3.11 \(\int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{2 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{b} \]

[Out]

((-2*I)*EllipticF[(I/2)*(a + b*x), 2])/b

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Rubi [A]  time = 0.0086202, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2641} \[ -\frac{2 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Cosh[a + b*x]],x]

[Out]

((-2*I)*EllipticF[(I/2)*(a + b*x), 2])/b

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0257158, size = 20, normalized size = 1. \[ -\frac{2 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Cosh[a + b*x]],x]

[Out]

((-2*I)*EllipticF[(I/2)*(a + b*x), 2])/b

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Maple [B]  time = 0.036, size = 135, normalized size = 6.8 \begin{align*} 2\,{\frac{\sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) }{\sqrt{2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sinh \left ( 1/2\,bx+a/2 \right ) \sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cosh(b*x+a)^(1/2),x)

[Out]

2*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/
2*a)^2+1)^(1/2)/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))/s
inh(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cosh \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(cosh(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{\cosh \left (b x + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(cosh(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cosh{\left (a + b x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(cosh(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cosh \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(cosh(b*x + a)), x)