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

________________________________________________________________________________________

Rubi [A]  time = 0.013316, antiderivative size = 26, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0302019, 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

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a\cosh \left ( 1/2\,dx+c/2 \right ) \sinh \left ( 1/2\,dx+c/2 \right ) \sqrt{2}}{\sqrt{a \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}d}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.78412, size = 54, normalized size = 2.08 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.85192, size = 123, normalized size = 4.73 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cosh{\left (c + d x \right )} + a}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.15514, size = 47, normalized size = 1.81 \begin{align*} \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} \end{align*}

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