3.95 \(\int \frac{\cosh ^4(x)}{a+b \text{sech}(x)} \, dx\)

Optimal. Leaf size=146 \[ \frac{x \left (4 a^2 b^2+3 a^4+8 b^4\right )}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}-\frac{2 b^5 \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a^5 \sqrt{a-b} \sqrt{a+b}}+\frac{\left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{8 a^3}-\frac{b \sinh (x) \cosh ^2(x)}{3 a^2}+\frac{\sinh (x) \cosh ^3(x)}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.657326, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {3853, 4104, 3919, 3831, 2659, 205} \[ \frac{x \left (4 a^2 b^2+3 a^4+8 b^4\right )}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}-\frac{2 b^5 \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a^5 \sqrt{a-b} \sqrt{a+b}}+\frac{\left (3 a^2+4 b^2\right ) \sinh (x) \cosh (x)}{8 a^3}-\frac{b \sinh (x) \cosh ^2(x)}{3 a^2}+\frac{\sinh (x) \cosh ^3(x)}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^4/(a + b*Sech[x]),x]

[Out]

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

Rule 3853

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(a*f*n), x] - Dist[1/(a*d*n), Int[((d*Csc[e + f*x])^(n + 1)*Simp[b*n - a*(n + 1)*Csc[e
+ f*x] - b*(n + 1)*Csc[e + f*x]^2, x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 -
b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]

Rule 4104

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m +
1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 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{\cosh ^4(x)}{a+b \text{sech}(x)} \, dx &=\frac{\cosh ^3(x) \sinh (x)}{4 a}+\frac{\int \frac{\cosh ^3(x) \left (-4 b+3 a \text{sech}(x)+3 b \text{sech}^2(x)\right )}{a+b \text{sech}(x)} \, dx}{4 a}\\ &=-\frac{b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac{\cosh ^3(x) \sinh (x)}{4 a}-\frac{\int \frac{\cosh ^2(x) \left (-3 \left (3 a^2+4 b^2\right )-a b \text{sech}(x)+8 b^2 \text{sech}^2(x)\right )}{a+b \text{sech}(x)} \, dx}{12 a^2}\\ &=\frac{\left (3 a^2+4 b^2\right ) \cosh (x) \sinh (x)}{8 a^3}-\frac{b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac{\cosh ^3(x) \sinh (x)}{4 a}+\frac{\int \frac{\cosh (x) \left (-8 b \left (2 a^2+3 b^2\right )+a \left (9 a^2-4 b^2\right ) \text{sech}(x)+3 b \left (3 a^2+4 b^2\right ) \text{sech}^2(x)\right )}{a+b \text{sech}(x)} \, dx}{24 a^3}\\ &=-\frac{b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}+\frac{\left (3 a^2+4 b^2\right ) \cosh (x) \sinh (x)}{8 a^3}-\frac{b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac{\cosh ^3(x) \sinh (x)}{4 a}-\frac{\int \frac{-3 \left (3 a^4+4 a^2 b^2+8 b^4\right )-3 a b \left (3 a^2+4 b^2\right ) \text{sech}(x)}{a+b \text{sech}(x)} \, dx}{24 a^4}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}+\frac{\left (3 a^2+4 b^2\right ) \cosh (x) \sinh (x)}{8 a^3}-\frac{b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac{\cosh ^3(x) \sinh (x)}{4 a}-\frac{b^5 \int \frac{\text{sech}(x)}{a+b \text{sech}(x)} \, dx}{a^5}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}+\frac{\left (3 a^2+4 b^2\right ) \cosh (x) \sinh (x)}{8 a^3}-\frac{b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac{\cosh ^3(x) \sinh (x)}{4 a}-\frac{b^4 \int \frac{1}{1+\frac{a \cosh (x)}{b}} \, dx}{a^5}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}+\frac{\left (3 a^2+4 b^2\right ) \cosh (x) \sinh (x)}{8 a^3}-\frac{b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac{\cosh ^3(x) \sinh (x)}{4 a}-\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}-\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^5}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{2 b^5 \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a^5 \sqrt{a-b} \sqrt{a+b}}-\frac{b \left (2 a^2+3 b^2\right ) \sinh (x)}{3 a^4}+\frac{\left (3 a^2+4 b^2\right ) \cosh (x) \sinh (x)}{8 a^3}-\frac{b \cosh ^2(x) \sinh (x)}{3 a^2}+\frac{\cosh ^3(x) \sinh (x)}{4 a}\\ \end{align*}

Mathematica [A]  time = 0.275261, size = 126, normalized size = 0.86 \[ \frac{12 x \left (4 a^2 b^2+3 a^4+8 b^4\right )-24 a b \left (3 a^2+4 b^2\right ) \sinh (x)+24 a^2 \left (a^2+b^2\right ) \sinh (2 x)+\frac{192 b^5 \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}-8 a^3 b \sinh (3 x)+3 a^4 \sinh (4 x)}{96 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^4/(a + b*Sech[x]),x]

[Out]

(12*(3*a^4 + 4*a^2*b^2 + 8*b^4)*x + (192*b^5*ArcTan[((-a + b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 2
4*a*b*(3*a^2 + 4*b^2)*Sinh[x] + 24*a^2*(a^2 + b^2)*Sinh[2*x] - 8*a^3*b*Sinh[3*x] + 3*a^4*Sinh[4*x])/(96*a^5)

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Maple [B]  time = 0.037, size = 406, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^4/(a+b*sech(x)),x)

[Out]

-2*b^5/a^5/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))-1/4/a/(tanh(1/2*x)+1)^4+1/2/a/(ta
nh(1/2*x)+1)^3-7/8/a/(tanh(1/2*x)+1)^2+5/8/a/(tanh(1/2*x)+1)+3/8/a*ln(tanh(1/2*x)+1)+1/4/a/(tanh(1/2*x)-1)^4+1
/2/a/(tanh(1/2*x)-1)^3+7/8/a/(tanh(1/2*x)-1)^2+5/8/a/(tanh(1/2*x)-1)-3/8/a*ln(tanh(1/2*x)-1)+1/2/a^3/(tanh(1/2
*x)-1)^2*b^2-1/2/a^3*ln(tanh(1/2*x)-1)*b^2-1/a^5*ln(tanh(1/2*x)-1)*b^4+1/a^2/(tanh(1/2*x)-1)*b+1/2/a^3/(tanh(1
/2*x)-1)*b^2+1/a^4/(tanh(1/2*x)-1)*b^3+1/a^2/(tanh(1/2*x)+1)*b+1/2/a^3/(tanh(1/2*x)+1)*b^2+1/a^4/(tanh(1/2*x)+
1)*b^3+1/3/a^2/(tanh(1/2*x)-1)^3*b+1/2/a^2/(tanh(1/2*x)-1)^2*b+1/2/a^3*ln(tanh(1/2*x)+1)*b^2+1/a^5*ln(tanh(1/2
*x)+1)*b^4+1/3/a^2/(tanh(1/2*x)+1)^3*b-1/2/a^2/(tanh(1/2*x)+1)^2*b-1/2/a^3/(tanh(1/2*x)+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(cosh(x)^4/(a+b*sech(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.81342, size = 5667, normalized size = 38.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^4/(a+b*sech(x)),x, algorithm="fricas")

[Out]

[1/192*(3*(a^6 - a^4*b^2)*cosh(x)^8 + 3*(a^6 - a^4*b^2)*sinh(x)^8 - 8*(a^5*b - a^3*b^3)*cosh(x)^7 - 8*(a^5*b -
a^3*b^3 - 3*(a^6 - a^4*b^2)*cosh(x))*sinh(x)^7 + 24*(a^6 - a^2*b^4)*cosh(x)^6 + 4*(6*a^6 - 6*a^2*b^4 + 21*(a^
6 - a^4*b^2)*cosh(x)^2 - 14*(a^5*b - a^3*b^3)*cosh(x))*sinh(x)^6 - 3*a^6 + 3*a^4*b^2 + 24*(3*a^6 + a^4*b^2 + 4
*a^2*b^4 - 8*b^6)*x*cosh(x)^4 - 24*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^5 - 24*(3*a^5*b + a^3*b^3 - 4*a*b^5 -
7*(a^6 - a^4*b^2)*cosh(x)^3 + 7*(a^5*b - a^3*b^3)*cosh(x)^2 - 6*(a^6 - a^2*b^4)*cosh(x))*sinh(x)^5 + 2*(105*(
a^6 - a^4*b^2)*cosh(x)^4 - 140*(a^5*b - a^3*b^3)*cosh(x)^3 + 180*(a^6 - a^2*b^4)*cosh(x)^2 + 12*(3*a^6 + a^4*b
^2 + 4*a^2*b^4 - 8*b^6)*x - 60*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x))*sinh(x)^4 + 24*(3*a^5*b + a^3*b^3 - 4*a*
b^5)*cosh(x)^3 + 8*(9*a^5*b + 3*a^3*b^3 - 12*a*b^5 + 21*(a^6 - a^4*b^2)*cosh(x)^5 - 35*(a^5*b - a^3*b^3)*cosh(
x)^4 + 60*(a^6 - a^2*b^4)*cosh(x)^3 + 12*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x*cosh(x) - 30*(3*a^5*b + a^3*b
^3 - 4*a*b^5)*cosh(x)^2)*sinh(x)^3 - 24*(a^6 - a^2*b^4)*cosh(x)^2 + 12*(7*(a^6 - a^4*b^2)*cosh(x)^6 - 2*a^6 +
2*a^2*b^4 - 14*(a^5*b - a^3*b^3)*cosh(x)^5 + 30*(a^6 - a^2*b^4)*cosh(x)^4 + 12*(3*a^6 + a^4*b^2 + 4*a^2*b^4 -
8*b^6)*x*cosh(x)^2 - 20*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^3 + 6*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x))*sin
h(x)^2 - 192*(b^5*cosh(x)^4 + 4*b^5*cosh(x)^3*sinh(x) + 6*b^5*cosh(x)^2*sinh(x)^2 + 4*b^5*cosh(x)*sinh(x)^3 +
b^5*sinh(x)^4)*sqrt(-a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) - a^2 + 2*b^2 + 2*(a^2*cosh
(x) + a*b)*sinh(x) + 2*sqrt(-a^2 + b^2)*(a*cosh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x)
+ 2*(a*cosh(x) + b)*sinh(x) + a)) + 8*(a^5*b - a^3*b^3)*cosh(x) + 8*(3*(a^6 - a^4*b^2)*cosh(x)^7 - 7*(a^5*b -
a^3*b^3)*cosh(x)^6 + a^5*b - a^3*b^3 + 18*(a^6 - a^2*b^4)*cosh(x)^5 + 12*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)
*x*cosh(x)^3 - 15*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^4 + 9*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^2 - 6*(a^6
- a^2*b^4)*cosh(x))*sinh(x))/((a^7 - a^5*b^2)*cosh(x)^4 + 4*(a^7 - a^5*b^2)*cosh(x)^3*sinh(x) + 6*(a^7 - a^5*
b^2)*cosh(x)^2*sinh(x)^2 + 4*(a^7 - a^5*b^2)*cosh(x)*sinh(x)^3 + (a^7 - a^5*b^2)*sinh(x)^4), 1/192*(3*(a^6 - a
^4*b^2)*cosh(x)^8 + 3*(a^6 - a^4*b^2)*sinh(x)^8 - 8*(a^5*b - a^3*b^3)*cosh(x)^7 - 8*(a^5*b - a^3*b^3 - 3*(a^6
- a^4*b^2)*cosh(x))*sinh(x)^7 + 24*(a^6 - a^2*b^4)*cosh(x)^6 + 4*(6*a^6 - 6*a^2*b^4 + 21*(a^6 - a^4*b^2)*cosh(
x)^2 - 14*(a^5*b - a^3*b^3)*cosh(x))*sinh(x)^6 - 3*a^6 + 3*a^4*b^2 + 24*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*
x*cosh(x)^4 - 24*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^5 - 24*(3*a^5*b + a^3*b^3 - 4*a*b^5 - 7*(a^6 - a^4*b^2)
*cosh(x)^3 + 7*(a^5*b - a^3*b^3)*cosh(x)^2 - 6*(a^6 - a^2*b^4)*cosh(x))*sinh(x)^5 + 2*(105*(a^6 - a^4*b^2)*cos
h(x)^4 - 140*(a^5*b - a^3*b^3)*cosh(x)^3 + 180*(a^6 - a^2*b^4)*cosh(x)^2 + 12*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8
*b^6)*x - 60*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x))*sinh(x)^4 + 24*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^3 + 8
*(9*a^5*b + 3*a^3*b^3 - 12*a*b^5 + 21*(a^6 - a^4*b^2)*cosh(x)^5 - 35*(a^5*b - a^3*b^3)*cosh(x)^4 + 60*(a^6 - a
^2*b^4)*cosh(x)^3 + 12*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x*cosh(x) - 30*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh
(x)^2)*sinh(x)^3 - 24*(a^6 - a^2*b^4)*cosh(x)^2 + 12*(7*(a^6 - a^4*b^2)*cosh(x)^6 - 2*a^6 + 2*a^2*b^4 - 14*(a^
5*b - a^3*b^3)*cosh(x)^5 + 30*(a^6 - a^2*b^4)*cosh(x)^4 + 12*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x*cosh(x)^2
- 20*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^3 + 6*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x))*sinh(x)^2 + 384*(b^5*
cosh(x)^4 + 4*b^5*cosh(x)^3*sinh(x) + 6*b^5*cosh(x)^2*sinh(x)^2 + 4*b^5*cosh(x)*sinh(x)^3 + b^5*sinh(x)^4)*sqr
t(a^2 - b^2)*arctan(-(a*cosh(x) + a*sinh(x) + b)/sqrt(a^2 - b^2)) + 8*(a^5*b - a^3*b^3)*cosh(x) + 8*(3*(a^6 -
a^4*b^2)*cosh(x)^7 - 7*(a^5*b - a^3*b^3)*cosh(x)^6 + a^5*b - a^3*b^3 + 18*(a^6 - a^2*b^4)*cosh(x)^5 + 12*(3*a^
6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x*cosh(x)^3 - 15*(3*a^5*b + a^3*b^3 - 4*a*b^5)*cosh(x)^4 + 9*(3*a^5*b + a^3*b
^3 - 4*a*b^5)*cosh(x)^2 - 6*(a^6 - a^2*b^4)*cosh(x))*sinh(x))/((a^7 - a^5*b^2)*cosh(x)^4 + 4*(a^7 - a^5*b^2)*c
osh(x)^3*sinh(x) + 6*(a^7 - a^5*b^2)*cosh(x)^2*sinh(x)^2 + 4*(a^7 - a^5*b^2)*cosh(x)*sinh(x)^3 + (a^7 - a^5*b^
2)*sinh(x)^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{4}{\left (x \right )}}{a + b \operatorname{sech}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**4/(a+b*sech(x)),x)

[Out]

Integral(cosh(x)**4/(a + b*sech(x)), x)

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Giac [A]  time = 1.14688, size = 246, normalized size = 1.68 \begin{align*} -\frac{2 \, b^{5} \arctan \left (\frac{a e^{x} + b}{\sqrt{a^{2} - b^{2}}}\right )}{\sqrt{a^{2} - b^{2}} a^{5}} + \frac{3 \, a^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (3 \, x\right )} + 24 \, a^{3} e^{\left (2 \, x\right )} + 24 \, a b^{2} e^{\left (2 \, x\right )} - 72 \, a^{2} b e^{x} - 96 \, b^{3} e^{x}}{192 \, a^{4}} + \frac{{\left (3 \, a^{4} + 4 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} + \frac{{\left (8 \, a^{3} b e^{x} - 3 \, a^{4} + 24 \,{\left (3 \, a^{3} b + 4 \, a b^{3}\right )} e^{\left (3 \, x\right )} - 24 \,{\left (a^{4} + a^{2} b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-4 \, x\right )}}{192 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^4/(a+b*sech(x)),x, algorithm="giac")

[Out]

-2*b^5*arctan((a*e^x + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^5) + 1/192*(3*a^3*e^(4*x) - 8*a^2*b*e^(3*x) + 24
*a^3*e^(2*x) + 24*a*b^2*e^(2*x) - 72*a^2*b*e^x - 96*b^3*e^x)/a^4 + 1/8*(3*a^4 + 4*a^2*b^2 + 8*b^4)*x/a^5 + 1/1
92*(8*a^3*b*e^x - 3*a^4 + 24*(3*a^3*b + 4*a*b^3)*e^(3*x) - 24*(a^4 + a^2*b^2)*e^(2*x))*e^(-4*x)/a^5