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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.42, size = 40, normalized size = 1.48 \[ \frac {2 \, \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{a d} \]

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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giac [A]  time = 0.12, size = 17, normalized size = 0.63 \[ \frac {2 \, e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{\sqrt {a} d} \]

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)

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maple [A]  time = 0.02, size = 26, normalized size = 0.96 \[ \frac {2}{d \sqrt {a \cosh \left (d x +c \right )-a \sinh \left (d x +c \right )}} \]

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 = 0.35, size = 17, normalized size = 0.63 \[ \frac {2 \, e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{\sqrt {a} d} \]

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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mupad [B]  time = 0.17, size = 27, normalized size = 1.00 \[ \frac {2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {a\,{\mathrm {e}}^{-c-d\,x}}}{a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- a \sinh {\left (c + d x \right )} + a \cosh {\left (c + d x \right )}}}\, dx \]

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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