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

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

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.38, size = 24, normalized size = 0.92 \[ \frac {2 \, \sqrt {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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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giac [A]  time = 0.12, size = 17, normalized size = 0.65 \[ \frac {2 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d} \]

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

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 = 1.58, size = 15, normalized size = 0.58 \[ \frac {2\,\sqrt {a\,{\mathrm {e}}^{c+d\,x}}}{d} \]

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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