3.776 \(\int \sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 (b \sinh (x)+c \cosh (x))}{\sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

[Out]

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

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Rubi [A]  time = 0.036615, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3112} \[ \frac{2 (b \sinh (x)+c \cosh (x))}{\sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]],x]

[Out]

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

Rule 3112

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Simp[(-2*(c*Cos[d
+ e*x] - b*Sin[d + e*x]))/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]), x] /; FreeQ[{a, b, c, d, e}, x] && E
qQ[a^2 - b^2 - c^2, 0]

Rubi steps

\begin{align*} \int \sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx &=\frac{2 (c \cosh (x)+b \sinh (x))}{\sqrt{-\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)}}\\ \end{align*}

Mathematica [C]  time = 73.3303, size = 9771, normalized size = 250.54 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]],x]

[Out]

Result too large to show

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Maple [B]  time = 0.444, size = 202, normalized size = 5.2 \begin{align*}{ \left ( -{b}^{2}+{c}^{2} \right ) \cosh \left ( x \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{-{(\sinh \left ( x \right ){b}^{2}-\sinh \left ( x \right ){c}^{2}+{b}^{2}-{c}^{2}){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}}-{\frac{1}{\sinh \left ( x \right ) }\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) +1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\arctan \left ({\cosh \left ( x \right ) \sqrt{\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) +1 \right ) }{\frac{1}{\sqrt{-\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) +1 \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \right ) \sqrt{{b}^{2}-{c}^{2}}{\frac{1}{\sqrt{\sqrt{{b}^{2}-{c}^{2}} \left ( \sinh \left ( x \right ) +1 \right ) }}}{\frac{1}{\sqrt{-{(\sinh \left ( x \right ){b}^{2}-\sinh \left ( x \right ){c}^{2}+{b}^{2}-{c}^{2}){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.12198, size = 211, normalized size = 5.41 \begin{align*} -\frac{\sqrt{2} \sqrt{-2 \, \sqrt{b + c} \sqrt{b - c} e^{\left (-x\right )} +{\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c} \sqrt{b + c} \sqrt{b - c} e^{\left (\frac{1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} - \sqrt{b + c} \sqrt{b - c}} - \frac{\sqrt{2} \sqrt{-2 \, \sqrt{b + c} \sqrt{b - c} e^{\left (-x\right )} +{\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c}{\left (b - c\right )} e^{\left (-\frac{1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} - \sqrt{b + c} \sqrt{b - c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt(-2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x) + b + c)*sqrt(b + c)*sqrt(b - c)*e^(1/2*x)/
((b - c)*e^(-x) - sqrt(b + c)*sqrt(b - c)) - sqrt(2)*sqrt(-2*sqrt(b + c)*sqrt(b - c)*e^(-x) + (b - c)*e^(-2*x)
+ b + c)*(b - c)*e^(-1/2*x)/((b - c)*e^(-x) - sqrt(b + c)*sqrt(b - c))

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Fricas [B]  time = 2.61099, size = 468, normalized size = 12. \begin{align*} \frac{2 \, \sqrt{\frac{1}{2}}{\left ({\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{b^{2} - c^{2}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c\right )} \sqrt{\frac{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{b^{2} - c^{2}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + b - c}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{{\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right )^{2} - b + c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(1/2)*((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 + 2*sqrt(b^2 - c^2)*(cosh(x) +
sinh(x)) + b - c)*sqrt(((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - 2*sqrt(b^2 - c^2)*
(cosh(x) + sinh(x)) + b - c)/(cosh(x) + sinh(x)))/((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sin
h(x)^2 - b + c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh{\left (x \right )} + c \sinh{\left (x \right )} - \sqrt{b^{2} - c^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*cosh(x) + c*sinh(x) - sqrt(b**2 - c**2)), x)

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Giac [B]  time = 1.20489, size = 139, normalized size = 3.56 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{b^{2} - c^{2}} e^{\left (\frac{1}{2} \, x\right )} \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} + b - c\right ) +{\left (b \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} + b - c\right ) - c \mathrm{sgn}\left (-\sqrt{b^{2} - c^{2}} e^{x} + b - c\right )\right )} e^{\left (-\frac{1}{2} \, x\right )}\right )}}{\sqrt{b - c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*(sqrt(b^2 - c^2)*e^(1/2*x)*sgn(-sqrt(b^2 - c^2)*e^x + b - c) + (b*sgn(-sqrt(b^2 - c^2)*e^x + b - c) -
c*sgn(-sqrt(b^2 - c^2)*e^x + b - c))*e^(-1/2*x))/sqrt(b - c)