3.1.26 \(\int \sqrt {-1-\text {csch}^2(x)} \, dx\) [26]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 16, 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*} \tanh (x) \sqrt {-\coth ^2(x)} \log (\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Csch[x]^2],x]

[Out]

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

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

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

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Csch[x]^2],x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(14)=28\).
time = 1.52, size = 81, normalized size = 5.06

method	result	size
risch	\(-\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, x}{1+{\mathrm e}^{2 x}}+\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 22, normalized size = 1.38 \begin {gather*} -i \, x - i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*x - I*log(e^(-x) + 1) - I*log(e^(-x) - 1)

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Fricas [C] Result contains complex when optimal does not.
time = 0.38, size = 13, normalized size = 0.81 \begin {gather*} -i \, x + i \, \log \left (e^{\left (2 \, x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*x + I*log(e^(2*x) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-csch(x)**2 - 1), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.39, size = 32, normalized size = 2.00 \begin {gather*} i \, x \mathrm {sgn}\left (-e^{\left (4 \, x\right )} + 1\right ) - i \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\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-csch(x)^2)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \sqrt {-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- 1/sinh(x)^2 - 1)^(1/2),x)

[Out]

int((- 1/sinh(x)^2 - 1)^(1/2), x)

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