### 3.593 $$\int \frac{1}{\sqrt{a \cosh (x)+b \sinh (x)}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0268969, 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, 2641} $-\frac{2 i \sqrt{\frac{a \cosh (x)+b \sinh (x)}{\sqrt{a^2-b^2}}} F\left (\left .\frac{1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\sqrt{a \cosh (x)+b \sinh (x)}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[a*Cosh[x] + b*Sinh[x]],x]

[Out]

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

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 2641

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

Rubi steps

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

Mathematica [C]  time = 0.0926322, size = 81, normalized size = 1.25 $\frac{2 \sqrt{a \cosh (x)+b \sinh (x)} \sqrt{\cosh ^2\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right )} \text{sech}\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},-\sinh ^2\left (\tanh ^{-1}\left (\frac{a}{b}\right )+x\right )\right )}{b \sqrt{1-\frac{a^2}{b^2}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[Cosh[x + ArcTanh[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, -Sinh[x + ArcTanh[a/b]]^2]*Sech[x + Arc
Tanh[a/b]]*Sqrt[a*Cosh[x] + b*Sinh[x]])/(Sqrt[1 - a^2/b^2]*b)

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Maple [A]  time = 0.19, size = 97, normalized size = 1.5 \begin{align*}{\frac{1}{\sinh \left ( x \right ) }\sqrt{-\sqrt{{a}^{2}-{b}^{2}} \left ( \sinh \left ( x \right ) \right ) ^{3}}\arctan \left ({\cosh \left ( x \right ) \sqrt{\sinh \left ( x \right ) \sqrt{{a}^{2}-{b}^{2}}}{\frac{1}{\sqrt{-\sqrt{{a}^{2}-{b}^{2}} \left ( \sinh \left ( x \right ) \right ) ^{3}}}}} \right ){\frac{1}{\sqrt{\sinh \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(1/(a*cosh(x)+b*sinh(x))^(1/2),x)

[Out]

(-(a^2-b^2)^(1/2)*sinh(x)^3)^(1/2)/(sinh(x)*(a^2-b^2)^(1/2))^(1/2)*arctan((sinh(x)*(a^2-b^2)^(1/2))^(1/2)*cosh
(x)/(-(a^2-b^2)^(1/2)*sinh(x)^3)^(1/2))/sinh(x)/(-sinh(x)*(a^2-b^2)^(1/2))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(a*cosh(x)+b*sinh(x))^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(a*cosh(x)+b*sinh(x))**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(a*cosh(x)+b*sinh(x))^(1/2),x, algorithm="giac")

[Out]

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