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

Optimal. Leaf size=26 \[ -\frac {2}{d \sqrt {a \sinh (c+d x)+a \cosh (c+d x)}} \]

[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 = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3071} \[ -\frac {2}{d \sqrt {a \sinh (c+d x)+a \cosh (c+d x)}} \]

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 = 0.04, size = 24, normalized size = 0.92 \[ -\frac {2}{d \sqrt {a (\sinh (c+d x)+\cosh (c+d x))}} \]

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 = 0.40, size = 42, normalized size = 1.62 \[ -\frac {2 \, \sqrt {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )} \]

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

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giac [A]  time = 0.13, size = 17, normalized size = 0.65 \[ -\frac {2 \, e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{\sqrt {a} d} \]

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 = 25, 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 = 0.36, size = 17, normalized size = 0.65 \[ -\frac {2 \, e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{\sqrt {a} d} \]

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 = 1.57, size = 15, normalized size = 0.58 \[ -\frac {2}{d\,\sqrt {a\,{\mathrm {e}}^{c+d\,x}}} \]

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/(d*(a*exp(c + d*x))^(1/2))

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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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