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

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

[Out]

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

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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 = {3182} \[ -i F\left (\left .i x+\frac {\pi }{2}\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Cosh[x]^2],x]

[Out]

(-I)*EllipticF[Pi/2 + I*x, -1]

Rule 3182

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

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 18, normalized size = 1.06 \[ -\frac {i F\left (i x\left |\frac {1}{2}\right .\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Cosh[x]^2],x]

[Out]

((-I)*EllipticF[I*x, 1/2])/Sqrt[2]

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {\cosh \relax (x)^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(cosh(x)^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(cosh(x)^2 + 1), x)

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maple [B]  time = 0.29, size = 45, normalized size = 2.65 \[ -\frac {i \sqrt {\left (1+\cosh ^{2}\relax (x )\right ) \left (\sinh ^{2}\relax (x )\right )}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \EllipticF \left (i \cosh \relax (x ), i\right )}{\sqrt {\cosh ^{4}\relax (x )-1}\, \sinh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(cosh(x)^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2 + 1)^(1/2),x)

[Out]

int(1/(cosh(x)^2 + 1)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(cosh(x)**2 + 1), x)

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