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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2646} \[ \frac {2 a \sinh (c+d x)}{d \sqrt {a \cosh (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

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

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 29, normalized size = 1.12 \[ \frac {2 \tanh \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cosh (c+d x)+1)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a*(1 + Cosh[c + d*x])]*Tanh[(c + d*x)/2])/d

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fricas [A]  time = 0.50, size = 41, normalized size = 1.58 \[ \frac {2 \, \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 35, normalized size = 1.35 \[ \frac {\sqrt {2} {\left (\sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(sqrt(a)*e^(1/2*d*x + 1/2*c) - sqrt(a)*e^(-1/2*d*x - 1/2*c))/d

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maple [A]  time = 0.14, size = 43, normalized size = 1.65 \[ \frac {2 a \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}}{\sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 40, normalized size = 1.54 \[ \frac {\sqrt {2} \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d} - \frac {\sqrt {2} \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sqrt(a)*e^(1/2*d*x + 1/2*c)/d - sqrt(2)*sqrt(a)*e^(-1/2*d*x - 1/2*c)/d

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mupad [B]  time = 0.11, size = 26, normalized size = 1.00 \[ \frac {2\,\mathrm {tanh}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cosh(c + d*x))^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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