3.764 \(\int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx\)

Optimal. Leaf size=102 \[ -\frac {2 i \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \]

[Out]

-2*I*(cos(1/2*I*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*EllipticF(sin(1/2*I*x-1/2*arcta
n(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1/2)))^(1/2))*((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^
(1/2)/(a+b*cosh(x)+c*sinh(x))^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 102, 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 = {3127, 2661} \[ -\frac {2 i \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Cosh[x] + c*Sinh[x]],x]

[Out]

((-2*I)*EllipticF[(I*x - ArcTan[b, (-I)*c])/2, (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x]
+ c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/Sqrt[a + b*Cosh[x] + c*Sinh[x]]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3127

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rubi steps

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

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Mathematica [C]  time = 0.52, size = 237, normalized size = 2.32 \[ \frac {2 \text {sech}\left (\tanh ^{-1}\left (\frac {b}{c}\right )+x\right ) \sqrt {a+b \cosh (x)+c \sinh (x)} \sqrt {-\frac {-i c \sqrt {1-\frac {b^2}{c^2}}+b \cosh (x)+c \sinh (x)}{a+i c \sqrt {1-\frac {b^2}{c^2}}}} \sqrt {-\frac {i c \sqrt {1-\frac {b^2}{c^2}}+b \cosh (x)+c \sinh (x)}{a-i c \sqrt {1-\frac {b^2}{c^2}}}} F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};\frac {a+b \cosh (x)+c \sinh (x)}{a+i \sqrt {1-\frac {b^2}{c^2}} c},\frac {a+b \cosh (x)+c \sinh (x)}{a-i \sqrt {1-\frac {b^2}{c^2}} c}\right )}{c \sqrt {1-\frac {b^2}{c^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*AppellF1[1/2, 1/2, 1/2, 3/2, (a + b*Cosh[x] + c*Sinh[x])/(a + I*Sqrt[1 - b^2/c^2]*c), (a + b*Cosh[x] + c*Si
nh[x])/(a - I*Sqrt[1 - b^2/c^2]*c)]*Sech[x + ArcTanh[b/c]]*Sqrt[a + b*Cosh[x] + c*Sinh[x]]*Sqrt[-(((-I)*Sqrt[1
- b^2/c^2]*c + b*Cosh[x] + c*Sinh[x])/(a + I*Sqrt[1 - b^2/c^2]*c))]*Sqrt[-((I*Sqrt[1 - b^2/c^2]*c + b*Cosh[x]
+ c*Sinh[x])/(a - I*Sqrt[1 - b^2/c^2]*c))])/(Sqrt[1 - b^2/c^2]*c)

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \cosh \relax (x) + c \sinh \relax (x) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(b*cosh(x) + c*sinh(x) + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \cosh \relax (x) + c \sinh \relax (x) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*cosh(x) + c*sinh(x) + a), x)

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maple [A]  time = 0.64, size = 248, normalized size = 2.43 \[ \frac {\sqrt {\frac {\left (-\sinh \relax (x ) b^{2}+\sinh \relax (x ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\relax (x )\right )}{\sqrt {b^{2}-c^{2}}}}\, \ln \left (\frac {\cosh \relax (x ) \sinh \relax (x ) \left (-b^{2}+c^{2}\right )+\cosh \relax (x ) \sqrt {b^{2}-c^{2}}\, a +\sqrt {\frac {\left (-b^{2}+c^{2}\right ) \left (\sinh ^{3}\relax (x )\right )}{\sqrt {b^{2}-c^{2}}}+a \left (\sinh ^{2}\relax (x )\right )}\, \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \relax (x )}{\sqrt {b^{2}-c^{2}}}+a}}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \relax (x ) b^{2}+\sinh \relax (x ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\right ) \sqrt {b^{2}-c^{2}}}{\left (-\sinh \relax (x ) b^{2}+\sinh \relax (x ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-sinh(x)*b^2+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*ln((cosh(x)*sinh(x)*(-b^2+c^2)+
cosh(x)*(b^2-c^2)^(1/2)*a+((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(1/2)*((-b^2+c^2)
/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))/(b^2-c^2)^(1/2)/((-sinh(x)*b^2+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/
2))^(1/2))/(-sinh(x)*b^2+sinh(x)*c^2+a*(b^2-c^2)^(1/2))*(b^2-c^2)^(1/2)/sinh(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*cosh(x) + c*sinh(x) + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\mathrm {cosh}\relax (x)+c\,\mathrm {sinh}\relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \cosh {\relax (x )} + c \sinh {\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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