3.8.76 \(\int \sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx\) [776]

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

[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 = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {3191} \begin {gather*} \frac {2 (b \sinh (x)+c \cosh (x))}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \end {gather*}

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 3191

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] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 80.54, size = 4196, normalized size = 107.59 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*b*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]])/c - (8*b*c*Sqrt[b^2 - c^2]*(-b^2 + c^2)*(EllipticF[ArcSin
[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1] - 2*El
lipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x
/2])))]], 1])*Sqrt[-Sqrt[(b - c)*(b + c)] + b*Cosh[x] + c*Sinh[x]]*(-1 + Tanh[x/2])*(-(((b + c + Sqrt[b^2 - c^
2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2]))))^(3/2)*(c + (b + Sqrt[b^2 - c^2])*Tanh[x/2]
)*(-1 + Tanh[x/2]^2))/((b + c + Sqrt[b^2 - c^2])^3*(-b^2 + c^2 + b*Sqrt[b^2 - c^2])*(1 + Cosh[x])*Sqrt[(-Sqrt[
(b - c)*(b + c)] + b*Cosh[x] + c*Sinh[x])/(1 + Cosh[x])^2]*(1 + Tanh[x/2])^2*Sqrt[-((-1 + Tanh[x/2]^2)*(2*c*Ta
nh[x/2] + Sqrt[b^2 - c^2]*(-1 + Tanh[x/2]^2) + b*(1 + Tanh[x/2]^2)))]) - (4*(b - c)*(b + c)*Sqrt[-Sqrt[(b - c)
*(b + c)] + b*Cosh[x] + c*Sinh[x]]*(2*b^3*c^2 + 3*b^2*c^3 - c^5 + 2*b^2*c^2*Sqrt[b^2 - c^2] + 3*b*c^3*Sqrt[b^2
- c^2] + c^4*Sqrt[b^2 - c^2] + 8*b^4*c*Tanh[x/2] + 12*b^3*c^2*Tanh[x/2] - 2*b^2*c^3*Tanh[x/2] - 8*b*c^4*Tanh[
x/2] - 2*c^5*Tanh[x/2] + 8*b^3*c*Sqrt[b^2 - c^2]*Tanh[x/2] + 12*b^2*c^2*Sqrt[b^2 - c^2]*Tanh[x/2] + 2*b*c^3*Sq
rt[b^2 - c^2]*Tanh[x/2] - 2*c^4*Sqrt[b^2 - c^2]*Tanh[x/2] + 8*b^5*Tanh[x/2]^2 + 12*b^4*c*Tanh[x/2]^2 - 4*b^3*c
^2*Tanh[x/2]^2 - 11*b^2*c^3*Tanh[x/2]^2 - 2*b*c^4*Tanh[x/2]^2 + c^5*Tanh[x/2]^2 + 8*b^4*Sqrt[b^2 - c^2]*Tanh[x
/2]^2 + 12*b^3*c*Sqrt[b^2 - c^2]*Tanh[x/2]^2 - 5*b*c^3*Sqrt[b^2 - c^2]*Tanh[x/2]^2 - c^4*Sqrt[b^2 - c^2]*Tanh[
x/2]^2 - 8*b^4*c*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 -
c^2])*(-1 + Tanh[x/2])))]], 1]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*
(-1 + Tanh[x/2])))] + 8*b^2*c^3*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b -
c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sq
rt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 8*b^3*c*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 -
c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*
(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 4*b*c^3*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSi
n[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Sqrt[
-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 16*b^5*Elliptic
Pi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))
]], 1]*Tanh[x/2]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2]
)))] + 8*b^4*c*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c
^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2
- c^2])*(-1 + Tanh[x/2])))] + 20*b^3*c^2*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2
]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[
x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 8*b^2*c^3*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[
b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((b + c + Sq
rt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 4*b*c^4*EllipticPi[-1, ArcSin
[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x
/2]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] - 16*b^4
*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 -
c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^
2 - c^2])*(-1 + Tanh[x/2])))] + 8*b^3*c*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2]
)*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((b + c + Sqrt[b^2 - c
^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 12*b^2*c^2*Sqrt[b^2 - c^2]*EllipticPi[-
1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]],
1]*Tanh[x/2]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]
- 4*b*c^3*Sqrt[b^2 - c^2]*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c +
Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 + Tanh[x/2]))/((b - c
+ Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))] + 16*b^5*EllipticPi[-1, ArcSin[Sqrt[-(((b + c + Sqrt[b^2 - c^2])*(1 +
Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1 + Tanh[x/2])))]], 1]*Tanh[x/2]^2*Sqrt[-(((b + c + Sqrt[b^2 - c^2])*
(1 + Tanh[x/2]))/((b - c + Sqrt[b^2 - c^2])*(-1...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(35)=70\).
time = 2.78, size = 202, normalized size = 5.18

method	result	size
default	\(\frac {\cosh \left (x \right ) \left (-b^{2}+c^{2}\right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}-\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}}\right ) \sqrt {b^{2}-c^{2}}}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(202\)
risch	\(-\frac {\sqrt {-\left (-b \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x} c +2 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x}-b +c \right ) {\mathrm e}^{-x}}\, \sqrt {-\left (-b \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x} c +2 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x}-b +c \right ) {\mathrm e}^{x}}\, \sqrt {2}\, \left ({\mathrm e}^{x} b +c \,{\mathrm e}^{x}+\sqrt {b^{2}-c^{2}}\right ) \left ({\mathrm e}^{x} b +c \,{\mathrm e}^{x}-\sqrt {b^{2}-c^{2}}\right )}{\left (-b \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x} c +2 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x}-b +c \right ) \left (b +c \right ) \sqrt {\left (b \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} c -2 \sqrt {b^{2}-c^{2}}\, {\mathrm e}^{x}+b -c \right ) {\mathrm e}^{x}}}\)	\(208\)

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,method=_RETURNVERBOSE)

[Out]

cosh(x)*(-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)-(-(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] Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (35) = 70\).
time = 0.52, size = 156, normalized size = 4.00 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (35) = 70\).
time = 0.41, size = 143, normalized size = 3.67 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {b \cosh {\left (x \right )} + c \sinh {\left (x \right )} - \sqrt {b^{2} - c^{2}}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (35) = 70\).
time = 0.41, size = 81, normalized size = 2.08 \begin {gather*} -\frac {\sqrt {2} {\left ({\left (b - c\right )} e^{\left (-\frac {1}{2} \, x\right )} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right ) + \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 )\right )}}{\sqrt {b - c}} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {b\,\mathrm {cosh}\left (x\right )-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\left (x\right )} \,d x \end {gather*}

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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