3.1.33 \(\int \frac {1}{(a+b \cosh ^2(x))^2} \, dx\) [33]

Optimal. Leaf size=65 \[ \frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \cosh (x) \sinh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )} \]

[Out]

1/2*(2*a+b)*arctanh(a^(1/2)*tanh(x)/(a+b)^(1/2))/a^(3/2)/(a+b)^(3/2)-1/2*b*cosh(x)*sinh(x)/a/(a+b)/(a+b*cosh(x
)^2)

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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3263, 12, 3260, 214} \begin {gather*} \frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \sinh (x) \cosh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^2)^(-2),x]

[Out]

((2*a + b)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b)^(3/2)) - (b*Cosh[x]*Sinh[x])/(2*a*(a + b
)*(a + b*Cosh[x]^2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 3263

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si
n[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^
(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && N
eQ[a + b, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \cosh ^2(x)\right )^2} \, dx &=-\frac {b \cosh (x) \sinh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}-\frac {\int \frac {-2 a-b}{a+b \cosh ^2(x)} \, dx}{2 a (a+b)}\\ &=-\frac {b \cosh (x) \sinh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}+\frac {(2 a+b) \int \frac {1}{a+b \cosh ^2(x)} \, dx}{2 a (a+b)}\\ &=-\frac {b \cosh (x) \sinh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}+\frac {(2 a+b) \text {Subst}\left (\int \frac {1}{a-(a+b) x^2} \, dx,x,\coth (x)\right )}{2 a (a+b)}\\ &=\frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \cosh (x) \sinh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 68, normalized size = 1.05 \begin {gather*} \frac {(2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {b \sinh (2 x)}{2 a (a+b) (2 a+b+b \cosh (2 x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^2)^(-2),x]

[Out]

((2*a + b)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b)^(3/2)) - (b*Sinh[2*x])/(2*a*(a + b)*(2*a
+ b + b*Cosh[2*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(53)=106\).
time = 0.53, size = 168, normalized size = 2.58

method	result	size
default	\(-\frac {2 \left (\frac {b \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2 a \left (a +b \right )}+\frac {b \tanh \left (\frac {x}{2}\right )}{2 a \left (a +b \right )}\right )}{a \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b}-\frac {\left (2 a +b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}\right )}{a \left (a +b \right )}\)	\(168\)
risch	\(\frac {2 \,{\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+b}{a \left (a +b \right ) \left ({\mathrm e}^{4 x} b +4 \,{\mathrm e}^{2 x} a +2 b \,{\mathrm e}^{2 x}+b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right ) b}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) a}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right ) b}{4 \sqrt {a^{2}+a b}\, \left (a +b \right ) a}\)	\(330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(1/2*b/a/(a+b)*tanh(1/2*x)^3+1/2*b/a/(a+b)*tanh(1/2*x))/(a*tanh(1/2*x)^4+b*tanh(1/2*x)^4-2*a*tanh(1/2*x)^2+
2*b*tanh(1/2*x)^2+a+b)-(2*a+b)/a/(a+b)*(-1/4/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))+1/4/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. 134 vs. \(2 (53) = 106\).
time = 0.49, size = 134, normalized size = 2.06 \begin {gather*} -\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} {\left (a^{2} + a b\right )}} - \frac {{\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} + b}{a^{2} b + a b^{2} + 2 \, {\left (2 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} b + a b^{2}\right )} e^{\left (-4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^2)^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)))/(sqr
t((a + b)*a)*(a^2 + a*b)) - ((2*a + b)*e^(-2*x) + b)/(a^2*b + a*b^2 + 2*(2*a^3 + 3*a^2*b + a*b^2)*e^(-2*x) + (
a^2*b + a*b^2)*e^(-4*x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (53) = 106\).
time = 0.40, size = 1239, normalized size = 19.06 \begin {gather*} \left [\frac {4 \, a^{2} b + 4 \, a b^{2} + 4 \, {\left (2 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} \cosh \left (x\right )^{2} + 8 \, {\left (2 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, {\left (2 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} \sinh \left (x\right )^{2} + {\left ({\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (4 \, a^{2} + 4 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 4 \, a^{2} + 4 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, a b + b^{2} + 4 \, {\left ({\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + {\left (4 \, a^{2} + 4 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} + a b} \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 )}{4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a^{5} + 5 \, a^{4} b + 4 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (2 \, a^{5} + 5 \, a^{4} b + 4 \, a^{3} b^{2} + a^{2} b^{3} + 3 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{3} + {\left (2 \, a^{5} + 5 \, a^{4} b + 4 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}}, \frac {2 \, a^{2} b + 2 \, a b^{2} + 2 \, {\left (2 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (2 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, {\left (2 \, a^{3} + 3 \, a^{2} b + a b^{2}\right )} \sinh \left (x\right )^{2} + {\left ({\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (4 \, a^{2} + 4 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 4 \, a^{2} + 4 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, a b + b^{2} + 4 \, {\left ({\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + {\left (4 \, a^{2} + 4 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \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 )}{2 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a^{5} + 5 \, a^{4} b + 4 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (2 \, a^{5} + 5 \, a^{4} b + 4 \, a^{3} b^{2} + a^{2} b^{3} + 3 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )^{3} + {\left (2 \, a^{5} + 5 \, a^{4} b + 4 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*a^2*b + 4*a*b^2 + 4*(2*a^3 + 3*a^2*b + a*b^2)*cosh(x)^2 + 8*(2*a^3 + 3*a^2*b + a*b^2)*cosh(x)*sinh(x)
+ 4*(2*a^3 + 3*a^2*b + a*b^2)*sinh(x)^2 + ((2*a*b + b^2)*cosh(x)^4 + 4*(2*a*b + b^2)*cosh(x)*sinh(x)^3 + (2*a*
b + b^2)*sinh(x)^4 + 2*(4*a^2 + 4*a*b + b^2)*cosh(x)^2 + 2*(3*(2*a*b + b^2)*cosh(x)^2 + 4*a^2 + 4*a*b + b^2)*s
inh(x)^2 + 2*a*b + b^2 + 4*((2*a*b + b^2)*cosh(x)^3 + (4*a^2 + 4*a*b + b^2)*cosh(x))*sinh(x))*sqrt(a^2 + 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*(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)*cosh(x
))*sinh(x) + b)))/(a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*cosh(x)^4 + 4*(a^4*b + 2*a^3*b^
2 + a^2*b^3)*cosh(x)*sinh(x)^3 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*sinh(x)^4 + 2*(2*a^5 + 5*a^4*b + 4*a^3*b^2 + a^
2*b^3)*cosh(x)^2 + 2*(2*a^5 + 5*a^4*b + 4*a^3*b^2 + a^2*b^3 + 3*(a^4*b + 2*a^3*b^2 + a^2*b^3)*cosh(x)^2)*sinh(
x)^2 + 4*((a^4*b + 2*a^3*b^2 + a^2*b^3)*cosh(x)^3 + (2*a^5 + 5*a^4*b + 4*a^3*b^2 + a^2*b^3)*cosh(x))*sinh(x)),
1/2*(2*a^2*b + 2*a*b^2 + 2*(2*a^3 + 3*a^2*b + a*b^2)*cosh(x)^2 + 4*(2*a^3 + 3*a^2*b + a*b^2)*cosh(x)*sinh(x)
+ 2*(2*a^3 + 3*a^2*b + a*b^2)*sinh(x)^2 + ((2*a*b + b^2)*cosh(x)^4 + 4*(2*a*b + b^2)*cosh(x)*sinh(x)^3 + (2*a*
b + b^2)*sinh(x)^4 + 2*(4*a^2 + 4*a*b + b^2)*cosh(x)^2 + 2*(3*(2*a*b + b^2)*cosh(x)^2 + 4*a^2 + 4*a*b + b^2)*s
inh(x)^2 + 2*a*b + b^2 + 4*((2*a*b + b^2)*cosh(x)^3 + (4*a^2 + 4*a*b + b^2)*cosh(x))*sinh(x))*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^4*b
+ 2*a^3*b^2 + a^2*b^3 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*cosh(x)^4 + 4*(a^4*b + 2*a^3*b^2 + a^2*b^3)*cosh(x)*sinh
(x)^3 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*sinh(x)^4 + 2*(2*a^5 + 5*a^4*b + 4*a^3*b^2 + a^2*b^3)*cosh(x)^2 + 2*(2*a
^5 + 5*a^4*b + 4*a^3*b^2 + a^2*b^3 + 3*(a^4*b + 2*a^3*b^2 + a^2*b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a^4*b + 2*a^3*
b^2 + a^2*b^3)*cosh(x)^3 + (2*a^5 + 5*a^4*b + 4*a^3*b^2 + a^2*b^3)*cosh(x))*sinh(x))]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)**2)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.42, size = 104, normalized size = 1.60 \begin {gather*} \frac {{\left (2 \, a + b\right )} \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{2 \, {\left (a^{2} + a b\right )} \sqrt {-a^{2} - a b}} + \frac {2 \, a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + b}{{\left (a^{2} + a b\right )} {\left (b e^{\left (4 \, x\right )} + 4 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a + b)*arctan(1/2*(b*e^(2*x) + 2*a + b)/sqrt(-a^2 - a*b))/((a^2 + a*b)*sqrt(-a^2 - a*b)) + (2*a*e^(2*x)
+ b*e^(2*x) + b)/((a^2 + a*b)*(b*e^(4*x) + 4*a*e^(2*x) + 2*b*e^(2*x) + b))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (b\,{\mathrm {cosh}\left (x\right )}^2+a\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x)^2)^2,x)

[Out]

int(1/(a + b*cosh(x)^2)^2, x)

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