3.586 \(\int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx\)

Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]

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Rubi [A]  time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3181, 208} \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]

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 208

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

Rule 3181

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 &=\operatorname {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 = 0.07, size = 31, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 1.40, size = 288, normalized size = 9.29 \[ \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b^{4} \cosh \relax (x)^{4} + 4 \, b^{4} \cosh \relax (x) \sinh \relax (x)^{3} + b^{4} \sinh \relax (x)^{4} + 8 \, a^{4} + 8 \, a^{2} b^{2} + b^{4} + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{4} \cosh \relax (x)^{2} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)^{2} + 4 \, {\left (b^{4} \cosh \relax (x)^{3} + {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 4 \, {\left (a b^{2} \cosh \relax (x)^{2} + 2 \, a b^{2} \cosh \relax (x) \sinh \relax (x) + a b^{2} \sinh \relax (x)^{2} + 2 \, a^{3} + a b^{2}\right )} \sqrt {a^{2} + b^{2}}}{b^{2} \cosh \relax (x)^{4} + 4 \, b^{2} \cosh \relax (x) \sinh \relax (x)^{3} + b^{2} \sinh \relax (x)^{4} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} + 2 \, a^{2} + b^{2}\right )} \sinh \relax (x)^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \relax (x)^{3} + {\left (2 \, a^{2} + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}\right )}{2 \, {\left (a^{3} + a b^{2}\right )}} \]

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)*sqr
t(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)

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giac [B]  time = 0.60, size = 79, normalized size = 2.55 \[ \frac {\log \left (\frac {b^{2} e^{\left (2 \, x\right )} + 2 \, a^{2} + b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}}{b^{2} e^{\left (2 \, x\right )} + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.17, size = 98, normalized size = 3.16

method	result	size
default	\(\frac {\ln \left (\sqrt {a^{2}+b^{2}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 a \tanh \left (\frac {x}{2}\right )+\sqrt {a^{2}+b^{2}}\right )}{2 a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left (\sqrt {a^{2}+b^{2}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a \tanh \left (\frac {x}{2}\right )+\sqrt {a^{2}+b^{2}}\right )}{2 a \sqrt {a^{2}+b^{2}}}\)	\(98\)
risch	\(\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}+b^{2} \sqrt {a^{2}+b^{2}}-2 a^{3}-2 b^{2} a}{b^{2} \sqrt {a^{2}+b^{2}}}\right )}{2 a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}+b^{2} \sqrt {a^{2}+b^{2}}+2 a^{3}+2 b^{2} a}{b^{2} \sqrt {a^{2}+b^{2}}}\right )}{2 a \sqrt {a^{2}+b^{2}}}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.98, size = 76, normalized size = 2.45 \[ -\frac {\log \left (\frac {b^{2} e^{\left (-2 \, x\right )} + 2 \, a^{2} + b^{2} - 2 \, \sqrt {a^{2} + b^{2}} a}{b^{2} e^{\left (-2 \, x\right )} + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} a}\right )}{2 \, \sqrt {a^{2} + b^{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.65, size = 109, normalized size = 3.52 \[ \frac {\mathrm {atan}\left (\frac {2\,a^2\,{\left (-a^4-a^2\,b^2\right )}^{3/2}+b^2\,{\left (-a^4-a^2\,b^2\right )}^{3/2}+b^2\,{\mathrm {e}}^{2\,x}\,{\left (-a^4-a^2\,b^2\right )}^{3/2}}{2\,a^8+4\,a^6\,b^2+2\,a^4\,b^4}\right )}{\sqrt {-a^4-a^2\,b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b^2*cosh(x)^2 + a^2),x)

[Out]

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

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sympy [A]  time = 46.11, size = 605, normalized size = 19.52 \[ \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 = - i b \vee a = i 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 {a \log {\left (- \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} - \frac {a \log {\left (\sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} - \frac {i b \log {\left (- \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} + \frac {i b \log {\left (\sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{3} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 a b^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} + \frac {\sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} \log {\left (- \sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 i a b \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} - \frac {\sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} \log {\left (\sqrt {\frac {a}{a - i b} + \frac {i b}{a - i b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{- 2 a^{2} \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}} - 2 i a b \sqrt {\frac {a}{a + i b} - \frac {i b}{a + i b}}} & \text {otherwise} \end {cases} \]

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, I*b) | Eq(a, -I*b)), (2*tanh(x/2)/(b**2*(tanh(x/2)**2 + 1)), Eq(a, 0)), (a*log(-sqrt(a/(a + I*b) - I*b
/(a + I*b)) + tanh(x/2))/(-2*a**3*sqrt(a/(a + I*b) - I*b/(a + I*b)) - 2*a*b**2*sqrt(a/(a + I*b) - I*b/(a + I*b
))) - a*log(sqrt(a/(a + I*b) - I*b/(a + I*b)) + tanh(x/2))/(-2*a**3*sqrt(a/(a + I*b) - I*b/(a + I*b)) - 2*a*b*
*2*sqrt(a/(a + I*b) - I*b/(a + I*b))) - I*b*log(-sqrt(a/(a + I*b) - I*b/(a + I*b)) + tanh(x/2))/(-2*a**3*sqrt(
a/(a + I*b) - I*b/(a + I*b)) - 2*a*b**2*sqrt(a/(a + I*b) - I*b/(a + I*b))) + I*b*log(sqrt(a/(a + I*b) - I*b/(a
+ I*b)) + tanh(x/2))/(-2*a**3*sqrt(a/(a + I*b) - I*b/(a + I*b)) - 2*a*b**2*sqrt(a/(a + I*b) - I*b/(a + I*b)))
+ sqrt(a/(a - I*b) + I*b/(a - I*b))*sqrt(a/(a + I*b) - I*b/(a + I*b))*log(-sqrt(a/(a - I*b) + I*b/(a - I*b))
+ tanh(x/2))/(-2*a**2*sqrt(a/(a + I*b) - I*b/(a + I*b)) - 2*I*a*b*sqrt(a/(a + I*b) - I*b/(a + I*b))) - sqrt(a/
(a - I*b) + I*b/(a - I*b))*sqrt(a/(a + I*b) - I*b/(a + I*b))*log(sqrt(a/(a - I*b) + I*b/(a - I*b)) + tanh(x/2)
)/(-2*a**2*sqrt(a/(a + I*b) - I*b/(a + I*b)) - 2*I*a*b*sqrt(a/(a + I*b) - I*b/(a + I*b))), True))

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