3.45 \(\int \frac{1}{\sqrt{a+a \cosh (c+d x)}} \, dx\)

Optimal. Leaf size=46 \[ \frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a \cosh (c+d x)+a}}\right )}{\sqrt{a} d} \]

[Out]

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

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Rubi [A]  time = 0.0225107, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2649, 206} \[ \frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a \cosh (c+d x)+a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

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

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+a \cosh (c+d x)}} \, dx &=\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{i a \sinh (c+d x)}{\sqrt{a+a \cosh (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a+a \cosh (c+d x)}}\right )}{\sqrt{a} d}\\ \end{align*}

Mathematica [A]  time = 0.0154185, size = 40, normalized size = 0.87 \[ \frac{2 \cosh \left (\frac{1}{2} (c+d x)\right ) \tan ^{-1}\left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )}{d \sqrt{a (\cosh (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

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

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Maple [B]  time = 0.036, size = 103, normalized size = 2.2 \begin{align*} -{\frac{\sqrt{2}}{d}\cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sqrt{ \left ( \sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}\ln \left ( 2\,{\frac{\sqrt{ \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}\sqrt{-a}-a}{\cosh \left ( 1/2\,dx+c/2 \right ) }} \right ){\frac{1}{\sqrt{-a}}} \left ( \sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cosh(1/2*d*x+1/2*c)*(sinh(1/2*d*x+1/2*c)^2*a)^(1/2)/(-a)^(1/2)*ln(2/cosh(1/2*d*x+1/2*c)*((sinh(1/2*d*x+1/2*c)
^2*a)^(1/2)*(-a)^(1/2)-a))/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 [B]  time = 1.83332, size = 116, normalized size = 2.52 \begin{align*} 2 \, \sqrt{2}{\left (\frac{\arctan \left (e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}\right )}{\sqrt{a} d} + \frac{e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{{\left (\sqrt{a} e^{\left (d x + c\right )} + \sqrt{a}\right )} d}\right )} - \frac{2 \, \sqrt{2} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{\sqrt{a} d e^{\left (d x + c\right )} + \sqrt{a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.93087, size = 456, normalized size = 9.91 \begin{align*} \left [\frac{\sqrt{2} \sqrt{-\frac{1}{a}} \log \left (-\frac{2 \, \sqrt{2} \sqrt{\frac{1}{2}} \sqrt{\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} \sqrt{-\frac{1}{a}}{\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + \cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1}\right )}{d}, \frac{2 \, \sqrt{2} \arctan \left (\frac{\sqrt{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 )\right )}}{\sqrt{a}}\right )}{\sqrt{a} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(2)*sqrt(-1/a)*log(-(2*sqrt(2)*sqrt(1/2)*sqrt(a/(cosh(d*x + c) + sinh(d*x + c)))*sqrt(-1/a)*(cosh(d*x + c
) + sinh(d*x + c)) + cosh(d*x + c) + sinh(d*x + c) - 1)/(cosh(d*x + c) + sinh(d*x + c) + 1))/d, 2*sqrt(2)*arct
an(sqrt(2)*sqrt(1/2)*sqrt(a/(cosh(d*x + c) + sinh(d*x + c)))*(cosh(d*x + c) + sinh(d*x + c))/sqrt(a))/(sqrt(a)
*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \cosh{\left (c + d x \right )} + a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*cosh(c + d*x) + a), x)

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Giac [A]  time = 1.19836, size = 28, normalized size = 0.61 \begin{align*} \frac{2 \, \sqrt{2} \arctan \left (e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}\right )}{\sqrt{a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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