3.8.77 \(\int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx\) [777]

Optimal. Leaf size=102 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{\sqrt [4]{b^2-c^2}} \]

[Out]

-arctanh(1/2*(b^2-c^2)^(1/4)*sinh(x+I*arctan(b,-I*c))*2^(1/2)/(-(b^2-c^2)^(1/2)+cosh(x+I*arctan(b,-I*c))*(b^2-
c^2)^(1/2))^(1/2))*2^(1/2)/(b^2-c^2)^(1/4)

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Rubi [A]
time = 0.07, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3194, 2728, 210} \begin {gather*} -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{\sqrt [4]{b^2-c^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]],x]

[Out]

-((Sqrt[2]*ArcTanh[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 - c^2] + Sqrt[b^2
- c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])])/(b^2 - c^2)^(1/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3194

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Int[1/Sqrt[a +
Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx &=\int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}} \, dx\\ &=2 i \text {Subst}\left (\int \frac {1}{-2 \sqrt {b^2-c^2}-x^2} \, dx,x,-\frac {i \sqrt {b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{\sqrt [4]{b^2-c^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 48.64, size = 52609, normalized size = 515.77 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]],x]

[Out]

Result too large to show

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Maple [A]
time = 2.63, size = 129, normalized size = 1.26

method	result	size
default	\(\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \arctan \left (\frac {\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \cosh \left (x \right )}{\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}}\right )}{\sqrt {\sqrt {b^{2}-c^{2}}\, \left (\sinh \left (x \right )+1\right )}\, \sinh \left (x \right ) \sqrt {-\frac {\sinh \left (x \right ) b^{2}-\sinh \left (x \right ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(b^2-c^2)^(1/2)*(sinh(x)+1)*sinh(x)^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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.43, size = 680, normalized size = 6.67 \begin {gather*} \left [\frac {\sqrt {2} \log \left (-\frac {{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} - \frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} {\left (2 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, {\left (b^{2} - c^{2}\right )} \sinh \left (x\right )^{2} + {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} + {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} + b - c\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}\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^{2} - c^{2}\right )}^{\frac {1}{4}}} - b^{2} + 2 \, b c - c^{2} + 2 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3} - {\left (b - c\right )} \cosh \left (x\right ) + {\left (3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} - b + c\right )} \sinh \left (x\right )\right )} \sqrt {b^{2} - c^{2}}}{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} - b^{2} + c^{2}\right )} \sinh \left (x\right )^{2} + b^{2} - 2 \, b c + c^{2} + 4 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{3} - {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right )}{{\left (b^{2} - c^{2}\right )}^{\frac {1}{4}}}, 2 \, \sqrt {2} \sqrt {-\frac {1}{\sqrt {b^{2} - c^{2}}}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} {\left (\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 )}} \sqrt {-\frac {1}{\sqrt {b^{2} - c^{2}}}}}{{\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}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(2)*log(-((b^2 + 2*b*c + c^2)*cosh(x)^4 + 4*(b^2 + 2*b*c + c^2)*cosh(x)^3*sinh(x) + 6*(b^2 + 2*b*c + c^2)
*cosh(x)^2*sinh(x)^2 + 4*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^3 + (b^2 + 2*b*c + c^2)*sinh(x)^4 - 2*sqrt(2)*sqr
t(1/2)*(2*(b^2 - c^2)*cosh(x)^2 + 4*(b^2 - c^2)*cosh(x)*sinh(x) + 2*(b^2 - c^2)*sinh(x)^2 + ((b + c)*cosh(x)^3
+ 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 + (b - c)*cosh(x) + (3*(b + c)*cosh(x)^2 + b - c)*sinh(x))*
sqrt(b^2 - c^2))*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^2 - c^2)^(1/4) - b^2 + 2*b*c - c^2 + 2*((b + c)*cosh(x)^3
+ 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 - (b - c)*cosh(x) + (3*(b + c)*cosh(x)^2 - b + c)*sinh(x))*s
qrt(b^2 - c^2))/((b^2 + 2*b*c + c^2)*cosh(x)^4 + 4*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^3 + (b^2 + 2*b*c + c^2)
*sinh(x)^4 - 2*(b^2 - c^2)*cosh(x)^2 + 2*(3*(b^2 + 2*b*c + c^2)*cosh(x)^2 - b^2 + c^2)*sinh(x)^2 + b^2 - 2*b*c
+ c^2 + 4*((b^2 + 2*b*c + c^2)*cosh(x)^3 - (b^2 - c^2)*cosh(x))*sinh(x)))/(b^2 - c^2)^(1/4), 2*sqrt(2)*sqrt(-
1/sqrt(b^2 - c^2))*arctan(sqrt(2)*sqrt(1/2)*(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)))*sqrt(-1/sqrt(b^2 - c^2))/((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - b +
c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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(1/(b*cosh(x)+c*sinh(x)-(b**2-c**2)**(1/2))**(1/2),x)

[Out]

Integral(1/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. 546 vs. \(2 (83) = 166\).
time = 0.63, size = 546, normalized size = 5.35 \begin {gather*} \frac {2 \, \sqrt {2} {\left (b^{2} - c^{2} - b + c\right )} \sqrt {b + c} \arctan \left (\frac {b^{3} e^{\left (\frac {1}{2} \, x\right )} + b^{2} c e^{\left (\frac {1}{2} \, x\right )} - b c^{2} e^{\left (\frac {1}{2} \, x\right )} - c^{3} e^{\left (\frac {1}{2} \, x\right )} - b^{2} e^{\left (\frac {1}{2} \, x\right )} + c^{2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {-\sqrt {b^{2} - c^{2}} b^{5} - \sqrt {b^{2} - c^{2}} b^{4} c + 2 \, \sqrt {b^{2} - c^{2}} b^{3} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} b^{2} c^{3} - \sqrt {b^{2} - c^{2}} b c^{4} - \sqrt {b^{2} - c^{2}} c^{5} + 2 \, \sqrt {b^{2} - c^{2}} b^{4} - 4 \, \sqrt {b^{2} - c^{2}} b^{2} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} c^{4} - \sqrt {b^{2} - c^{2}} b^{3} + \sqrt {b^{2} - c^{2}} b^{2} c + \sqrt {b^{2} - c^{2}} b c^{2} - \sqrt {b^{2} - c^{2}} c^{3}}}\right )}{\sqrt {-\sqrt {b^{2} - c^{2}} b^{5} - \sqrt {b^{2} - c^{2}} b^{4} c + 2 \, \sqrt {b^{2} - c^{2}} b^{3} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} b^{2} c^{3} - \sqrt {b^{2} - c^{2}} b c^{4} - \sqrt {b^{2} - c^{2}} c^{5} + 2 \, \sqrt {b^{2} - c^{2}} b^{4} - 4 \, \sqrt {b^{2} - c^{2}} b^{2} c^{2} + 2 \, \sqrt {b^{2} - c^{2}} c^{4} - \sqrt {b^{2} - c^{2}} b^{3} + \sqrt {b^{2} - c^{2}} b^{2} c + \sqrt {b^{2} - c^{2}} b c^{2} - \sqrt {b^{2} - c^{2}} c^{3}} \mathrm {sgn}\left (-\sqrt {b^{2} - c^{2}} e^{x} + b - c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2)*(b^2 - c^2 - b + c)*sqrt(b + c)*arctan((b^3*e^(1/2*x) + b^2*c*e^(1/2*x) - b*c^2*e^(1/2*x) - c^3*e^(1
/2*x) - b^2*e^(1/2*x) + c^2*e^(1/2*x))/sqrt(-sqrt(b^2 - c^2)*b^5 - sqrt(b^2 - c^2)*b^4*c + 2*sqrt(b^2 - c^2)*b
^3*c^2 + 2*sqrt(b^2 - c^2)*b^2*c^3 - sqrt(b^2 - c^2)*b*c^4 - sqrt(b^2 - c^2)*c^5 + 2*sqrt(b^2 - c^2)*b^4 - 4*s
qrt(b^2 - c^2)*b^2*c^2 + 2*sqrt(b^2 - c^2)*c^4 - sqrt(b^2 - c^2)*b^3 + sqrt(b^2 - c^2)*b^2*c + sqrt(b^2 - c^2)
*b*c^2 - sqrt(b^2 - c^2)*c^3))/(sqrt(-sqrt(b^2 - c^2)*b^5 - sqrt(b^2 - c^2)*b^4*c + 2*sqrt(b^2 - c^2)*b^3*c^2
+ 2*sqrt(b^2 - c^2)*b^2*c^3 - sqrt(b^2 - c^2)*b*c^4 - sqrt(b^2 - c^2)*c^5 + 2*sqrt(b^2 - c^2)*b^4 - 4*sqrt(b^2
- c^2)*b^2*c^2 + 2*sqrt(b^2 - c^2)*c^4 - sqrt(b^2 - c^2)*b^3 + sqrt(b^2 - c^2)*b^2*c + sqrt(b^2 - c^2)*b*c^2
- sqrt(b^2 - c^2)*c^3)*sgn(-sqrt(b^2 - c^2)*e^x + b - c))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\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(1/(b*cosh(x) - (b^2 - c^2)^(1/2) + c*sinh(x))^(1/2),x)

[Out]

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

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