3.9.39 \(\int \frac {\cosh ^2(x)}{a \cosh ^2(x)+b \sinh ^2(x)} \, dx\) [839]

Optimal. Leaf size=38 \[ \frac {x}{a+b}+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)} \]

[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 = 0.08, 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 = {400, 212, 211} \begin {gather*} \frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)}+\frac {x}{a+b} \end {gather*}

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 211

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

Rule 212

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

Rule 400

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 &=\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right )}{a+b}+\frac {b \text {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 = 0.04, size = 33, normalized size = 0.87 \begin {gather*} \frac {x+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a}}\right )}{\sqrt {a}}}{a+b} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(30)=60\).
time = 1.22, size = 186, normalized size = 4.89

method	result	size
risch	\(\frac {x}{a +b}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 x}+\frac {a +2 \sqrt {-a b}-b}{a +b}\right )}{2 a \left (a +b \right )}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{2 a \left (a +b \right )}\)	\(92\)
default	\(\frac {2 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 b +2 a}-\frac {2 b a \left (\frac {\left (a +\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 \sqrt {b \left (a +b \right )}\, a \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 \sqrt {b \left (a +b \right )}\, a \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{a +b}-\frac {2 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b +2 a}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (30) = 60\).
time = 0.48, size = 74, normalized size = 1.95 \begin {gather*} -\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} \end {gather*}

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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Fricas [A]
time = 0.37, size = 363, normalized size = 9.55 \begin {gather*} \left [\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - 6 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, {\left ({\left (a^{2} + a b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} + a b\right )} \sinh \left (x\right )^{2} + a^{2} - a b\right )} \sqrt {-\frac {b}{a}}}{{\left (a + b\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a + b\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (x\right )^{2} + a - b\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (x\right )^{3} + {\left (a - b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + a + b}\right ) + 2 \, x}{2 \, {\left (a + b\right )}}, \frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b\right )} \sqrt {\frac {b}{a}}}{2 \, b}\right ) + x}{a + b}\right ] \end {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (32) = 64\).
time = 0.72, size = 224, normalized size = 5.89 \begin {gather*} \begin {cases} \tilde {\infty } \left (x - \frac {\cosh {\left (x \right )}}{\sinh {\left (x \right )}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {x \sinh ^{2}{\left (x \right )}}{- 2 b \sinh ^{2}{\left (x \right )} + 2 b \cosh ^{2}{\left (x \right )}} - \frac {x \cosh ^{2}{\left (x \right )}}{- 2 b \sinh ^{2}{\left (x \right )} + 2 b \cosh ^{2}{\left (x \right )}} - \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{- 2 b \sinh ^{2}{\left (x \right )} + 2 b \cosh ^{2}{\left (x \right )}} & \text {for}\: a = - b \\\frac {x}{a} & \text {for}\: b = 0 \\\frac {x - \frac {\cosh {\left (x \right )}}{\sinh {\left (x \right )}}}{b} & \text {for}\: a = 0 \\\frac {2 a x \sqrt {- \frac {b}{a}}}{2 a^{2} \sqrt {- \frac {b}{a}} + 2 a b \sqrt {- \frac {b}{a}}} - \frac {b \log {\left (- \sqrt {- \frac {b}{a}} \sinh {\left (x \right )} + \cosh {\left (x \right )} \right )}}{2 a^{2} \sqrt {- \frac {b}{a}} + 2 a b \sqrt {- \frac {b}{a}}} + \frac {b \log {\left (\sqrt {- \frac {b}{a}} \sinh {\left (x \right )} + \cosh {\left (x \right )} \right )}}{2 a^{2} \sqrt {- \frac {b}{a}} + 2 a b \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \end {gather*}

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*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)), ((x - cosh(x)/sinh(x))/b, Eq(a, 0)), (2*a*x*sqrt(-b/a)/(2*a**2*sqrt(-b/a) + 2*a*b*sqrt(-
b/a)) - b*log(-sqrt(-b/a)*sinh(x) + cosh(x))/(2*a**2*sqrt(-b/a) + 2*a*b*sqrt(-b/a)) + b*log(sqrt(-b/a)*sinh(x)
+ cosh(x))/(2*a**2*sqrt(-b/a) + 2*a*b*sqrt(-b/a)), True))

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Giac [A]
time = 0.42, size = 45, normalized size = 1.18 \begin {gather*} \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} \end {gather*}

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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Mupad [B]
time = 1.95, size = 208, normalized size = 5.47 \begin {gather*} \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}} \end {gather*}

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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