3.8.58 \(\int \frac {1}{(\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^2} \, dx\) [758]

Optimal. Leaf size=100 \[ \frac {c \cosh (x)+b \sinh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {c+\sqrt {b^2-c^2} \sinh (x)}{3 c \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))} \]

[Out]

1/3*(c*cosh(x)+b*sinh(x))/(b^2-c^2)^(1/2)/(b*cosh(x)+c*sinh(x)+(b^2-c^2)^(1/2))^2+1/3*(-c-sinh(x)*(b^2-c^2)^(1
/2))/c/(c*cosh(x)+b*sinh(x))/(b^2-c^2)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3195, 3193} \begin {gather*} \frac {b \sinh (x)+c \cosh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {\sqrt {b^2-c^2} \sinh (x)+c}{3 c \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Cosh[x] + b*Sinh[x])/(3*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (c + Sqrt[b^2 - c^2]
*Sinh[x])/(3*c*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))

Rule 3193

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

Rule 3195

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

Rubi steps

\begin {align*} \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx &=\frac {c \cosh (x)+b \sinh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac {\int \frac {1}{\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx}{3 \sqrt {b^2-c^2}}\\ &=\frac {c \cosh (x)+b \sinh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {c+\sqrt {b^2-c^2} \sinh (x)}{3 c \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 68, normalized size = 0.68 \begin {gather*} -\frac {-2 c \sqrt {b^2-c^2}+2 b c \cosh ^3(x)+2 c^2 \sinh (x)+c^2 \cosh ^2(x) \sinh (x)+b^2 \sinh ^3(x)}{3 c (c \cosh (x)+b \sinh (x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(-2*c*Sqrt[b^2 - c^2] + 2*b*c*Cosh[x]^3 + 2*c^2*Sinh[x] + c^2*Cosh[x]^2*Sinh[x] + b^2*Sinh[x]^3)/(c*(c*Co
sh[x] + b*Sinh[x])^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(216\) vs. \(2(91)=182\).
time = 2.06, size = 217, normalized size = 2.17

method	result	size
risch	\(-\frac {2 \left (3 \,{\mathrm e}^{x} b +3 c \,{\mathrm e}^{x}+\sqrt {b^{2}-c^{2}}\right )}{3 \left ({\mathrm e}^{x} b +c \,{\mathrm e}^{x}+\sqrt {b^{2}-c^{2}}\right )^{3}}\)	\(47\)
default	\(\frac {2 \left (\sqrt {b^{2}-c^{2}}+b \right ) \left (\frac {\left (\sqrt {b^{2}-c^{2}}+b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{c^{2}}+\frac {\left (2 b^{2}-c^{2}+2 \sqrt {b^{2}-c^{2}}\, b \right ) \tanh \left (\frac {x}{2}\right )}{c^{3}}+\frac {\frac {4 \sqrt {b^{2}-c^{2}}\, b^{2}}{3}-\frac {2 \sqrt {b^{2}-c^{2}}\, c^{2}}{3}+\frac {4 b^{3}}{3}-\frac {4 b \,c^{2}}{3}}{c^{4}}\right )}{c^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+\frac {2 \sqrt {\left (b -c \right ) \left (b +c \right )}\, \tanh \left (\frac {x}{2}\right )}{c}+\frac {2 b \tanh \left (\frac {x}{2}\right )}{c}+\frac {2 \sqrt {\left (b -c \right ) \left (b +c \right )}\, b}{c^{2}}+\frac {2 b^{2}}{c^{2}}-1\right ) \left (\tanh \left (\frac {x}{2}\right )+\frac {\sqrt {\left (b -c \right ) \left (b +c \right )}}{c}+\frac {b}{c}\right )}\)	\(217\)

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))^2,x,method=_RETURNVERBOSE)

[Out]

2*((b^2-c^2)^(1/2)+b)/c^2*(((b^2-c^2)^(1/2)+b)/c^2*tanh(1/2*x)^2+(2*b^2-c^2+2*(b^2-c^2)^(1/2)*b)/c^3*tanh(1/2*
x)+2/3*(2*(b^2-c^2)^(1/2)*b^2-(b^2-c^2)^(1/2)*c^2+2*b^3-2*b*c^2)/c^4)/(tanh(1/2*x)^2+2/c*((b-c)*(b+c))^(1/2)*t
anh(1/2*x)+2/c*b*tanh(1/2*x)+2/c^2*((b-c)*(b+c))^(1/2)*b+2*b^2/c^2-1)/(tanh(1/2*x)+1/c*((b-c)*(b+c))^(1/2)+b/c
)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (88) = 176\).
time = 0.42, size = 660, normalized size = 6.60 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{4} + 12 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{4} + 6 \, {\left (b^{2} - c^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\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} + 12 \, {\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 ) - 8 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{3} + 3 \, {\left (b + c\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (b + c\right )} \sinh \left (x\right )^{3}\right )} \sqrt {b^{2} - c^{2}}\right )}}{3 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{6} + 6 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \sinh \left (x\right )^{6} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{4} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4} - 5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} - b^{4} + 2 \, b^{3} c - 2 \, b c^{3} + c^{4} + 4 \, {\left (5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{4} + b^{4} - 2 \, b^{2} c^{2} + c^{4} - 6 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 6 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \left (x\right )^{5} - 2 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \left (x\right )^{3} + {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\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))^2,x, algorithm="fricas")

[Out]

-2/3*(3*(b^2 + 2*b*c + c^2)*cosh(x)^4 + 12*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^3 + 3*(b^2 + 2*b*c + c^2)*sinh(
x)^4 + 6*(b^2 - c^2)*cosh(x)^2 + 6*(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
+ 12*((b^2 + 2*b*c + c^2)*cosh(x)^3 + (b^2 - c^2)*cosh(x))*sinh(x) - 8*((b + c)*cosh(x)^3 + 3*(b + c)*cosh(x)
^2*sinh(x) + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3)*sqrt(b^2 - c^2))/((b^4 + 4*b^3*c + 6*b^2*c^2 + 4
*b*c^3 + c^4)*cosh(x)^6 + 6*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)*sinh(x)^5 + (b^4 + 4*b^3*c + 6
*b^2*c^2 + 4*b*c^3 + c^4)*sinh(x)^6 - 3*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^4 - 3*(b^4 + 2*b^3*c - 2*b*c^3
- c^4 - 5*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^2)*sinh(x)^4 - b^4 + 2*b^3*c - 2*b*c^3 + c^4 +
4*(5*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^3 - 3*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x))*sinh(x
)^3 + 3*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^2 + 3*(5*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^4 + b^4 -
2*b^2*c^2 + c^4 - 6*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^2)*sinh(x)^2 + 6*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*
b*c^3 + c^4)*cosh(x)^5 - 2*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^3 + (b^4 - 2*b^2*c^2 + c^4)*cosh(x))*sinh(x
))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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))**2,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[2,0]%%%}+%%%{2,[1,1]%%%}+%%%{1,[0,2]%%%},[4]%%%}+
%%%{%%{[%%%

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

[Out]

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

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