3.1.45 \(\int \sqrt {-1-\cosh ^2(x)} \, dx\) [45]

Optimal. Leaf size=39 \[ -\frac {i \sqrt {-1-\cosh ^2(x)} E\left (\left .\frac {\pi }{2}+i x\right |-1\right )}{\sqrt {1+\cosh ^2(x)}} \]

[Out]

(-sinh(x)^2)^(1/2)/sinh(x)*EllipticE(cosh(x),I)*(-1-cosh(x)^2)^(1/2)/(1+cosh(x)^2)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3257, 3256} \begin {gather*} -\frac {i \sqrt {-\cosh ^2(x)-1} E\left (\left .i x+\frac {\pi }{2}\right |-1\right )}{\sqrt {\cosh ^2(x)+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Cosh[x]^2],x]

[Out]

((-I)*Sqrt[-1 - Cosh[x]^2]*EllipticE[Pi/2 + I*x, -1])/Sqrt[1 + Cosh[x]^2]

Rule 3256

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]/f)*EllipticE[e + f*x, -b/a], x] /
; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3257

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin
[e + f*x]^2/a)], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \sqrt {-1-\cosh ^2(x)} \, dx &=\frac {\sqrt {-1-\cosh ^2(x)} \int \sqrt {1+\cosh ^2(x)} \, dx}{\sqrt {1+\cosh ^2(x)}}\\ &=-\frac {i \sqrt {-1-\cosh ^2(x)} E\left (\left .\frac {\pi }{2}+i x\right |-1\right )}{\sqrt {1+\cosh ^2(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 40, normalized size = 1.03 \begin {gather*} \frac {i \sqrt {2} \sqrt {3+\cosh (2 x)} E\left (i x\left |\frac {1}{2}\right .\right )}{\sqrt {-3-\cosh (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Cosh[x]^2],x]

[Out]

(I*Sqrt[2]*Sqrt[3 + Cosh[2*x]]*EllipticE[I*x, 1/2])/Sqrt[-3 - Cosh[2*x]]

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Maple [A]
time = 0.83, size = 62, normalized size = 1.59

method	result	size
default	\(-\frac {\sqrt {-\left (\cosh ^{2}\left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \sqrt {-\left (\sinh ^{2}\left (x \right )\right )}\, \sqrt {\cosh ^{2}\left (x \right )+1}\, \EllipticE \left (\cosh \left (x \right ), i\right )}{\sqrt {1-\left (\cosh ^{4}\left (x \right )\right )}\, \sinh \left (x \right ) \sqrt {-1-\left (\cosh ^{2}\left (x \right )\right )}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-(cosh(x)^2+1)*sinh(x)^2)^(1/2)*(-sinh(x)^2)^(1/2)*(cosh(x)^2+1)^(1/2)*EllipticE(cosh(x),I)/(1-cosh(x)^4)^(1
/2)/sinh(x)/(-1-cosh(x)^2)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-cosh(x)^2 - 1), x)

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Fricas [F]
time = 0.07, size = 108, normalized size = 2.77 \begin {gather*} \frac {2 \, {\left (e^{\left (2 \, x\right )} - e^{x}\right )} {\rm integral}\left (\frac {4 \, \sqrt {-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1} {\left (e^{\left (2 \, x\right )} + 1\right )}}{e^{\left (6 \, x\right )} - 2 \, e^{\left (5 \, x\right )} + 7 \, e^{\left (4 \, x\right )} - 12 \, e^{\left (3 \, x\right )} + 7 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1}, x\right ) + \sqrt {-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1} {\left (e^{x} + 1\right )}}{2 \, {\left (e^{\left (2 \, x\right )} - e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(e^(2*x) - e^x)*integral(4*sqrt(-e^(4*x) - 6*e^(2*x) - 1)*(e^(2*x) + 1)/(e^(6*x) - 2*e^(5*x) + 7*e^(4*x
) - 12*e^(3*x) + 7*e^(2*x) - 2*e^x + 1), x) + sqrt(-e^(4*x) - 6*e^(2*x) - 1)*(e^x + 1))/(e^(2*x) - e^x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-cosh(x)**2 - 1), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-cosh(x)^2 - 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {-{\mathrm {cosh}\left (x\right )}^2-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- cosh(x)^2 - 1)^(1/2),x)

[Out]

int((- cosh(x)^2 - 1)^(1/2), x)

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