3.48 \(\int (1-\cosh ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{3} \left (-\sinh ^2(x)\right )^{3/2} \coth (x)+\frac{2}{3} \sqrt{-\sinh ^2(x)} \coth (x) \]

[Out]

(2*Coth[x]*Sqrt[-Sinh[x]^2])/3 + (Coth[x]*(-Sinh[x]^2)^(3/2))/3

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Rubi [A]  time = 0.028454, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {3176, 3203, 3207, 2638} \[ \frac{1}{3} \left (-\sinh ^2(x)\right )^{3/2} \coth (x)+\frac{2}{3} \sqrt{-\sinh ^2(x)} \coth (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cosh[x]^2)^(3/2),x]

[Out]

(2*Coth[x]*Sqrt[-Sinh[x]^2])/3 + (Coth[x]*(-Sinh[x]^2)^(3/2))/3

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3203

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^p)/(2*f*p), x]
+ Dist[(b*(2*p - 1))/(2*p), Int[(b*Sin[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] &&  !IntegerQ[p] &&
GtQ[p, 1]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0305086, size = 25, normalized size = 0.76 \[ -\frac{1}{12} \sqrt{-\sinh ^2(x)} (\cosh (3 x)-9 \cosh (x)) \text{csch}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cosh[x]^2)^(3/2),x]

[Out]

-((-9*Cosh[x] + Cosh[3*x])*Csch[x]*Sqrt[-Sinh[x]^2])/12

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Maple [A]  time = 0.101, size = 21, normalized size = 0.6 \begin{align*}{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) \left ( \left ( \sinh \left ( x \right ) \right ) ^{2}-2 \right ) }{3}{\frac{1}{\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sinh(x)*cosh(x)*(sinh(x)^2-2)/(-sinh(x)^2)^(1/2)

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Maxima [C]  time = 2.1126, size = 31, normalized size = 0.94 \begin{align*} \frac{1}{24} i \, e^{\left (3 \, x\right )} - \frac{3}{8} i \, e^{\left (-x\right )} + \frac{1}{24} i \, e^{\left (-3 \, x\right )} - \frac{3}{8} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*I*e^(3*x) - 3/8*I*e^(-x) + 1/24*I*e^(-3*x) - 3/8*I*e^x

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Fricas [A]  time = 2.25603, size = 4, normalized size = 0.12 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.25674, size = 89, normalized size = 2.7 \begin{align*} -\frac{1}{24} i \,{\left (9 \, e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) - \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )\right )} e^{\left (-3 \, x\right )} + \frac{1}{24} i \, e^{\left (3 \, x\right )} \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) - \frac{3}{8} i \, e^{x} \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*I*(9*e^(2*x)*sgn(-e^(3*x) + e^x) - sgn(-e^(3*x) + e^x))*e^(-3*x) + 1/24*I*e^(3*x)*sgn(-e^(3*x) + e^x) -
3/8*I*e^x*sgn(-e^(3*x) + e^x)