3.50 \(\int \sqrt{a-a \cosh (c+d x)} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 a \sinh (c+d x)}{d \sqrt{a-a \cosh (c+d x)}} \]

[Out]

(-2*a*Sinh[c + d*x])/(d*Sqrt[a - a*Cosh[c + d*x]])

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Rubi [A]  time = 0.0144789, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2646} \[ -\frac{2 a \sinh (c+d x)}{d \sqrt{a-a \cosh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Cosh[c + d*x]],x]

[Out]

(-2*a*Sinh[c + d*x])/(d*Sqrt[a - a*Cosh[c + d*x]])

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \sqrt{a-a \cosh (c+d x)} \, dx &=-\frac{2 a \sinh (c+d x)}{d \sqrt{a-a \cosh (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0321064, size = 30, normalized size = 1.11 \[ \frac{2 \coth \left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \cosh (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Cosh[c + d*x]],x]

[Out]

(2*Sqrt[a - a*Cosh[c + d*x]]*Coth[(c + d*x)/2])/d

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Maple [A]  time = 0.034, size = 41, normalized size = 1.5 \begin{align*} -4\,{\frac{\sinh \left ( 1/2\,dx+c/2 \right ) a\cosh \left ( 1/2\,dx+c/2 \right ) }{\sqrt{-2\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*cosh(d*x+c))^(1/2),x)

[Out]

-4*sinh(1/2*d*x+1/2*c)*a*cosh(1/2*d*x+1/2*c)/(-2*sinh(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [B]  time = 1.59228, size = 78, normalized size = 2.89 \begin{align*} -\frac{\sqrt{2} \sqrt{a} e^{\left (-d x - c\right )}}{d \sqrt{-e^{\left (-d x - c\right )}}} - \frac{\sqrt{2} \sqrt{a}}{d \sqrt{-e^{\left (-d x - c\right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt(a)*e^(-d*x - c)/(d*sqrt(-e^(-d*x - c))) - sqrt(2)*sqrt(a)/(d*sqrt(-e^(-d*x - c)))

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Fricas [A]  time = 1.74937, size = 124, normalized size = 4.59 \begin{align*} \frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}{\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(1/2)*sqrt(-a/(cosh(d*x + c) + sinh(d*x + c)))*(cosh(d*x + c) + sinh(d*x + c) + 1)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(-a*cosh(c + d*x) + a), x)

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Giac [B]  time = 1.19146, size = 85, normalized size = 3.15 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{-a e^{\left (d x + c\right )}} a \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - \frac{a^{2} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right )}{\sqrt{-a e^{\left (d x + c\right )}}}\right )}}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*(sqrt(-a*e^(d*x + c))*a*sgn(-e^(d*x + c) + 1) - a^2*sgn(-e^(d*x + c) + 1)/sqrt(-a*e^(d*x + c)))/(a*d)