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

Optimal. Leaf size=37 \[ \frac {2 (b \sinh (x)+c \cosh (x))}{\sqrt {\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, 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*}

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Mathematica [C]  time = 74.46, size = 10054, normalized size = 271.73 \[ \text {Result too large to show} \]

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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fricas [B]  time = 0.45, size = 143, normalized size = 3.86 \[ \frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + b - c\right )} \sqrt {\frac {{\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {b^{2} - c^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + b - c}{\cosh \relax (x) + \sinh \relax (x)}}}{{\left (b + c\right )} \cosh \relax (x)^{2} + 2 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b + c\right )} \sinh \relax (x)^{2} - b + c} \]

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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giac [B]  time = 0.14, size = 104, normalized size = 2.81 \[ -\frac {\sqrt {2} {\left (\sqrt {b^{2} - c^{2}} e^{\left (\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - {\left (b \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right ) - c \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} - b + c\right )\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )}}{\sqrt {b - c}} \]

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)

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maple [B]  time = 0.81, size = 201, normalized size = 5.43 \[ \frac {\left (-b^{2}+c^{2}\right ) \cosh \relax (x )}{\sqrt {b^{2}-c^{2}}\, \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right ) \left (\sinh ^{2}\relax (x )\right )}\, \sqrt {b^{2}-c^{2}}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right )}\, \cosh \relax (x )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right ) \left (\sinh ^{2}\relax (x )\right )}}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )-1\right )}\, \sinh \relax (x ) \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}} \]

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)*(b^2-c^2)^(1/2)/((b^2-c^2)^(1/2)*(sinh(x)-1))^(1/2)*arctan(((b^2-c^2)^(1/2)*(si
nh(x)-1))^(1/2)*cosh(x)/(-(b^2-c^2)^(1/2)*(sinh(x)-1)*sinh(x)^2)^(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 = 0.83, size = 153, normalized size = 4.14 \[ \frac {\sqrt {2} \sqrt {2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c} \sqrt {b + c} \sqrt {b - c} e^{\left (\frac {1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} + \sqrt {b + c} \sqrt {b - c}} - \frac {\sqrt {2} \sqrt {2 \, \sqrt {b + c} \sqrt {b - c} e^{\left (-x\right )} + {\left (b - c\right )} e^{\left (-2 \, x\right )} + b + c} {\left (b - c\right )} e^{\left (-\frac {1}{2} \, x\right )}}{{\left (b - c\right )} e^{\left (-x\right )} + \sqrt {b + c} \sqrt {b - c}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {b\,\mathrm {cosh}\relax (x)+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cosh {\relax (x )} + c \sinh {\relax (x )} + \sqrt {b^{2} - c^{2}}}\, dx \]

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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