3.32 \(\int \frac{\text{sech}^5(x)}{a+b \cosh ^2(x)} \, dx\)

Optimal. Leaf size=90 \[ \frac{\left (3 a^2-4 a b+8 b^2\right ) \tan ^{-1}(\sinh (x))}{8 a^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{a^3 \sqrt{a+b}}+\frac{(3 a-4 b) \tanh (x) \text{sech}(x)}{8 a^2}+\frac{\tanh (x) \text{sech}^3(x)}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.138285, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3186, 414, 527, 522, 203, 205} \[ \frac{\left (3 a^2-4 a b+8 b^2\right ) \tan ^{-1}(\sinh (x))}{8 a^3}-\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (x)}{\sqrt{a+b}}\right )}{a^3 \sqrt{a+b}}+\frac{(3 a-4 b) \tanh (x) \text{sech}(x)}{8 a^2}+\frac{\tanh (x) \text{sech}^3(x)}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^5/(a + b*Cosh[x]^2),x]

[Out]

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

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

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

Rule 203

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

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]

Rubi steps

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

Mathematica [A]  time = 0.291266, size = 86, normalized size = 0.96 \[ \frac{2 \left (3 a^2-4 a b+8 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+2 a^2 \tanh (x) \text{sech}^3(x)+\frac{8 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \text{csch}(x)}{\sqrt{b}}\right )}{\sqrt{a+b}}+a (3 a-4 b) \tanh (x) \text{sech}(x)}{8 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^5/(a + b*Cosh[x]^2),x]

[Out]

((8*b^(5/2)*ArcTan[(Sqrt[a + b]*Csch[x])/Sqrt[b]])/Sqrt[a + b] + 2*(3*a^2 - 4*a*b + 8*b^2)*ArcTan[Tanh[x/2]] +
a*(3*a - 4*b)*Sech[x]*Tanh[x] + 2*a^2*Sech[x]^3*Tanh[x])/(8*a^3)

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Maple [B]  time = 0.048, size = 274, normalized size = 3. \begin{align*} -{\frac{1}{{a}^{3}}{b}^{{\frac{5}{2}}}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( x/2 \right ) \sqrt{a+b}+2\,\sqrt{a} \right ){\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{a+b}}}}-{\frac{1}{{a}^{3}}{b}^{{\frac{5}{2}}}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( x/2 \right ) \sqrt{a+b}-2\,\sqrt{a} \right ){\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{a+b}}}}-{\frac{5}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{7} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{b}{{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{7} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{3}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{b}{{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}-{\frac{3}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}-{\frac{b}{{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{5}{4\,a}\tanh \left ({\frac{x}{2}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}-{\frac{b}{{a}^{2}}\tanh \left ({\frac{x}{2}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-4}}+{\frac{3}{4\,a}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{b}{{a}^{2}}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+2\,{\frac{\arctan \left ( \tanh \left ( x/2 \right ) \right ){b}^{2}}{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^5/(a+b*cosh(x)^2),x)

[Out]

-b^(5/2)/a^3/(a+b)^(1/2)*arctan(1/2*(2*tanh(1/2*x)*(a+b)^(1/2)+2*a^(1/2))/b^(1/2))-b^(5/2)/a^3/(a+b)^(1/2)*arc
tan(1/2*(2*tanh(1/2*x)*(a+b)^(1/2)-2*a^(1/2))/b^(1/2))-5/4/a/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^7+1/a^2/(tanh(1/2
*x)^2+1)^4*tanh(1/2*x)^7*b+3/4/a/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5+1/a^2/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*b-3
/4/a/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3-1/a^2/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3*b+5/4/a/(tanh(1/2*x)^2+1)^4*tan
h(1/2*x)-1/a^2/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)*b+3/4/a*arctan(tanh(1/2*x))-1/a^2*b*arctan(tanh(1/2*x))+2/a^3*a
rctan(tanh(1/2*x))*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, a - 4 \, b\right )} e^{\left (7 \, x\right )} +{\left (11 \, a - 4 \, b\right )} e^{\left (5 \, x\right )} -{\left (11 \, a - 4 \, b\right )} e^{\left (3 \, x\right )} -{\left (3 \, a - 4 \, b\right )} e^{x}}{4 \,{\left (a^{2} e^{\left (8 \, x\right )} + 4 \, a^{2} e^{\left (6 \, x\right )} + 6 \, a^{2} e^{\left (4 \, x\right )} + 4 \, a^{2} e^{\left (2 \, x\right )} + a^{2}\right )}} + \frac{{\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \arctan \left (e^{x}\right )}{4 \, a^{3}} - 32 \, \int \frac{b^{3} e^{\left (3 \, x\right )} + b^{3} e^{x}}{16 \,{\left (a^{3} b e^{\left (4 \, x\right )} + a^{3} b + 2 \,{\left (2 \, a^{4} + a^{3} b\right )} e^{\left (2 \, x\right )}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(a+b*cosh(x)^2),x, algorithm="maxima")

[Out]

1/4*((3*a - 4*b)*e^(7*x) + (11*a - 4*b)*e^(5*x) - (11*a - 4*b)*e^(3*x) - (3*a - 4*b)*e^x)/(a^2*e^(8*x) + 4*a^2
*e^(6*x) + 6*a^2*e^(4*x) + 4*a^2*e^(2*x) + a^2) + 1/4*(3*a^2 - 4*a*b + 8*b^2)*arctan(e^x)/a^3 - 32*integrate(1
/16*(b^3*e^(3*x) + b^3*e^x)/(a^3*b*e^(4*x) + a^3*b + 2*(2*a^4 + a^3*b)*e^(2*x)), x)

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Fricas [B]  time = 2.70816, size = 8682, normalized size = 96.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(a+b*cosh(x)^2),x, algorithm="fricas")

[Out]

[1/4*((3*a^2 - 4*a*b)*cosh(x)^7 + 7*(3*a^2 - 4*a*b)*cosh(x)*sinh(x)^6 + (3*a^2 - 4*a*b)*sinh(x)^7 + (11*a^2 -
4*a*b)*cosh(x)^5 + (21*(3*a^2 - 4*a*b)*cosh(x)^2 + 11*a^2 - 4*a*b)*sinh(x)^5 + 5*(7*(3*a^2 - 4*a*b)*cosh(x)^3
+ (11*a^2 - 4*a*b)*cosh(x))*sinh(x)^4 - (11*a^2 - 4*a*b)*cosh(x)^3 + (35*(3*a^2 - 4*a*b)*cosh(x)^4 + 10*(11*a^
2 - 4*a*b)*cosh(x)^2 - 11*a^2 + 4*a*b)*sinh(x)^3 + (21*(3*a^2 - 4*a*b)*cosh(x)^5 + 10*(11*a^2 - 4*a*b)*cosh(x)
^3 - 3*(11*a^2 - 4*a*b)*cosh(x))*sinh(x)^2 + 2*(b^2*cosh(x)^8 + 8*b^2*cosh(x)*sinh(x)^7 + b^2*sinh(x)^8 + 4*b^
2*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 + b^2)*sinh(x)^6 + 6*b^2*cosh(x)^4 + 8*(7*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh
(x)^5 + 2*(35*b^2*cosh(x)^4 + 30*b^2*cosh(x)^2 + 3*b^2)*sinh(x)^4 + 4*b^2*cosh(x)^2 + 8*(7*b^2*cosh(x)^5 + 10*
b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 4*(7*b^2*cosh(x)^6 + 15*b^2*cosh(x)^4 + 9*b^2*cosh(x)^2 + b^2)*sinh
(x)^2 + b^2 + 8*(b^2*cosh(x)^7 + 3*b^2*cosh(x)^5 + 3*b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(-b/(a + b))*lo
g((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a + 3*b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 - 2*a - 3*b)
*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a + 3*b)*cosh(x))*sinh(x) - 4*((a + b)*cosh(x)^3 + 3*(a + b)*cosh(x)*sinh(x)^
2 + (a + b)*sinh(x)^3 - (a + b)*cosh(x) + (3*(a + b)*cosh(x)^2 - a - b)*sinh(x))*sqrt(-b/(a + b)) + 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)) + ((3*a^2 - 4*a*b + 8*b^2)*cosh(x)^8 + 8*(3*a^2 - 4*a*b + 8*
b^2)*cosh(x)*sinh(x)^7 + (3*a^2 - 4*a*b + 8*b^2)*sinh(x)^8 + 4*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^6 + 4*(7*(3*a^2
- 4*a*b + 8*b^2)*cosh(x)^2 + 3*a^2 - 4*a*b + 8*b^2)*sinh(x)^6 + 8*(7*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^3 + 3*(3
*a^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x)^5 + 6*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^4 + 2*(35*(3*a^2 - 4*a*b + 8*b^2)
*cosh(x)^4 + 30*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 9*a^2 - 12*a*b + 24*b^2)*sinh(x)^4 + 8*(7*(3*a^2 - 4*a*b +
8*b^2)*cosh(x)^5 + 10*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^3 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x)^3 + 4*(3
*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 4*(7*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^6 + 15*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^
4 + 9*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 3*a^2 - 4*a*b + 8*b^2)*sinh(x)^2 + 3*a^2 - 4*a*b + 8*b^2 + 8*((3*a^2
- 4*a*b + 8*b^2)*cosh(x)^7 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^5 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^3 + (3*a
^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) - (3*a^2 - 4*a*b)*cosh(x) + (7*(3*a^2 - 4*a*b)
*cosh(x)^6 + 5*(11*a^2 - 4*a*b)*cosh(x)^4 - 3*(11*a^2 - 4*a*b)*cosh(x)^2 - 3*a^2 + 4*a*b)*sinh(x))/(a^3*cosh(x
)^8 + 8*a^3*cosh(x)*sinh(x)^7 + a^3*sinh(x)^8 + 4*a^3*cosh(x)^6 + 6*a^3*cosh(x)^4 + 4*(7*a^3*cosh(x)^2 + a^3)*
sinh(x)^6 + 8*(7*a^3*cosh(x)^3 + 3*a^3*cosh(x))*sinh(x)^5 + 4*a^3*cosh(x)^2 + 2*(35*a^3*cosh(x)^4 + 30*a^3*cos
h(x)^2 + 3*a^3)*sinh(x)^4 + 8*(7*a^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + 3*a^3*cosh(x))*sinh(x)^3 + a^3 + 4*(7*a^3*
cosh(x)^6 + 15*a^3*cosh(x)^4 + 9*a^3*cosh(x)^2 + a^3)*sinh(x)^2 + 8*(a^3*cosh(x)^7 + 3*a^3*cosh(x)^5 + 3*a^3*c
osh(x)^3 + a^3*cosh(x))*sinh(x)), 1/4*((3*a^2 - 4*a*b)*cosh(x)^7 + 7*(3*a^2 - 4*a*b)*cosh(x)*sinh(x)^6 + (3*a^
2 - 4*a*b)*sinh(x)^7 + (11*a^2 - 4*a*b)*cosh(x)^5 + (21*(3*a^2 - 4*a*b)*cosh(x)^2 + 11*a^2 - 4*a*b)*sinh(x)^5
+ 5*(7*(3*a^2 - 4*a*b)*cosh(x)^3 + (11*a^2 - 4*a*b)*cosh(x))*sinh(x)^4 - (11*a^2 - 4*a*b)*cosh(x)^3 + (35*(3*a
^2 - 4*a*b)*cosh(x)^4 + 10*(11*a^2 - 4*a*b)*cosh(x)^2 - 11*a^2 + 4*a*b)*sinh(x)^3 + (21*(3*a^2 - 4*a*b)*cosh(x
)^5 + 10*(11*a^2 - 4*a*b)*cosh(x)^3 - 3*(11*a^2 - 4*a*b)*cosh(x))*sinh(x)^2 - 4*(b^2*cosh(x)^8 + 8*b^2*cosh(x)
*sinh(x)^7 + b^2*sinh(x)^8 + 4*b^2*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 + b^2)*sinh(x)^6 + 6*b^2*cosh(x)^4 + 8*(7*b^
2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^5 + 2*(35*b^2*cosh(x)^4 + 30*b^2*cosh(x)^2 + 3*b^2)*sinh(x)^4 + 4*b^2*cos
h(x)^2 + 8*(7*b^2*cosh(x)^5 + 10*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 4*(7*b^2*cosh(x)^6 + 15*b^2*cosh(x
)^4 + 9*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 8*(b^2*cosh(x)^7 + 3*b^2*cosh(x)^5 + 3*b^2*cosh(x)^3 + b^2*cosh
(x))*sinh(x))*sqrt(b/(a + b))*arctan(1/2*sqrt(b/(a + b))*(cosh(x) + sinh(x))) - 4*(b^2*cosh(x)^8 + 8*b^2*cosh(
x)*sinh(x)^7 + b^2*sinh(x)^8 + 4*b^2*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 + b^2)*sinh(x)^6 + 6*b^2*cosh(x)^4 + 8*(7*
b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^5 + 2*(35*b^2*cosh(x)^4 + 30*b^2*cosh(x)^2 + 3*b^2)*sinh(x)^4 + 4*b^2*c
osh(x)^2 + 8*(7*b^2*cosh(x)^5 + 10*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 4*(7*b^2*cosh(x)^6 + 15*b^2*cosh
(x)^4 + 9*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 8*(b^2*cosh(x)^7 + 3*b^2*cosh(x)^5 + 3*b^2*cosh(x)^3 + b^2*co
sh(x))*sinh(x))*sqrt(b/(a + b))*arctan(1/2*(b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)^3 + (4*a + 3*b)*co
sh(x) + (3*b*cosh(x)^2 + 4*a + 3*b)*sinh(x))*sqrt(b/(a + b))/b) + ((3*a^2 - 4*a*b + 8*b^2)*cosh(x)^8 + 8*(3*a^
2 - 4*a*b + 8*b^2)*cosh(x)*sinh(x)^7 + (3*a^2 - 4*a*b + 8*b^2)*sinh(x)^8 + 4*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^6
+ 4*(7*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 3*a^2 - 4*a*b + 8*b^2)*sinh(x)^6 + 8*(7*(3*a^2 - 4*a*b + 8*b^2)*co
sh(x)^3 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x)^5 + 6*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^4 + 2*(35*(3*a^2 -
4*a*b + 8*b^2)*cosh(x)^4 + 30*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 9*a^2 - 12*a*b + 24*b^2)*sinh(x)^4 + 8*(7*(3
*a^2 - 4*a*b + 8*b^2)*cosh(x)^5 + 10*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^3 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x))*si
nh(x)^3 + 4*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 4*(7*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^6 + 15*(3*a^2 - 4*a*b + 8
*b^2)*cosh(x)^4 + 9*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 3*a^2 - 4*a*b + 8*b^2)*sinh(x)^2 + 3*a^2 - 4*a*b + 8*b
^2 + 8*((3*a^2 - 4*a*b + 8*b^2)*cosh(x)^7 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^5 + 3*(3*a^2 - 4*a*b + 8*b^2)*co
sh(x)^3 + (3*a^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) - (3*a^2 - 4*a*b)*cosh(x) + (7*(
3*a^2 - 4*a*b)*cosh(x)^6 + 5*(11*a^2 - 4*a*b)*cosh(x)^4 - 3*(11*a^2 - 4*a*b)*cosh(x)^2 - 3*a^2 + 4*a*b)*sinh(x
))/(a^3*cosh(x)^8 + 8*a^3*cosh(x)*sinh(x)^7 + a^3*sinh(x)^8 + 4*a^3*cosh(x)^6 + 6*a^3*cosh(x)^4 + 4*(7*a^3*cos
h(x)^2 + a^3)*sinh(x)^6 + 8*(7*a^3*cosh(x)^3 + 3*a^3*cosh(x))*sinh(x)^5 + 4*a^3*cosh(x)^2 + 2*(35*a^3*cosh(x)^
4 + 30*a^3*cosh(x)^2 + 3*a^3)*sinh(x)^4 + 8*(7*a^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + 3*a^3*cosh(x))*sinh(x)^3 + a
^3 + 4*(7*a^3*cosh(x)^6 + 15*a^3*cosh(x)^4 + 9*a^3*cosh(x)^2 + a^3)*sinh(x)^2 + 8*(a^3*cosh(x)^7 + 3*a^3*cosh(
x)^5 + 3*a^3*cosh(x)^3 + a^3*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(sech(x)**5/(a+b*cosh(x)**2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(a+b*cosh(x)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError