3.1.89 \(\int \sqrt {-1-\sinh ^2(x)} \, dx\) [89]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 13, 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 = {3255, 3286, 2717} \begin {gather*} \sqrt {-\cosh ^2(x)} \tanh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Sinh[x]^2],x]

[Out]

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3255

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 3286

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
]])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \sqrt {-\cosh ^2(x)} \tanh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Sinh[x]^2],x]

[Out]

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

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Maple [A]
time = 0.76, size = 15, normalized size = 1.15

method	result	size
default	\(-\frac {\sinh \left (x \right ) \cosh \left (x \right )}{\sqrt {-\left (\cosh ^{2}\left (x \right )\right )}}\)	\(15\)
risch	\(\frac {\sqrt {-\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}\, {\mathrm e}^{2 x}}{2+2 \,{\mathrm e}^{2 x}}-\frac {\sqrt {-\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{2 \left (1+{\mathrm e}^{2 x}\right )}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 0.48, size = 25, normalized size = 1.92 \begin {gather*} -\frac {e^{\left (-2 \, x\right )}}{2 \, \sqrt {-e^{\left (-2 \, x\right )}}} + \frac {1}{2 \, \sqrt {-e^{\left (-2 \, x\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-2*x)/sqrt(-e^(-2*x)) + 1/2/sqrt(-e^(-2*x))

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Fricas [C] Result contains complex when optimal does not.
time = 0.41, size = 14, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, {\left (i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(I*e^(2*x) - I)*e^(-x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-sinh(x)**2 - 1), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2} i \, e^{\left (-x\right )} + \frac {1}{2} i \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*e^(-x) + 1/2*I*e^x

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Mupad [B]
time = 0.17, size = 5, normalized size = 0.38 \begin {gather*} \mathrm {sinh}\left (x\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- sinh(x)^2 - 1)^(1/2),x)

[Out]

sinh(x)*1i

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