\(\int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx\) [763]

Optimal result

Integrand size = 14, antiderivative size = 102 \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=-\frac {2 i E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]

[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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3198, 2732} \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=-\frac {2 i \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{\sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}} \]

[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 2732

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

Rule 3198

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]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], 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*} \text {integral}& = \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*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.09 (sec) , antiderivative size = 1401, normalized size of antiderivative = 13.74 \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\frac {2 b \sqrt {a+b \cosh (x)+c \sinh (x)}}{c}+\frac {2 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},-\frac {i \left (a+\sqrt {1-\frac {b^2}{c^2}} c \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )\right )}{\sqrt {1-\frac {b^2}{c^2}} \left (1-\frac {i a}{\sqrt {1-\frac {b^2}{c^2}} c}\right ) c},-\frac {i \left (a+\sqrt {1-\frac {b^2}{c^2}} c \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )\right )}{\sqrt {1-\frac {b^2}{c^2}} \left (-1-\frac {i a}{\sqrt {1-\frac {b^2}{c^2}} c}\right ) c}\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {b}{c}\right )\right ) \sqrt {-1+i \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )} \sqrt {\frac {c \sqrt {\frac {-b^2+c^2}{c^2}}-i c \sqrt {\frac {-b^2+c^2}{c^2}} \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )}{i a+c \sqrt {\frac {-b^2+c^2}{c^2}}}} \sqrt {\frac {c \sqrt {\frac {-b^2+c^2}{c^2}}+i c \sqrt {\frac {-b^2+c^2}{c^2}} \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )}{-i a+c \sqrt {\frac {-b^2+c^2}{c^2}}}} \sqrt {a+c \sqrt {\frac {-b^2+c^2}{c^2}} \sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )}}{\sqrt {1-\frac {b^2}{c^2}} c \sqrt {i \left (i+\sinh \left (x+\text {arctanh}\left (\frac {b}{c}\right )\right )\right )}}-\frac {b^2 \left (\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (-1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}-b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2-c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2-c^2}{b^2}}}}}-\frac {-\frac {2 b \left (a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )\right )}{b^2-c^2}+\frac {c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}}\right )}{c}+c \left (\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )},\frac {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \left (-1+\frac {a}{b \sqrt {1-\frac {c^2}{b^2}}}\right )}\right ) \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}-b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{a+b \sqrt {\frac {b^2-c^2}{b^2}}}} \sqrt {a+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )} \sqrt {\frac {b \sqrt {\frac {b^2-c^2}{b^2}}+b \sqrt {\frac {b^2-c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{-a+b \sqrt {\frac {b^2-c^2}{b^2}}}}}-\frac {-\frac {2 b \left (a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )\right )}{b^2-c^2}+\frac {c \sinh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}{b \sqrt {1-\frac {c^2}{b^2}}}}{\sqrt {a+b \sqrt {1-\frac {c^2}{b^2}} \cosh \left (x+\text {arctanh}\left (\frac {c}{b}\right )\right )}}\right ) \]

[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]]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(316\) vs. \(2(125)=250\).

Time = 1.82 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.11

method	result	size
default	\(\frac {\left (-b^{2}+c^{2}\right ) \cosh \left (x \right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {\frac {\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )^{2}}{\sqrt {b^{2}-c^{2}}}}\, a \ln \left (\frac {-\sinh \left (x \right ) \cosh \left (x \right ) b^{2}+\sinh \left (x \right ) \cosh \left (x \right ) c^{2}+\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, a +\sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )^{3}}{\sqrt {b^{2}-c^{2}}}+a \sinh \left (x \right )^{2}}\, \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )}{\sqrt {b^{2}-c^{2}}}+a}}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\right ) \sqrt {b^{2}-c^{2}}}{\left (-b^{2} \sinh \left (x \right )+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )}\)	\(317\)
risch	\(\text {Expression too large to display}\)	\(1093\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.08 \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\frac {2 \, {\left (\sqrt {2} a \sqrt {b + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right ) - 3 \, \sqrt {2} {\left (b + c\right )}^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right )\right ) - 3 \, \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a} {\left (b + c\right )}\right )}}{3 \, {\left (b + c\right )}} \]

[In]

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

[Out]

2/3*(sqrt(2)*a*sqrt(b + c)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 -
9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x) + 3*(b + c)*sinh(x) + 2*a)/(b + c)
) - 3*sqrt(2)*(b + c)^(3/2)*weierstrassZeta(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*
a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c
+ c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x) + 3*(b + c)
*sinh(x) + 2*a)/(b + c))) - 3*sqrt(b*cosh(x) + c*sinh(x) + a)*(b + c))/(b + c)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\int { \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a} \,d x } \]

[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)

Giac [F]

\[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\int { \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx=\int \sqrt {a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )} \,d x \]

[In]

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

[Out]

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