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)

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Rubi [A]
time = 0.02, 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*}

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Mathematica [A]
time = 0.03, 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])

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 20.19, size = 1175, normalized size = 33.57 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\text {Tanh}\left [x\right ]\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{-\frac {\text {Tanh}\left [x\right ]}{b^2},a\text {==}0\right \},\left \{\frac {\frac {1}{\text {Tanh}\left [\frac {x}{2}\right ]}+\text {Tanh}\left [\frac {x}{2}\right ]}{2 b^2},a\text {==}b\text {$\vert $$\vert $}a\text {==}-b\right \}\right \},-\frac {a \text {Log}\left [\text {Tanh}\left [\frac {x}{2}\right ]-\sqrt {\frac {a}{a+b}-\frac {b}{a+b}}\right ] \sqrt {\frac {a}{a-b}+\frac {b}{a-b}}}{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 \text {Log}\left [\text {Tanh}\left [\frac {x}{2}\right ]-\sqrt {\frac {a}{a-b}+\frac {b}{a-b}}\right ] \sqrt {\frac {a}{a+b}-\frac {b}{a+b}}}{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 \text {Log}\left [\sqrt {\frac {a}{a+b}-\frac {b}{a+b}}+\text {Tanh}\left [\frac {x}{2}\right ]\right ] \sqrt {\frac {a}{a-b}+\frac {b}{a-b}}}{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 \text {Log}\left [\sqrt {\frac {a}{a-b}+\frac {b}{a-b}}+\text {Tanh}\left [\frac {x}{2}\right ]\right ] \sqrt {\frac {a}{a+b}-\frac {b}{a+b}}}{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 \text {Log}\left [\sqrt {\frac {a}{a+b}-\frac {b}{a+b}}+\text {Tanh}\left [\frac {x}{2}\right ]\right ] \sqrt {\frac {a}{a-b}+\frac {b}{a-b}}}{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 \text {Log}\left [\text {Tanh}\left [\frac {x}{2}\right ]-\sqrt {\frac {a}{a-b}+\frac {b}{a-b}}\right ] \sqrt {\frac {a}{a+b}-\frac {b}{a+b}}}{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 \text {Log}\left [\sqrt {\frac {a}{a-b}+\frac {b}{a-b}}+\text {Tanh}\left [\frac {x}{2}\right ]\right ] \sqrt {\frac {a}{a+b}-\frac {b}{a+b}}}{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 \text {Log}\left [\text {Tanh}\left [\frac {x}{2}\right ]-\sqrt {\frac {a}{a+b}-\frac {b}{a+b}}\right ] \sqrt {\frac {a}{a-b}+\frac {b}{a-b}}}{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}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{DirectedInfinity[Tanh[x]], a == 0 && b == 0}, {-Tanh[x] / b ^ 2, a == 0}, {(1 / Tanh[x / 2] + Tanh
[x / 2]) / (2 b ^ 2), a == b || a == -b}}, -a Log[Tanh[x / 2] - Sqrt[a / (a + b) - b / (a + b)]] Sqrt[a / (a -
b) + b / (a - b)] / (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 Log[Tanh[x / 2] - Sqrt[a / (a - b) + b / (a -
b)]] Sqrt[a / (a + b) - b / (a + b)] / (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 Log[Sqrt[a / (a + b) - b /
(a + b)] + Tanh[x / 2]] Sqrt[a / (a - b) + b / (a - b)] / (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 Log[Sqr
t[a / (a - b) + b / (a - b)] + Tanh[x / 2]] Sqrt[a / (a + b) - b / (a + b)] / (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 Log[Sqrt[a / (a + b) - b / (a + b)] + Tanh[x / 2]] Sqrt[a / (a - b) + b / (a - b)] / (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 Log[Tanh[x / 2] - Sqrt[a / (a - b) + b / (a - b)]] Sqrt[a / (a + b) - b / (a +
b)] / (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 Log[Sqrt[a / (a - b) + b / (a - b)] + Tanh[x / 2]] Sqrt[a /
(a + b) - b / (a + b)] / (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 Log[Tanh[x / 2] - Sqrt[a / (a + b) - b /
(a + b)]] Sqrt[a / (a - b) + b / (a - b)] / (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)])]

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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))

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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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (31) = 62\).
time = 0.32, 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)]

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Sympy [A]
time = 16.06, size = 874, normalized size = 24.97

result too large to display

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)), (-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)), Eq(a, b) | Eq(a, -b)), (-a*sqrt(a/(a - b) + b/(a - b))*lo
g(-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*sq
rt(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*s
qrt(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))

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Giac [A]
time = 0.00, size = 54, normalized size = 1.54 \begin {gather*} -\frac {\arctan \left (\frac {-2 a^{2}+b^{2} \left (\mathrm {e}^{x}\right )^{2}+b^{2}}{2 a \sqrt {-a^{2}+b^{2}}}\right )}{a \sqrt {-a^{2}+b^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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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)

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