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

Optimal. Leaf size=38 \[ \frac {x}{a+b}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {391, 206, 205} \[ \frac {x}{a+b}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 205

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 391

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 33, normalized size = 0.87 \[ \frac {\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {a}}+x}{a+b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 363, normalized size = 9.55 \[ \left [\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \relax (x)^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{2} + a^{2} - b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - 6 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 4 \, {\left ({\left (a^{2} + a b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{2} + a b\right )} \sinh \relax (x)^{2} + a^{2} - a b\right )} \sqrt {-\frac {b}{a}}}{{\left (a + b\right )} \cosh \relax (x)^{4} + 4 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a + b\right )} \sinh \relax (x)^{4} + 2 \, {\left (a - b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \relax (x)^{2} + a - b\right )} \sinh \relax (x)^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \relax (x)^{3} + {\left (a - b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + a + b}\right ) + 2 \, x}{2 \, {\left (a + b\right )}}, \frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} + a - b\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) + x}{a + b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(-b/a)*log(((a^2 + 2*a*b + b^2)*cosh(x)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)^3 + (a^2 + 2*a*b +
b^2)*sinh(x)^4 + 2*(a^2 - b^2)*cosh(x)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(x)^2 + a^2 - b^2)*sinh(x)^2 + a^2 -
6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(x)^3 + (a^2 - b^2)*cosh(x))*sinh(x) + 4*((a^2 + a*b)*cosh(x)^2 + 2*(
a^2 + a*b)*cosh(x)*sinh(x) + (a^2 + a*b)*sinh(x)^2 + a^2 - a*b)*sqrt(-b/a))/((a + b)*cosh(x)^4 + 4*(a + b)*cos
h(x)*sinh(x)^3 + (a + b)*sinh(x)^4 + 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 + a - b)*sinh(x)^2 + 4*((a +
b)*cosh(x)^3 + (a - b)*cosh(x))*sinh(x) + a + b)) + 2*x)/(a + b), (sqrt(b/a)*arctan(1/2*((a + b)*cosh(x)^2 +
2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)*sqrt(b/a)/b) + x)/(a + b)]

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giac [A]  time = 0.14, size = 45, normalized size = 1.18 \[ \frac {b \arctan \left (\frac {a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} {\left (a + b\right )}} + \frac {x}{a + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.21, size = 389, normalized size = 10.24 \[ -\frac {2 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b +2 a}+\frac {2 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b +2 a}-\frac {b a \arctanh \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{\left (a +b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {b \arctanh \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{\left (a +b \right ) \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {b^{2} \arctanh \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{\left (a +b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {b a \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{\left (a +b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {b \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{\left (a +b \right ) \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {b^{2} \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{\left (a +b \right ) \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.44, size = 74, normalized size = 1.95 \[ -\frac {{\left (a - b\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (2 \, x\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (a + b\right )}} - \frac {\arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, x\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b}} + \frac {x}{a + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.95, size = 208, normalized size = 5.47 \[ \frac {x}{a+b}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{2\,x}\,\left (\frac {4\,b}{{\left (a+b\right )}^4}+\frac {\left (a^2-b^2\right )\,\left (a-b\right )}{{\left (a+b\right )}^3\,\sqrt {a\,{\left (a+b\right )}^2}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}}\right )+\frac {\left (a-b\right )\,\left (a^2+2\,a\,b+b^2\right )}{{\left (a+b\right )}^3\,\sqrt {a\,{\left (a+b\right )}^2}\,\sqrt {a^3+2\,a^2\,b+a\,b^2}}\right )\,\left (a^2\,\sqrt {a^3+2\,a^2\,b+a\,b^2}+b^2\,\sqrt {a^3+2\,a^2\,b+a\,b^2}+2\,a\,b\,\sqrt {a^3+2\,a^2\,b+a\,b^2}\right )}{2\,\sqrt {b}}\right )}{\sqrt {a^3+2\,a^2\,b+a\,b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.69, size = 265, normalized size = 6.97 \[ \begin {cases} \tilde {\infty } \left (x - \frac {\cosh {\relax (x )}}{\sinh {\relax (x )}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {x - \frac {\cosh {\relax (x )}}{\sinh {\relax (x )}}}{b} & \text {for}\: a = 0 \\\frac {x \sinh ^{2}{\relax (x )}}{- 2 b \sinh ^{2}{\relax (x )} + 2 b \cosh ^{2}{\relax (x )}} - \frac {x \cosh ^{2}{\relax (x )}}{- 2 b \sinh ^{2}{\relax (x )} + 2 b \cosh ^{2}{\relax (x )}} - \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{- 2 b \sinh ^{2}{\relax (x )} + 2 b \cosh ^{2}{\relax (x )}} & \text {for}\: a = - b \\\frac {x}{a} & \text {for}\: b = 0 \\\frac {2 i a \sqrt {b} x \sqrt {\frac {1}{a}}}{2 i a^{2} \sqrt {b} \sqrt {\frac {1}{a}} + 2 i a b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} - \frac {b \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} \sinh {\relax (x )} + \cosh {\relax (x )} \right )}}{2 i a^{2} \sqrt {b} \sqrt {\frac {1}{a}} + 2 i a b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} + \frac {b \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} \sinh {\relax (x )} + \cosh {\relax (x )} \right )}}{2 i a^{2} \sqrt {b} \sqrt {\frac {1}{a}} + 2 i a b^{\frac {3}{2}} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(x - cosh(x)/sinh(x)), Eq(a, 0) & Eq(b, 0)), ((x - cosh(x)/sinh(x))/b, Eq(a, 0)), (x*sinh(x)**2
/(-2*b*sinh(x)**2 + 2*b*cosh(x)**2) - x*cosh(x)**2/(-2*b*sinh(x)**2 + 2*b*cosh(x)**2) - sinh(x)*cosh(x)/(-2*b*
sinh(x)**2 + 2*b*cosh(x)**2), Eq(a, -b)), (x/a, Eq(b, 0)), (2*I*a*sqrt(b)*x*sqrt(1/a)/(2*I*a**2*sqrt(b)*sqrt(1
/a) + 2*I*a*b**(3/2)*sqrt(1/a)) - b*log(-I*sqrt(b)*sqrt(1/a)*sinh(x) + cosh(x))/(2*I*a**2*sqrt(b)*sqrt(1/a) +
2*I*a*b**(3/2)*sqrt(1/a)) + b*log(I*sqrt(b)*sqrt(1/a)*sinh(x) + cosh(x))/(2*I*a**2*sqrt(b)*sqrt(1/a) + 2*I*a*b
**(3/2)*sqrt(1/a)), True))

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