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

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

[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))

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Rubi [A]  time = 0.122786, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 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]

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 12

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

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]

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*}

Mathematica [A]  time = 0.588333, 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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Maple [B]  time = 0.047, size = 468, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

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)*tan
h(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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.82081, size = 11926, normalized size = 111.46 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [B]  time = 1.59237, size = 308, normalized size = 2.88 \begin{align*} \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}} \end{align*}

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)