3.1.16 \(\int \frac {1}{a+b \cosh ^2(x)} \, dx\) [16]

Optimal. Leaf size=29 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3260, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^2)^(-1),x]

[Out]

ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b])

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

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Mathematica [A]
time = 0.05, size = 29, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^2)^(-1),x]

[Out]

ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(21)=42\).
time = 0.45, size = 81, normalized size = 2.79

method	result	size
default	\(\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{2 \sqrt {a}\, \sqrt {a +b}}-\frac {\ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}-\sqrt {a +b}\right )}{2 \sqrt {a}\, \sqrt {a +b}}\)	\(81\)
risch	\(\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (21) = 42\).
time = 0.48, size = 53, normalized size = 1.83 \begin {gather*} -\frac {\log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{2 \, \sqrt {{\left (a + b\right )} a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (21) = 42\).
time = 0.38, size = 293, normalized size = 10.10 \begin {gather*} \left [\frac {\log \left (\frac {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 b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + 8 \, a^{2} + 8 \, a b + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \, {\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, a + b\right )} \sqrt {a^{2} + a b}}{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (2 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}\right )}{2 \, \sqrt {a^{2} + a b}}, \frac {\sqrt {-a^{2} - a b} \arctan \left (\frac {{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, a + b\right )} \sqrt {-a^{2} - a b}}{2 \, {\left (a^{2} + a b\right )}}\right )}{a^{2} + a b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 10924 vs. \(2 (27) = 54\).
time = 30.58, size = 10924, normalized size = 376.69 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*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*(tanh(x/2)**2 + 1)), Eq(a,
0)), (-tanh(x/2)/(2*b) - 1/(2*b*tanh(x/2)), Eq(a, -b)), (-a**3*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a +
b))*log(-sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + tanh(x/2))/(2*a**4*sqrt(a/(a + b) - b/(a + b) -
2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(a/(a + b) - b/(a + b
) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a**3*sqrt(-a*b)*sqrt(a/(a + b
) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(a
/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqr
t(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a*b**2*
sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b
))) + a**3*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*log(sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a
+ b)) + tanh(x/2))/(2*a**4*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*
sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b)
+ 2*sqrt(-a*b)/(a + b)) + 8*a**3*sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b)
- b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(
a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(
a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a*b**2*sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/
(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b))) - a**3*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/
(a + b))*log(-sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b)) + tanh(x/2))/(2*a**4*sqrt(a/(a + b) - b/(a +
b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(a/(a + b) - b/(
a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a**3*sqrt(-a*b)*sqrt(a/(
a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*s
qrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**
3*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a*
b**2*sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(
a + b))) + a**3*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b))*log(sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*
b)/(a + b)) + tanh(x/2))/(2*a**4*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b)
+ 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a
+ b) + 2*sqrt(-a*b)/(a + b)) + 8*a**3*sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a
+ b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqr
t(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*
sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 8*a*b**2*sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-
a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b))) + 10*a**2*b*sqrt(a/(a + b) - b/(a + b) - 2*s
qrt(-a*b)/(a + b))*log(-sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + tanh(x/2))/(2*a**4*sqrt(a/(a + b)
- b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(a/(a
+ b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a**3*sqrt(-a*b
)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a
**2*b**2*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b))
+ 2*a*b**3*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a +
b)) - 8*a*b**2*sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sq
rt(-a*b)/(a + b))) - 10*a**2*b*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*log(sqrt(a/(a + b) - b/(a +
b) + 2*sqrt(-a*b)/(a + b)) + tanh(x/2))/(2*a**4*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/(a +
b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**3*b*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a + b))*sqrt(a/
(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 8*a**3*sqrt(-a*b)*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*b)/(a +
b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) - 10*a**2*b**2*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(-a*
b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*sqrt(-a*b)/(a + b)) + 2*a*b**3*sqrt(a/(a + b) - b/(a + b) - 2*sqrt(
-a*b)/(a + b))*sqrt(a/(a + b) - b/(a + b) + 2*s...

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Giac [A]
time = 0.41, size = 39, normalized size = 1.34 \begin {gather*} \frac {\arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.35, size = 267, normalized size = 9.21 \begin {gather*} -\frac {\mathrm {atan}\left (\frac {b^2\,{\mathrm {e}}^{2\,x}\,{\left (-a^2-b\,a\right )}^{3/2}\,\left (\frac {4\,\left (4\,a+2\,b\right )\,\left (8\,a^3+12\,a^2\,b+4\,a\,b^2\right )}{b^5\,{\left (-a^2-b\,a\right )}^{3/2}\,\sqrt {-a\,\left (a+b\right )}}+\frac {2\,\left (8\,a^2+8\,a\,b+b^2\right )\,\left (8\,a^2\,\sqrt {-a^2-b\,a}+b^2\,\sqrt {-a^2-b\,a}+8\,a\,b\,\sqrt {-a^2-b\,a}\right )}{a\,b^5\,\left (a+b\right )\,{\left (-a^2-b\,a\right )}^{3/2}}\right )}{4}+\frac {\left (2\,a^2\,b+2\,a\,b^2\right )\,\left (4\,a+2\,b\right )}{b^3\,\sqrt {-a\,\left (a+b\right )}}+\frac {\left (b^2\,\sqrt {-a^2-b\,a}+2\,a\,b\,\sqrt {-a^2-b\,a}\right )\,\left (8\,a^2+8\,a\,b+b^2\right )}{2\,a\,b^3\,\left (a+b\right )}\right )}{\sqrt {-a^2-b\,a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x)^2),x)

[Out]

-atan((b^2*exp(2*x)*(- a*b - a^2)^(3/2)*((4*(4*a + 2*b)*(4*a*b^2 + 12*a^2*b + 8*a^3))/(b^5*(- a*b - a^2)^(3/2)
*(-a*(a + b))^(1/2)) + (2*(8*a*b + 8*a^2 + b^2)*(8*a^2*(- a*b - a^2)^(1/2) + b^2*(- a*b - a^2)^(1/2) + 8*a*b*(
- a*b - a^2)^(1/2)))/(a*b^5*(a + b)*(- a*b - a^2)^(3/2))))/4 + ((2*a*b^2 + 2*a^2*b)*(4*a + 2*b))/(b^3*(-a*(a +
b))^(1/2)) + ((b^2*(- a*b - a^2)^(1/2) + 2*a*b*(- a*b - a^2)^(1/2))*(8*a*b + 8*a^2 + b^2))/(2*a*b^3*(a + b)))
/(- a*b - a^2)^(1/2)

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