3.1.45 \(\int \frac {1}{\sqrt {a+a \cosh (c+d x)}} \, dx\) [45]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {2728, 212} \begin {gather*} \frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a \cosh (c+d x)+a}}\right )}{\sqrt {a} d} \end {gather*}

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 212

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

Rule 2728

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+a \cosh (c+d x)}} \, dx &=\frac {(2 i) \text {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*}

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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.87 \begin {gather*} \frac {2 \text {ArcTan}\left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh \left (\frac {1}{2} (c+d x)\right )}{d \sqrt {a (1+\cosh (c+d x))}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(102\) vs. \(2(37)=74\).
time = 0.94, size = 103, normalized size = 2.24

method	result	size
default	\(-\frac {\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}\, \ln \left (\frac {2 \sqrt {-a}\, \sqrt {\left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}-2 a}{\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) \sqrt {2}}{\sqrt {-a}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*((-a)^(1/2)*(sinh(1/2*d*x+1/2*c)^2*a)^(1/
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (37) = 74\).
time = 0.53, size = 86, normalized size = 1.87 \begin {gather*} 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 {gather*}

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 = 0.42, size = 149, normalized size = 3.24 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a \cosh {\left (c + d x \right )} + a}}\, dx \end {gather*}

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 = 0.42, size = 21, normalized size = 0.46 \begin {gather*} \frac {2 \, \sqrt {2} \arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{\sqrt {a} d} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*cosh(c + d*x))^(1/2),x)

[Out]

int(1/(a + a*cosh(c + d*x))^(1/2), x)

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