3.613 \(\int \frac{1}{\sqrt{a \cosh (c+d x)-a \sinh (c+d x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.016832, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3071} \[ \frac{2}{d \sqrt{a \cosh (c+d x)-a \sinh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]],x]

[Out]

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

Rule 3071

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \cosh (c+d x)-a \sinh (c+d x)}} \, dx &=\frac{2}{d \sqrt{a \cosh (c+d x)-a \sinh (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0289077, size = 26, normalized size = 0.96 \[ \frac{2}{d \sqrt{a (\cosh (c+d x)-\sinh (c+d x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]],x]

[Out]

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

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Maple [A]  time = 0.001, size = 26, normalized size = 1. \begin{align*} 2\,{\frac{1}{d\sqrt{a\cosh \left ( dx+c \right ) -a\sinh \left ( dx+c \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02596, size = 23, normalized size = 0.85 \begin{align*} \frac{2 \, e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{\sqrt{a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d)

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Fricas [A]  time = 2.34977, size = 109, normalized size = 4.04 \begin{align*} \frac{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 )\right )}}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15991, size = 23, normalized size = 0.85 \begin{align*} \frac{2 \, e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{\sqrt{a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d)