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

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

[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 = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2646} \[ -\frac {2 a \sinh (c+d x)}{d \sqrt {a-a \cosh (c+d x)}} \]

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 = 30, normalized size = 1.11 \[ \frac {2 \coth \left (\frac {1}{2} (c+d x)\right ) \sqrt {a-a \cosh (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 42, normalized size = 1.56 \[ \frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right )}}{d} \]

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 [B]  time = 0.15, size = 61, normalized size = 2.26 \[ -\frac {\sqrt {2} {\left (\sqrt {-a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + \sqrt {-a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )\right )}}{d} \]

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

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maple [A]  time = 0.22, size = 41, normalized size = 1.52 \[ -\frac {4 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) a \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {-2 a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sinh(1/2*d*x+1/2*c)*a*cosh(1/2*d*x+1/2*c)/(-2*a*sinh(1/2*d*x+1/2*c)^2)^(1/2)/d

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maxima [B]  time = 0.43, size = 58, normalized size = 2.15 \[ -\frac {\sqrt {2} \sqrt {a} e^{\left (-d x - c\right )}}{d \sqrt {-e^{\left (-d x - c\right )}}} - \frac {\sqrt {2} \sqrt {a}}{d \sqrt {-e^{\left (-d x - c\right )}}} \]

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

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mupad [B]  time = 0.94, size = 27, normalized size = 1.00 \[ \frac {2\,\mathrm {coth}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a-a\,\mathrm {cosh}\left (c+d\,x\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*cosh(c + d*x))^(1/2),x)

[Out]

(2*coth(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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