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

Optimal. Leaf size=39 \[ \frac {x}{b}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{b \sqrt {a+b}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3171, 3181, 208} \[ \frac {x}{b}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{b \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^2/(a + b*Cosh[x]^2),x]

[Out]

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

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 3171

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(B*x
)/b, x] + Dist[(A*b - a*B)/b, Int[1/(a + b*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B}, x]

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

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Mathematica [A]  time = 0.08, size = 36, normalized size = 0.92 \[ \frac {x-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^2/(a + b*Cosh[x]^2),x]

[Out]

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

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fricas [A]  time = 0.49, size = 317, normalized size = 8.13 \[ \left [\frac {\sqrt {\frac {a}{a + b}} \log \left (\frac {b^{2} \cosh \relax (x)^{4} + 4 \, b^{2} \cosh \relax (x) \sinh \relax (x)^{3} + b^{2} \sinh \relax (x)^{4} + 2 \, {\left (2 \, a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} + 2 \, a b + b^{2}\right )} \sinh \relax (x)^{2} + 8 \, a^{2} + 8 \, a b + b^{2} + 4 \, {\left (b^{2} \cosh \relax (x)^{3} + {\left (2 \, a b + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 4 \, {\left ({\left (a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a b + b^{2}\right )} \sinh \relax (x)^{2} + 2 \, a^{2} + 3 \, a b + b^{2}\right )} \sqrt {\frac {a}{a + b}}}{b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} + 2 \, a + b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} + {\left (2 \, a + b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + b}\right ) + 2 \, x}{2 \, b}, -\frac {\sqrt {-\frac {a}{a + b}} \arctan \left (\frac {{\left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} + 2 \, a + b\right )} \sqrt {-\frac {a}{a + b}}}{2 \, a}\right ) - x}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(a/(a + b))*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*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 + 2*a^2 + 3*a*b
+ b^2)*sqrt(a/(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*co
sh(x)^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + 2*x)/b, -(sqrt(-a/(a + b))*
arctan(1/2*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 2*a + b)*sqrt(-a/(a + b))/a) - x)/b]

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giac [A]  time = 0.13, size = 50, normalized size = 1.28 \[ -\frac {a \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} b} + \frac {x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.10, size = 110, normalized size = 2.82 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}+\frac {\sqrt {a}\, \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {a}\, \tanh \left (\frac {x}{2}\right )-\sqrt {a +b}\right )}{2 b \sqrt {a +b}}-\frac {\sqrt {a}\, \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {a}\, \tanh \left (\frac {x}{2}\right )+\sqrt {a +b}\right )}{2 b \sqrt {a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2/(a+b*cosh(x)^2),x)

[Out]

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

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maxima [B]  time = 0.51, size = 120, normalized size = 3.08 \[ -\frac {{\left (2 \, a + b\right )} \log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{4 \, \sqrt {{\left (a + b\right )} a} b} - \frac {\log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{4 \, \sqrt {{\left (a + b\right )} a}} + \frac {x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a + b)*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)*b) - 1/4*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) + x/b

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mupad [B]  time = 1.38, size = 376, normalized size = 9.64 \[ \frac {x}{b}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\left (b^5\,\sqrt {-b^3-a\,b^2}+a\,b^4\,\sqrt {-b^3-a\,b^2}\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (\frac {2\,\left (8\,a^{5/2}\,\sqrt {-b^3-a\,b^2}+\sqrt {a}\,b^2\,\sqrt {-b^3-a\,b^2}+8\,a^{3/2}\,b\,\sqrt {-b^3-a\,b^2}\right )\,\left (8\,a^2+8\,a\,b+b^2\right )}{b^8\,{\left (a+b\right )}^2\,\sqrt {-b^3-a\,b^2}}+\frac {4\,\sqrt {a}\,\left (4\,a+2\,b\right )\,\left (8\,a^3\,b+12\,a^2\,b^2+4\,a\,b^3\right )}{b^7\,\left (a+b\right )\,\sqrt {-b^2\,\left (a+b\right )}\,\sqrt {-b^3-a\,b^2}}\right )+\frac {2\,\left (\sqrt {a}\,b^2\,\sqrt {-b^3-a\,b^2}+2\,a^{3/2}\,b\,\sqrt {-b^3-a\,b^2}\right )\,\left (8\,a^2+8\,a\,b+b^2\right )}{b^8\,{\left (a+b\right )}^2\,\sqrt {-b^3-a\,b^2}}+\frac {4\,\sqrt {a}\,\left (2\,a^2\,b^2+2\,a\,b^3\right )\,\left (4\,a+2\,b\right )}{b^7\,\left (a+b\right )\,\sqrt {-b^2\,\left (a+b\right )}\,\sqrt {-b^3-a\,b^2}}\right )}{4\,a}\right )}{\sqrt {-b^3-a\,b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2/(a + b*cosh(x)^2),x)

[Out]

x/b + (a^(1/2)*atan(((b^5*(- a*b^2 - b^3)^(1/2) + a*b^4*(- a*b^2 - b^3)^(1/2))*(exp(2*x)*((2*(8*a^(5/2)*(- a*b
^2 - b^3)^(1/2) + a^(1/2)*b^2*(- a*b^2 - b^3)^(1/2) + 8*a^(3/2)*b*(- a*b^2 - b^3)^(1/2))*(8*a*b + 8*a^2 + b^2)
)/(b^8*(a + b)^2*(- a*b^2 - b^3)^(1/2)) + (4*a^(1/2)*(4*a + 2*b)*(4*a*b^3 + 8*a^3*b + 12*a^2*b^2))/(b^7*(a + b
)*(-b^2*(a + b))^(1/2)*(- a*b^2 - b^3)^(1/2))) + (2*(a^(1/2)*b^2*(- a*b^2 - b^3)^(1/2) + 2*a^(3/2)*b*(- a*b^2
- b^3)^(1/2))*(8*a*b + 8*a^2 + b^2))/(b^8*(a + b)^2*(- a*b^2 - b^3)^(1/2)) + (4*a^(1/2)*(2*a*b^3 + 2*a^2*b^2)*
(4*a + 2*b))/(b^7*(a + b)*(-b^2*(a + b))^(1/2)*(- a*b^2 - b^3)^(1/2))))/(4*a)))/(- a*b^2 - b^3)^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**2/(a+b*cosh(x)**2),x)

[Out]

Timed out

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