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

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

[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))*EllipticE(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))^(1/2)/((a+b*cosh(x)+c*
sinh(x))/(a+(b^2-c^2)^(1/2)))^(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 = {3119, 2653} \[ -\frac {2 i \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2653

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

Rule 3119

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[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] /; FreeQ[{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 \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx &=\frac {\sqrt {a+b \cosh (x)+c \sinh (x)} \int \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 {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\\ &=-\frac {2 i E\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)}}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}\\ \end {align*}

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Mathematica [C]  time = 6.11, size = 1401, normalized size = 13.74 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*b*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/c + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sin
h[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*
Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c)]*Sech[x + ArcTanh[b/c]]*Sqrt
[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] - I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b
/c]])/(I*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + A
rcTanh[b/c]])/((-I)*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/
(Sqrt[1 - b^2/c^2]*c*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (b^2*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*S
qrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 -
c^2/b^2]*Cosh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])
/(b*Sqrt[1 - c^2/b^2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*S
qrt[(b^2 - c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2]
+ b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^
2/b^2]*Cosh[x + ArcTanh[c/b]]))/(b^2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqr
t[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]]]))/c + c*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Co
sh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + A
rcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^
2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2
])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2
)/b^2]*Cosh[x + ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + Arc
Tanh[c/b]]))/(b^2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh
[x + ArcTanh[c/b]]])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.80, size = 314, normalized size = 3.08 \[ \frac {\left (-b^{2}+c^{2}\right ) \cosh \relax (x )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \relax (x ) b^{2}+\sinh \relax (x ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {\frac {\left (-\sinh \relax (x ) b^{2}+\sinh \relax (x ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\relax (x )\right )}{\sqrt {b^{2}-c^{2}}}}\, a \ln \left (\frac {\cosh \relax (x ) \sinh \relax (x ) \left (-b^{2}+c^{2}\right )+\cosh \relax (x ) \sqrt {b^{2}-c^{2}}\, a +\sqrt {\frac {\left (-b^{2}+c^{2}\right ) \left (\sinh ^{3}\relax (x )\right )}{\sqrt {b^{2}-c^{2}}}+a \left (\sinh ^{2}\relax (x )\right )}\, \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \relax (x )}{\sqrt {b^{2}-c^{2}}}+a}}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \relax (x ) b^{2}+\sinh \relax (x ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\right ) \sqrt {b^{2}-c^{2}}}{\left (-\sinh \relax (x ) b^{2}+\sinh \relax (x ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b^2+c^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)*cosh(x)+((-sin
h(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)*a*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 \[ \int \sqrt {b \cosh \relax (x) + c \sinh \relax (x) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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 \[ \int \sqrt {a + b \cosh {\relax (x )} + c \sinh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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