3.2.71 \(\int \sqrt {1-\text {sech}^2(x)} \, dx\) [171]

Optimal. Leaf size=14 \[ \coth (x) \log (\cosh (x)) \sqrt {\tanh ^2(x)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4206, 3739, 3556} \begin {gather*} \sqrt {\tanh ^2(x)} \coth (x) \log (\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sech[x]^2],x]

[Out]

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

Rule 3556

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

Rule 3739

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 4206

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 = 14, normalized size = 1.00 \begin {gather*} \coth (x) \log (\cosh (x)) \sqrt {\tanh ^2(x)} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(12)=24\).
time = 1.50, size = 79, normalized size = 5.64

method	result	size
risch	\(-\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, x}{{\mathrm e}^{2 x}-1}+\frac {\left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [A]
time = 0.48, size = 13, normalized size = 0.93 \begin {gather*} -x - \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(e^(-2*x) + 1)

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Fricas [A]
time = 0.36, size = 18, normalized size = 1.29 \begin {gather*} -x + \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(2*cosh(x)/(cosh(x) - sinh(x)))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {1 - \operatorname {sech}^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 - sech(x)**2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.39, size = 26, normalized size = 1.86 \begin {gather*} -x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \sqrt {1-\frac {1}{{\mathrm {cosh}\left (x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 1/cosh(x)^2)^(1/2),x)

[Out]

int((1 - 1/cosh(x)^2)^(1/2), x)

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