3.174 \(\int \sqrt {-1+\text {sech}^2(x)} \, dx\)

Optimal. Leaf size=16 \[ \sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]

[Out]

coth(x)*ln(cosh(x))*(-tanh(x)^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4121, 3658, 3475} \[ \sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Sech[x]^2],x]

[Out]

Coth[x]*Log[Cosh[x]]*Sqrt[-Tanh[x]^2]

Rule 3475

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

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[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 4121

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

Rubi steps

\begin {align*} \int \sqrt {-1+\text {sech}^2(x)} \, dx &=\int \sqrt {-\tanh ^2(x)} \, dx\\ &=\left (\coth (x) \sqrt {-\tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt {-\tanh ^2(x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \sqrt {-\tanh ^2(x)} \coth (x) \log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Sech[x]^2],x]

[Out]

Coth[x]*Log[Cosh[x]]*Sqrt[-Tanh[x]^2]

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fricas [A]  time = 0.51, size = 1, normalized size = 0.06 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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giac [C]  time = 0.13, size = 31, normalized size = 1.94 \[ i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+sech(x)^2)^(1/2),x, algorithm="giac")

[Out]

I*x*sgn(-e^(4*x) + 1) - I*log(e^(2*x) + 1)*sgn(-e^(4*x) + 1)

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maple [B]  time = 0.32, size = 81, normalized size = 5.06 \[ -\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x}{{\mathrm e}^{2 x}-1}+\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+sech(x)^2)^(1/2),x)

[Out]

-1/(exp(2*x)-1)*(1+exp(2*x))*(-(exp(2*x)-1)^2/(1+exp(2*x))^2)^(1/2)*x+1/(exp(2*x)-1)*(1+exp(2*x))*(-(exp(2*x)-
1)^2/(1+exp(2*x))^2)^(1/2)*ln(1+exp(2*x))

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maxima [C]  time = 0.55, size = 13, normalized size = 0.81 \[ -i \, x - i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*x - I*log(e^(-2*x) + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \sqrt {\frac {1}{{\mathrm {cosh}\relax (x)}^2}-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(x)^2 - 1)^(1/2),x)

[Out]

int((1/cosh(x)^2 - 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {sech}^{2}{\relax (x )} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+sech(x)**2)**(1/2),x)

[Out]

Integral(sqrt(sech(x)**2 - 1), x)

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