3.6.87 \(\int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx\) [587]

Optimal. Leaf size=35 \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \]

[Out]

arctanh(a*tanh(x)/(a^2-b^2)^(1/2))/a/(a^2-b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3260, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 - b^2*Cosh[x]^2)^(-1),x]

[Out]

ArcTanh[(a*Tanh[x])/Sqrt[a^2 - b^2]]/(a*Sqrt[a^2 - b^2])

Rule 214

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

Rule 3260

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]

Rubi steps

\begin {align*} \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx &=\text {Subst}\left (\int \frac {1}{a^2-\left (a^2-b^2\right ) x^2} \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 35, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 - b^2*Cosh[x]^2)^(-1),x]

[Out]

ArcTanh[(a*Tanh[x])/Sqrt[a^2 - b^2]]/(a*Sqrt[a^2 - b^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(31)=62\).
time = 0.07, size = 74, normalized size = 2.11

method	result	size
default	\(\frac {\arctanh \left (\frac {\left (a +b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}\)	\(74\)
risch	\(\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a^{2} \sqrt {a^{2}-b^{2}}-b^{2} \sqrt {a^{2}-b^{2}}-2 a^{3}+2 a \,b^{2}}{b^{2} \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, a}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a^{2} \sqrt {a^{2}-b^{2}}-b^{2} \sqrt {a^{2}-b^{2}}+2 a^{3}-2 a \,b^{2}}{b^{2} \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, a}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a/((a+b)*(a-b))^(1/2)*arctanh((a+b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))+1/a/((a+b)*(a-b))^(1/2)*arctanh((a-b)*t
anh(1/2*x)/((a+b)*(a-b))^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (31) = 62\).
time = 0.97, size = 388, normalized size = 11.09 \begin {gather*} \left [\frac {\sqrt {a^{2} - b^{2}} \log \left (\frac {b^{4} \cosh \left (x\right )^{4} + 4 \, b^{4} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{4} \sinh \left (x\right )^{4} + 8 \, a^{4} - 8 \, a^{2} b^{2} + b^{4} - 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{4} \cosh \left (x\right )^{2} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{4} \cosh \left (x\right )^{3} - {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, {\left (a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} - 2 \, a^{3} + a b^{2}\right )} \sqrt {a^{2} - b^{2}}}{b^{2} \cosh \left (x\right )^{4} + 4 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{2} \sinh \left (x\right )^{4} - 2 \, {\left (2 \, a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} - 2 \, a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} - {\left (2 \, a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right )}{2 \, {\left (a^{3} - a b^{2}\right )}}, \frac {\sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {{\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} - 2 \, a^{2} + b^{2}\right )} \sqrt {-a^{2} + b^{2}}}{2 \, {\left (a^{3} - a b^{2}\right )}}\right )}{a^{3} - a b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(a^2 - b^2)*log((b^4*cosh(x)^4 + 4*b^4*cosh(x)*sinh(x)^3 + b^4*sinh(x)^4 + 8*a^4 - 8*a^2*b^2 + b^4 -
2*(2*a^2*b^2 - b^4)*cosh(x)^2 + 2*(3*b^4*cosh(x)^2 - 2*a^2*b^2 + b^4)*sinh(x)^2 + 4*(b^4*cosh(x)^3 - (2*a^2*b^
2 - b^4)*cosh(x))*sinh(x) + 4*(a*b^2*cosh(x)^2 + 2*a*b^2*cosh(x)*sinh(x) + a*b^2*sinh(x)^2 - 2*a^3 + a*b^2)*sq
rt(a^2 - b^2))/(b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*sinh(x)^4 - 2*(2*a^2 - b^2)*cosh(x)^2 + 2*(3*b^2
*cosh(x)^2 - 2*a^2 + b^2)*sinh(x)^2 + b^2 + 4*(b^2*cosh(x)^3 - (2*a^2 - b^2)*cosh(x))*sinh(x)))/(a^3 - a*b^2),
sqrt(-a^2 + b^2)*arctan(-1/2*(b^2*cosh(x)^2 + 2*b^2*cosh(x)*sinh(x) + b^2*sinh(x)^2 - 2*a^2 + b^2)*sqrt(-a^2
+ b^2)/(a^3 - a*b^2))/(a^3 - a*b^2)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (27) = 54\).
time = 18.15, size = 892, normalized size = 25.49 \begin {gather*} \begin {cases} \frac {\tilde {\infty } \tanh {\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} + 1} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\tanh {\left (\frac {x}{2} \right )}}{2 b^{2}} + \frac {1}{2 b^{2} \tanh {\left (\frac {x}{2} \right )}} & \text {for}\: a = - b \\- \frac {2 \tanh {\left (\frac {x}{2} \right )}}{b^{2} \left (\tanh ^{2}{\left (\frac {x}{2} \right )} + 1\right )} & \text {for}\: a = 0 \\\frac {\tanh {\left (\frac {x}{2} \right )}}{2 b^{2}} + \frac {1}{2 b^{2} \tanh {\left (\frac {x}{2} \right )}} & \text {for}\: a = b \\- \frac {a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (- \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} + \frac {a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\sqrt {\frac {a}{a + b} - \frac {b}{a + b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} - \frac {a \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} + \frac {a \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} + \frac {b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (- \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} - \frac {b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\sqrt {\frac {a}{a + b} - \frac {b}{a + b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} - \frac {b \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} + \frac {b \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{2 a^{3} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}} - 2 a b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \sqrt {\frac {a}{a + b} - \frac {b}{a + b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2-b**2*cosh(x)**2),x)

[Out]

Piecewise((zoo*tanh(x/2)/(tanh(x/2)**2 + 1), Eq(a, 0) & Eq(b, 0)), (tanh(x/2)/(2*b**2) + 1/(2*b**2*tanh(x/2)),
Eq(a, -b)), (-2*tanh(x/2)/(b**2*(tanh(x/2)**2 + 1)), Eq(a, 0)), (tanh(x/2)/(2*b**2) + 1/(2*b**2*tanh(x/2)), E
q(a, b)), (-a*sqrt(a/(a - b) + b/(a - b))*log(-sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/(2*a**3*sqrt(a/(a - b)
+ b/(a - b))*sqrt(a/(a + b) - b/(a + b)) - 2*a*b**2*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b)))
+ a*sqrt(a/(a - b) + b/(a - b))*log(sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/(2*a**3*sqrt(a/(a - b) + b/(a - b
))*sqrt(a/(a + b) - b/(a + b)) - 2*a*b**2*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b))) - a*sqrt(a/
(a + b) - b/(a + b))*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(2*a**3*sqrt(a/(a - b) + b/(a - b))*sqrt(a/
(a + b) - b/(a + b)) - 2*a*b**2*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b))) + a*sqrt(a/(a + b) -
b/(a + b))*log(sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(2*a**3*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b
/(a + b)) - 2*a*b**2*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b))) + b*sqrt(a/(a - b) + b/(a - b))*
log(-sqrt(a/(a + b) - b/(a + b)) + tanh(x/2))/(2*a**3*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b))
- 2*a*b**2*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b))) - b*sqrt(a/(a - b) + b/(a - b))*log(sqrt(a
/(a + b) - b/(a + b)) + tanh(x/2))/(2*a**3*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b)) - 2*a*b**2*
sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b))) - b*sqrt(a/(a + b) - b/(a + b))*log(-sqrt(a/(a - b) +
b/(a - b)) + tanh(x/2))/(2*a**3*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b)) - 2*a*b**2*sqrt(a/(a
- b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b))) + b*sqrt(a/(a + b) - b/(a + b))*log(sqrt(a/(a - b) + b/(a - b))
+ tanh(x/2))/(2*a**3*sqrt(a/(a - b) + b/(a - b))*sqrt(a/(a + b) - b/(a + b)) - 2*a*b**2*sqrt(a/(a - b) + b/(a
- b))*sqrt(a/(a + b) - b/(a + b))), True))

________________________________________________________________________________________

Giac [A]
time = 1.36, size = 50, normalized size = 1.43 \begin {gather*} -\frac {\arctan \left (\frac {b^{2} e^{\left (2 \, x\right )} - 2 \, a^{2} + b^{2}}{2 \, \sqrt {-a^{2} + b^{2}} a}\right )}{\sqrt {-a^{2} + b^{2}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.38, size = 106, normalized size = 3.03 \begin {gather*} -\frac {\mathrm {atan}\left (\frac {b^2\,{\left (a^2\,b^2-a^4\right )}^{3/2}-2\,a^2\,{\left (a^2\,b^2-a^4\right )}^{3/2}+b^2\,{\mathrm {e}}^{2\,x}\,{\left (a^2\,b^2-a^4\right )}^{3/2}}{2\,a^8-4\,a^6\,b^2+2\,a^4\,b^4}\right )}{\sqrt {a^2\,b^2-a^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(b^2*cosh(x)^2 - a^2),x)

[Out]

-atan((b^2*(a^2*b^2 - a^4)^(3/2) - 2*a^2*(a^2*b^2 - a^4)^(3/2) + b^2*exp(2*x)*(a^2*b^2 - a^4)^(3/2))/(2*a^8 +
2*a^4*b^4 - 4*a^6*b^2))/(a^2*b^2 - a^4)^(1/2)

________________________________________________________________________________________