3.42 \(\int \sqrt {1+\cosh ^2(x)} \, dx\)

Optimal. Leaf size=17 \[ -i E\left (\left .i x+\frac {\pi }{2}\right |-1\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3177} \[ -i E\left (\left .i x+\frac {\pi }{2}\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Cosh[x]^2],x]

[Out]

(-I)*EllipticE[Pi/2 + I*x, -1]

Rule 3177

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 18, normalized size = 1.06 \[ -i \sqrt {2} E\left (i x\left |\frac {1}{2}\right .\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Cosh[x]^2],x]

[Out]

(-I)*Sqrt[2]*EllipticE[I*x, 1/2]

________________________________________________________________________________________

fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\cosh \relax (x)^{2} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(cosh(x)^2 + 1), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cosh \relax (x)^{2} + 1}\,{d x} \]

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)

________________________________________________________________________________________

maple [B]  time = 0.35, size = 58, normalized size = 3.41 \[ -\frac {i \sqrt {\left (1+\cosh ^{2}\relax (x )\right ) \left (\sinh ^{2}\relax (x )\right )}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \left (2 \EllipticF \left (i \cosh \relax (x ), i\right )-\EllipticE \left (i \cosh \relax (x ), i\right )\right )}{\sqrt {\cosh ^{4}\relax (x )-1}\, \sinh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cosh(x)^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\cosh \relax (x)^{2} + 1}\,{d x} \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________