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

Optimal. Leaf size=48 \[ -\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a-a \cosh (c+d x)}}\right )}{\sqrt{a} d} \]

[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 = 0.0254998, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2649, 206} \[ -\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a-a \cosh (c+d x)}}\right )}{\sqrt{a} d} \]

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 = 0.0300706, size = 41, normalized size = 0.85 \[ \frac{2 \sinh \left (\frac{1}{2} (c+d x)\right ) \log \left (\tanh \left (\frac{1}{4} (c+d x)\right )\right )}{d \sqrt{a-a \cosh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Log[Tanh[(c + d*x)/4]]*Sinh[(c + d*x)/2])/(d*Sqrt[a - a*Cosh[c + d*x]])

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Maple [A]  time = 0.036, size = 41, normalized size = 0.9 \begin{align*} -2\,{\frac{\sinh \left ( 1/2\,dx+c/2 \right ){\it Artanh} \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \right ) }{\sqrt{-2\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}d}} \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]

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

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

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

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Fricas [A]  time = 1.9059, size = 458, normalized size = 9.54 \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 = 1.20268, size = 54, normalized size = 1.12 \begin{align*} -\frac{2 \, \sqrt{2} \arctan \left (\frac{\sqrt{-a e^{\left (d x + c\right )}}}{\sqrt{a}}\right )}{\sqrt{a} d \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\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="giac")

[Out]

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