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

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

[Out]

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

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Rubi [A]  time = 0.0161512, antiderivative size = 26, normalized size of antiderivative = 1., 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} \[ \frac{2 \sqrt{a \sinh (c+d x)+a \cosh (c+d x)}}{d} \]

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 = 0.0223419, size = 24, normalized size = 0.92 \[ \frac{2 \sqrt{a (\sinh (c+d x)+\cosh (c+d x))}}{d} \]

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 = 0.002, size = 25, 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 = 1.01627, size = 23, normalized size = 0.88 \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 = 2.00343, size = 61, normalized size = 2.35 \begin{align*} \frac{2 \, \sqrt{a \cosh \left (d x + c\right ) + a \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) + a*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 = 1.16006, size = 23, normalized size = 0.88 \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