3.43 \(\int \sqrt{1-\cosh ^2(x)} \, dx\)

Optimal. Leaf size=13 \[ \sqrt{-\sinh ^2(x)} \coth (x) \]

[Out]

Coth[x]*Sqrt[-Sinh[x]^2]

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Rubi [A]  time = 0.0218042, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3176, 3207, 2638} \[ \sqrt{-\sinh ^2(x)} \coth (x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Cosh[x]^2],x]

[Out]

Coth[x]*Sqrt[-Sinh[x]^2]

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

Mathematica [A]  time = 0.0046484, size = 13, normalized size = 1. \[ \sqrt{-\sinh ^2(x)} \coth (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Cosh[x]^2],x]

[Out]

Coth[x]*Sqrt[-Sinh[x]^2]

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Maple [A]  time = 0.073, size = 15, normalized size = 1.2 \begin{align*} -{\cosh \left ( x \right ) \sinh \left ( x \right ){\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)^(1/2),x)

[Out]

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

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Maxima [C]  time = 1.64138, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{2} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*e^(-x) - 1/2*I*e^x

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Fricas [A]  time = 2.20031, size = 4, normalized size = 0.31 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - cosh(x)**2), x)

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Giac [C]  time = 1.21349, size = 42, normalized size = 3.23 \begin{align*} -\frac{1}{2} i \, e^{\left (-x\right )} \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) - \frac{1}{2} 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)^(1/2),x, algorithm="giac")

[Out]

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