3.34 \(\int \frac {1}{(a+b \cosh ^2(x))^3} \, dx\)

Optimal. Leaf size=107 \[ -\frac {3 b (2 a+b) \sinh (x) \cosh (x)}{8 a^2 (a+b)^2 \left (a+b \cosh ^2(x)\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2}}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2} \]

[Out]

1/8*(8*a^2+8*a*b+3*b^2)*arctanh(a^(1/2)*tanh(x)/(a+b)^(1/2))/a^(5/2)/(a+b)^(5/2)-1/4*b*cosh(x)*sinh(x)/a/(a+b)
/(a+b*cosh(x)^2)^2-3/8*b*(2*a+b)*cosh(x)*sinh(x)/a^2/(a+b)^2/(a+b*cosh(x)^2)

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Rubi [A]  time = 0.12, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3184, 3173, 12, 3181, 208} \[ \frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2}}-\frac {3 b (2 a+b) \sinh (x) \cosh (x)}{8 a^2 (a+b)^2 \left (a+b \cosh ^2(x)\right )}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^2)^(-3),x]

[Out]

((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(8*a^(5/2)*(a + b)^(5/2)) - (b*Cosh[x]*Sinh[x
])/(4*a*(a + b)*(a + b*Cosh[x]^2)^2) - (3*b*(2*a + b)*Cosh[x]*Sinh[x])/(8*a^2*(a + b)^2*(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 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 3173

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

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]

Rule 3184

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
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] && NeQ
[a + b, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx &=-\frac {b \cosh (x) \sinh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}-\frac {\int \frac {-4 a-3 b+2 b \cosh ^2(x)}{\left (a+b \cosh ^2(x)\right )^2} \, dx}{4 a (a+b)}\\ &=-\frac {b \cosh (x) \sinh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}-\frac {3 b (2 a+b) \cosh (x) \sinh (x)}{8 a^2 (a+b)^2 \left (a+b \cosh ^2(x)\right )}-\frac {\int \frac {-8 a^2-8 a b-3 b^2}{a+b \cosh ^2(x)} \, dx}{8 a^2 (a+b)^2}\\ &=-\frac {b \cosh (x) \sinh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}-\frac {3 b (2 a+b) \cosh (x) \sinh (x)}{8 a^2 (a+b)^2 \left (a+b \cosh ^2(x)\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \int \frac {1}{a+b \cosh ^2(x)} \, dx}{8 a^2 (a+b)^2}\\ &=-\frac {b \cosh (x) \sinh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}-\frac {3 b (2 a+b) \cosh (x) \sinh (x)}{8 a^2 (a+b)^2 \left (a+b \cosh ^2(x)\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-(a+b) x^2} \, dx,x,\coth (x)\right )}{8 a^2 (a+b)^2}\\ &=\frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2}}-\frac {b \cosh (x) \sinh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}-\frac {3 b (2 a+b) \cosh (x) \sinh (x)}{8 a^2 (a+b)^2 \left (a+b \cosh ^2(x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 106, normalized size = 0.99 \[ \frac {\frac {\left (8 a^2+8 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {\sqrt {a} b \sinh (2 x) \left (16 a^2+3 b (2 a+b) \cosh (2 x)+16 a b+3 b^2\right )}{(a+b)^2 (2 a+b \cosh (2 x)+b)^2}}{8 a^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^2)^(-3),x]

[Out]

(((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(a + b)^(5/2) - (Sqrt[a]*b*(16*a^2 + 16*a*b
+ 3*b^2 + 3*b*(2*a + b)*Cosh[2*x])*Sinh[2*x])/((a + b)^2*(2*a + b + b*Cosh[2*x])^2))/(8*a^(5/2))

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fricas [B]  time = 0.60, size = 5117, normalized size = 47.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^6 + 24*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a
*b^4)*cosh(x)*sinh(x)^5 + 4*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*sinh(x)^6 + 24*a^3*b^2 + 36*a^2*b^3
+ 12*a*b^4 + 12*(16*a^5 + 40*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 + 3*a*b^4)*cosh(x)^4 + 12*(16*a^5 + 40*a^4*b + 38
*a^3*b^2 + 17*a^2*b^3 + 3*a*b^4 + 5*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^2)*sinh(x)^4 + 16*(5
*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^3 + 3*(16*a^5 + 40*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 + 3*
a*b^4)*cosh(x))*sinh(x)^3 + 4*(40*a^4*b + 80*a^3*b^2 + 49*a^2*b^3 + 9*a*b^4)*cosh(x)^2 + 4*(40*a^4*b + 80*a^3*
b^2 + 49*a^2*b^3 + 9*a*b^4 + 15*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^4 + 18*(16*a^5 + 40*a^4*
b + 38*a^3*b^2 + 17*a^2*b^3 + 3*a*b^4)*cosh(x)^2)*sinh(x)^2 + ((8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^8 + 8*(8*
a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)*sinh(x)^7 + (8*a^2*b^2 + 8*a*b^3 + 3*b^4)*sinh(x)^8 + 4*(16*a^3*b + 24*a^2*
b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^6 + 4*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4 + 7*(8*a^2*b^2 + 8*a*b^3 + 3*b
^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^3 + 3*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3
+ 3*b^4)*cosh(x))*sinh(x)^5 + 2*(64*a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4)*cosh(x)^4 + 2*(35*(8*a^2
*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^4 + 64*a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4 + 30*(16*a^3*b + 24*a^
2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^2)*sinh(x)^4 + 8*a^2*b^2 + 8*a*b^3 + 3*b^4 + 8*(7*(8*a^2*b^2 + 8*a*b^3 + 3*b
^4)*cosh(x)^5 + 10*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^3 + (64*a^4 + 128*a^3*b + 112*a^2*b^2 +
48*a*b^3 + 9*b^4)*cosh(x))*sinh(x)^3 + 4*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^2 + 4*(7*(8*a^2*b^
2 + 8*a*b^3 + 3*b^4)*cosh(x)^6 + 15*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^4 + 16*a^3*b + 24*a^2*b
^2 + 14*a*b^3 + 3*b^4 + 3*(64*a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((8*a
^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^7 + 3*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^5 + (64*a^4 + 128*a
^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4)*cosh(x)^3 + (16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*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))*s
inh(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*cos
h(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)) + 8*(3*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^5 + 6*(16*a^5
+ 40*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 + 3*a*b^4)*cosh(x)^3 + (40*a^4*b + 80*a^3*b^2 + 49*a^2*b^3 + 9*a*b^4)*cos
h(x))*sinh(x))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^8 + 8*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a
^3*b^5)*cosh(x)*sinh(x)^7 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*sinh(x)^8 + a^6*b^2 + 3*a^5*b^3 + 3*a^
4*b^4 + a^3*b^5 + 4*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^6 + 4*(2*a^7*b + 7*a^6*b^2
+ 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5 + 7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^2)*sinh(x)^6 + 8*(7
*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^3 + 3*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*
b^5)*cosh(x))*sinh(x)^5 + 2*(8*a^8 + 32*a^7*b + 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*cosh(x)^4 +
2*(8*a^8 + 32*a^7*b + 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5 + 35*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 +
a^3*b^5)*cosh(x)^4 + 30*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(
a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^5 + 10*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b
^5)*cosh(x)^3 + (8*a^8 + 32*a^7*b + 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*cosh(x))*sinh(x)^3 + 4*(
2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^2 + 4*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*
b^4 + a^3*b^5 + 7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^6 + 15*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3
+ 5*a^4*b^4 + a^3*b^5)*cosh(x)^4 + 3*(8*a^8 + 32*a^7*b + 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*cos
h(x)^2)*sinh(x)^2 + 8*((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^7 + 3*(2*a^7*b + 7*a^6*b^2 + 9*a^5*
b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^5 + (8*a^8 + 32*a^7*b + 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*c
osh(x)^3 + (2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x))*sinh(x)), 1/8*(2*(8*a^4*b + 16*a^3
*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^6 + 12*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)*sinh(x)^5 +
2*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*sinh(x)^6 + 12*a^3*b^2 + 18*a^2*b^3 + 6*a*b^4 + 6*(16*a^5 + 40
*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 + 3*a*b^4)*cosh(x)^4 + 6*(16*a^5 + 40*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 + 3*a*b
^4 + 5*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(5*(8*a^4*b + 16*a^3*b^2 + 11*a^
2*b^3 + 3*a*b^4)*cosh(x)^3 + 3*(16*a^5 + 40*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 + 3*a*b^4)*cosh(x))*sinh(x)^3 + 2*
(40*a^4*b + 80*a^3*b^2 + 49*a^2*b^3 + 9*a*b^4)*cosh(x)^2 + 2*(40*a^4*b + 80*a^3*b^2 + 49*a^2*b^3 + 9*a*b^4 + 1
5*(8*a^4*b + 16*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^4 + 18*(16*a^5 + 40*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 +
3*a*b^4)*cosh(x)^2)*sinh(x)^2 + ((8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^8 + 8*(8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cos
h(x)*sinh(x)^7 + (8*a^2*b^2 + 8*a*b^3 + 3*b^4)*sinh(x)^8 + 4*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x
)^6 + 4*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4 + 7*(8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(
7*(8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^3 + 3*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x))*sinh(x)^5 +
2*(64*a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4)*cosh(x)^4 + 2*(35*(8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x
)^4 + 64*a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4 + 30*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh
(x)^2)*sinh(x)^4 + 8*a^2*b^2 + 8*a*b^3 + 3*b^4 + 8*(7*(8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^5 + 10*(16*a^3*b +
24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^3 + (64*a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4)*cosh(x))*sin
h(x)^3 + 4*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^2 + 4*(7*(8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh(x)^6
+ 15*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^4 + 16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4 + 3*(64*
a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3 + 9*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cosh
(x)^7 + 3*(16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(x)^5 + (64*a^4 + 128*a^3*b + 112*a^2*b^2 + 48*a*b^3
+ 9*b^4)*cosh(x)^3 + (16*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*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)) + 4*(3*(8*a^4*b + 16
*a^3*b^2 + 11*a^2*b^3 + 3*a*b^4)*cosh(x)^5 + 6*(16*a^5 + 40*a^4*b + 38*a^3*b^2 + 17*a^2*b^3 + 3*a*b^4)*cosh(x)
^3 + (40*a^4*b + 80*a^3*b^2 + 49*a^2*b^3 + 9*a*b^4)*cosh(x))*sinh(x))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*
b^5)*cosh(x)^8 + 8*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)*sinh(x)^7 + (a^6*b^2 + 3*a^5*b^3 + 3*a^
4*b^4 + a^3*b^5)*sinh(x)^8 + a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5 + 4*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 +
5*a^4*b^4 + a^3*b^5)*cosh(x)^6 + 4*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5 + 7*(a^6*b^2 + 3*a^5
*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^3
+ 3*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x))*sinh(x)^5 + 2*(8*a^8 + 32*a^7*b + 51*a^6*
b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*cosh(x)^4 + 2*(8*a^8 + 32*a^7*b + 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*
b^4 + 3*a^3*b^5 + 35*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^4 + 30*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b
^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*cosh(x)^5 +
10*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^3 + (8*a^8 + 32*a^7*b + 51*a^6*b^2 + 41*a^5
*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*cosh(x))*sinh(x)^3 + 4*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*
cosh(x)^2 + 4*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5 + 7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^
3*b^5)*cosh(x)^6 + 15*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^4 + 3*(8*a^8 + 32*a^7*b
+ 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4
+ a^3*b^5)*cosh(x)^7 + 3*(2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*cosh(x)^5 + (8*a^8 + 32*a^7*
b + 51*a^6*b^2 + 41*a^5*b^3 + 17*a^4*b^4 + 3*a^3*b^5)*cosh(x)^3 + (2*a^7*b + 7*a^6*b^2 + 9*a^5*b^3 + 5*a^4*b^4
+ a^3*b^5)*cosh(x))*sinh(x))]

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giac [B]  time = 0.68, size = 228, normalized size = 2.13 \[ \frac {{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{8 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {-a^{2} - a b}} + \frac {8 \, a^{2} b e^{\left (6 \, x\right )} + 8 \, a b^{2} e^{\left (6 \, x\right )} + 3 \, b^{3} e^{\left (6 \, x\right )} + 48 \, a^{3} e^{\left (4 \, x\right )} + 72 \, a^{2} b e^{\left (4 \, x\right )} + 42 \, a b^{2} e^{\left (4 \, x\right )} + 9 \, b^{3} e^{\left (4 \, x\right )} + 40 \, a^{2} b e^{\left (2 \, x\right )} + 40 \, a b^{2} e^{\left (2 \, x\right )} + 9 \, b^{3} e^{\left (2 \, x\right )} + 6 \, a b^{2} + 3 \, b^{3}}{4 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (b e^{\left (4 \, x\right )} + 4 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + b\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^2)^3,x, algorithm="giac")

[Out]

1/8*(8*a^2 + 8*a*b + 3*b^2)*arctan(1/2*(b*e^(2*x) + 2*a + b)/sqrt(-a^2 - a*b))/((a^4 + 2*a^3*b + a^2*b^2)*sqrt
(-a^2 - a*b)) + 1/4*(8*a^2*b*e^(6*x) + 8*a*b^2*e^(6*x) + 3*b^3*e^(6*x) + 48*a^3*e^(4*x) + 72*a^2*b*e^(4*x) + 4
2*a*b^2*e^(4*x) + 9*b^3*e^(4*x) + 40*a^2*b*e^(2*x) + 40*a*b^2*e^(2*x) + 9*b^3*e^(2*x) + 6*a*b^2 + 3*b^3)/((a^4
+ 2*a^3*b + a^2*b^2)*(b*e^(4*x) + 4*a*e^(2*x) + 2*b*e^(2*x) + b)^2)

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maple [B]  time = 0.12, size = 477, normalized size = 4.46 \[ -\frac {2 \left (\frac {b \left (8 a +3 b \right ) \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{8 \left (a +b \right ) a^{2}}-\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{8 \left (a +b \right )^{2} a^{2}}-\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{8 \left (a +b \right )^{2} a^{2}}+\frac {b \left (8 a +3 b \right ) \tanh \left (\frac {x}{2}\right )}{8 \left (a +b \right ) a^{2}}\right )}{\left (\left (\tanh ^{4}\left (\frac {x}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )^{2}}-\frac {\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 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a}\, \sqrt {a +b}}-\frac {\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 ) b}{2 a^{\frac {3}{2}} \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a +b}}-\frac {3 \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 ) b^{2}}{16 a^{\frac {5}{2}} \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a +b}}+\frac {\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 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a}\, \sqrt {a +b}}+\frac {\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 ) b}{2 a^{\frac {3}{2}} \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a +b}}+\frac {3 \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 ) b^{2}}{16 a^{\frac {5}{2}} \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)^2)^3,x)

[Out]

-2*(1/8*b*(8*a+3*b)/(a+b)/a^2*tanh(1/2*x)^7-1/8*b*(8*a^2-13*a*b-9*b^2)/(a+b)^2/a^2*tanh(1/2*x)^5-1/8*b*(8*a^2-
13*a*b-9*b^2)/(a+b)^2/a^2*tanh(1/2*x)^3+1/8*b*(8*a+3*b)/(a+b)/a^2*tanh(1/2*x))/(tanh(1/2*x)^4*a+b*tanh(1/2*x)^
4-2*a*tanh(1/2*x)^2+2*tanh(1/2*x)^2*b+a+b)^2-1/2/(a^2+2*a*b+b^2)/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/a^(3/2)/(a^2+2*a*b+b^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))*b-3/16/a^(5/2)/(a^2+2*a*b+b^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))*b^2+1/2/(a^2+2*a*b+b^2)/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/a^(3/2)/(a^2+2*a*b+b^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))*b+3/16/a^(5/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2+2*a^(1/2)*tan
h(1/2*x)+(a+b)^(1/2))*b^2

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maxima [B]  time = 0.59, size = 344, normalized size = 3.21 \[ -\frac {{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{16 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {6 \, a b^{2} + 3 \, b^{3} + {\left (40 \, a^{2} b + 40 \, a b^{2} + 9 \, b^{3}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (16 \, a^{3} + 24 \, a^{2} b + 14 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (-4 \, x\right )} + {\left (8 \, a^{2} b + 8 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (-6 \, x\right )}}{4 \, {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + 4 \, {\left (2 \, a^{5} b + 5 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (8 \, a^{6} + 24 \, a^{5} b + 27 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 3 \, a^{2} b^{4}\right )} e^{\left (-4 \, x\right )} + 4 \, {\left (2 \, a^{5} b + 5 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)^2)^3,x, algorithm="maxima")

[Out]

-1/16*(8*a^2 + 8*a*b + 3*b^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)))/((a^4 + 2*a^3*b + a^2*b^2)*sqrt((a + b)*a)) - 1/4*(6*a*b^2 + 3*b^3 + (40*a^2*b + 40*a*b^2 + 9*b^3)*
e^(-2*x) + 3*(16*a^3 + 24*a^2*b + 14*a*b^2 + 3*b^3)*e^(-4*x) + (8*a^2*b + 8*a*b^2 + 3*b^3)*e^(-6*x))/(a^4*b^2
+ 2*a^3*b^3 + a^2*b^4 + 4*(2*a^5*b + 5*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*e^(-2*x) + 2*(8*a^6 + 24*a^5*b + 27*a^4*
b^2 + 14*a^3*b^3 + 3*a^2*b^4)*e^(-4*x) + 4*(2*a^5*b + 5*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*e^(-6*x) + (a^4*b^2 + 2
*a^3*b^3 + a^2*b^4)*e^(-8*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,{\mathrm {cosh}\relax (x)}^2+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x)^2)^3,x)

[Out]

int(1/(a + b*cosh(x)^2)^3, x)

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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(1/(a+b*cosh(x)**2)**3,x)

[Out]

Timed out

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