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

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Rubi [A]  time = 0.0158502, 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} $-\frac{2 \sqrt{a \cosh (c+d x)-a \sinh (c+d x)}}{d}$

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*}

Mathematica [A]  time = 0.0220003, 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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Maple [A]  time = 0., size = 26, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{a\cosh \left ( dx+c \right ) -a\sinh \left ( dx+c \right ) }}{d}} \end{align*}

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 = 1.03767, size = 23, normalized size = 0.85 \begin{align*} -\frac{2 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d} \end{align*}

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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Fricas [A]  time = 2.26464, size = 62, normalized size = 2.3 \begin{align*} -\frac{2 \, \sqrt{\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{d} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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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Giac [A]  time = 1.15519, size = 23, normalized size = 0.85 \begin{align*} -\frac{2 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d} \end{align*}

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