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

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

[Out]

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

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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} \[ -\frac {2 \sqrt {a \cosh (c+d x)-a \sinh (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 24, normalized size = 0.89 \[ -\frac {2 \, \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)))/d

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giac [A]  time = 0.14, size = 17, normalized size = 0.63 \[ -\frac {2 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a)*e^(-1/2*d*x - 1/2*c)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 17, normalized size = 0.63 \[ -\frac {2 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a)*e^(-1/2*d*x - 1/2*c)/d

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mupad [B]  time = 1.55, size = 18, normalized size = 0.67 \[ -\frac {2\,\sqrt {a\,{\mathrm {e}}^{-c-d\,x}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(a*exp(- c - d*x))^(1/2))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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