{\coth ^2(x)-2}}\right )+2 \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{\coth ^2(x)-2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0425849, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4128, 416, 523, 217, 206, 377, 203} \[ -\frac{1}{2} \coth (x) \sqrt{\coth ^2(x)-2}+\tan ^{-1}\left (\frac{\coth (x)}{\sqrt{\coth ^2(x)-2}}\right )+2 \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{\coth ^2(x)-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Csch[x]^2)^(3/2),x]

[Out]

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

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

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 206

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

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])

Rubi steps

\begin{align*} \int \left (-1+\text{csch}^2(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (-2+x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\frac{1}{2} \coth (x) \sqrt{-2+\coth ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-6+4 x^2}{\left (1-x^2\right ) \sqrt{-2+x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac{1}{2} \coth (x) \sqrt{-2+\coth ^2(x)}+2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-2+x^2}} \, dx,x,\coth (x)\right )+\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{-2+x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac{1}{2} \coth (x) \sqrt{-2+\coth ^2(x)}+2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\coth (x)}{\sqrt{-2+\coth ^2(x)}}\right )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\coth (x)}{\sqrt{-2+\coth ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac{\coth (x)}{\sqrt{-2+\coth ^2(x)}}\right )+2 \tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{-2+\coth ^2(x)}}\right )-\frac{1}{2} \coth (x) \sqrt{-2+\coth ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0948409, size = 90, normalized size = 1.91 \[ \frac{\sinh ^3(x) \left (\text{csch}^2(x)-1\right )^{3/2} \left (2 \sqrt{2} \left (\log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)-3}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \cosh (x)}{\sqrt{\cosh (2 x)-3}}\right )\right )+\sqrt{\cosh (2 x)-3} \coth (x) \text{csch}(x)\right )}{(\cosh (2 x)-3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Csch[x]^2)^(3/2),x]

[Out]

((-1 + Csch[x]^2)^(3/2)*(Sqrt[-3 + Cosh[2*x]]*Coth[x]*Csch[x] + 2*Sqrt[2]*(2*ArcTan[(Sqrt[2]*Cosh[x])/Sqrt[-3
+ Cosh[2*x]]] + Log[Sqrt[2]*Cosh[x] + Sqrt[-3 + Cosh[2*x]]]))*Sinh[x]^3)/(-3 + Cosh[2*x])^(3/2)

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int \left ( -1+ \left ({\rm csch} \left (x\right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\operatorname{csch}\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+csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((csch(x)^2 - 1)^(3/2), x)

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Fricas [B]  time = 2.26635, size = 2329, normalized size = 49.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2
*cosh(x)*sinh(x) + sinh(x)^2)) + (cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2
- 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2
- 1)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*cosh(x)*si
nh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 2)*sinh(x)^2 + 4*cosh(x)^2 + 4*(cosh(x)^3 + 2*cosh(x))*sinh(x) - 1)) +
(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - co
sh(x))*sinh(x) + 1)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-(cosh(x)^2 + sinh(x)^
2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 6*(cosh(x)^
2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x))*sinh(x) + 1)) - 2*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3
+ sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)*log((cosh(x)^
2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + sqrt(2)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) +
sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 2*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^
4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)*log((cosh(x)^2 + 2*cosh
(x)*sinh(x) + sinh(x)^2 - sqrt(2)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2
)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh
(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\operatorname{csch}^{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+csch(x)**2)**(3/2),x)

[Out]

Integral((csch(x)**2 - 1)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\operatorname{csch}\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+csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((csch(x)^2 - 1)^(3/2), x)