### 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)*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])]

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Rubi [A]  time = 0.0714839, antiderivative size = 102, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [C]  time = 6.10253, 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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Maple [B]  time = 0.373, size = 314, normalized size = 3.1 \begin{align*}{ \left ( -{b}^{2}+{c}^{2} \right ) \cosh \left ( x \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{{ \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}}+{\frac{a}{\sinh \left ( x \right ) }\sqrt{{ \left ( \sinh \left ( x \right ) \right ) ^{2} \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}\ln \left ({ \left ( \cosh \left ( x \right ) \sinh \left ( x \right ) \left ( -{b}^{2}+{c}^{2} \right ) +\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}a+\sqrt{{ \left ( -{b}^{2}+{c}^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{3}{\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}+a \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{{b}^{2}-{c}^{2}}\sqrt{{ \left ( -{b}^{2}+{c}^{2} \right ) \sinh \left ( x \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}+a} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{{ \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}} \right ) \sqrt{{b}^{2}-{c}^{2}} \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ) ^{-1}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \cosh{\left (x \right )} + c \sinh{\left (x \right )}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}\,{d x} \end{align*}

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)