### 3.590 $$\int \sqrt{a \cosh (x)+b \sinh (x)} \, dx$$

Optimal. Leaf size=65 $-\frac{2 i \sqrt{a \cosh (x)+b \sinh (x)} E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}$

[Out]

((-2*I)*EllipticE[(I*x - ArcTan[a, (-I)*b])/2, 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/Sqrt[(a*Cosh[x] + b*Sinh[x])/Sq
rt[a^2 - b^2]]

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Rubi [A]  time = 0.0273524, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {3078, 2639} $-\frac{2 i \sqrt{a \cosh (x)+b \sinh (x)} E\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[a*Cosh[x] + b*Sinh[x]],x]

[Out]

((-2*I)*EllipticE[(I*x - ArcTan[a, (-I)*b])/2, 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/Sqrt[(a*Cosh[x] + b*Sinh[x])/Sq
rt[a^2 - b^2]]

Rule 3078

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[c + d*x] +
b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n, Int[Cos[c + d*x - ArcTan[a, b]]^n, x]
, x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) &&  !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

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

Mathematica [C]  time = 0.702785, size = 206, normalized size = 3.17 $\frac{b \left (b^2-a^2\right ) \sinh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cosh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )\right )+\sqrt{-\sinh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )} \left (2 a^2 b \sqrt{1-\frac{b^2}{a^2}} \sinh (x)+2 a^3 \sqrt{1-\frac{b^2}{a^2}} \cosh (x)-2 a \left (a^2-b^2\right ) \cosh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )+a^2 b \sinh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )-b^3 \sinh \left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )\right )}{a b \sqrt{1-\frac{b^2}{a^2}} \sqrt{-\sinh ^2\left (\tanh ^{-1}\left (\frac{b}{a}\right )+x\right )} \sqrt{a \cosh (x)+b \sinh (x)}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*(-a^2 + b^2)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cosh[x + ArcTanh[b/a]]^2]*Sinh[x + ArcTanh[b/a]] + Sqrt
[-Sinh[x + ArcTanh[b/a]]^2]*(2*a^3*Sqrt[1 - b^2/a^2]*Cosh[x] - 2*a*(a^2 - b^2)*Cosh[x + ArcTanh[b/a]] + 2*a^2*
b*Sqrt[1 - b^2/a^2]*Sinh[x] + a^2*b*Sinh[x + ArcTanh[b/a]] - b^3*Sinh[x + ArcTanh[b/a]]))/(a*b*Sqrt[1 - b^2/a^
2]*Sqrt[a*Cosh[x] + b*Sinh[x]]*Sqrt[-Sinh[x + ArcTanh[b/a]]^2])

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Maple [A]  time = 0.129, size = 33, normalized size = 0.5 \begin{align*} -{\cosh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}{\frac{1}{\sqrt{-\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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