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]

-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 = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.53, size = 154, normalized size = 3.21 \[ \left [\frac {\sqrt {2} \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} - \cosh \left (d x + c\right ) - \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}\right )}{d}, \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{\sqrt {a}}\right )}{\sqrt {a} d}\right ] \]

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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giac [A]  time = 0.14, size = 40, normalized size = 0.83 \[ -\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a e^{\left (d x + c\right )}}}{\sqrt {a}}\right )}{\sqrt {a} d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} \]

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))

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maple [A]  time = 0.22, size = 41, normalized size = 0.85 \[ -\frac {2 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \arctanh \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\sqrt {-2 a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a \cosh \left (d x + c\right ) + a}}\,{d x} \]

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- a \cosh {\left (c + d x \right )} + a}}\, dx \]

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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