3.113 \(\int \frac{A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx\)

Optimal. Leaf size=197 \[ \frac{\left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}-\frac{\sinh (x) \left (11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right )}{6 \left (a^2-b^2\right )^3 (a+b \cosh (x))}-\frac{\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}-\frac{\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3} \]

[Out]

((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b
)^(7/2)) - ((A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sinh[x])/(
6*(a^2 - b^2)^2*(a + b*Cosh[x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sinh[x])/(6*(a^2 - b^2)^3*
(a + b*Cosh[x]))

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Rubi [A]  time = 0.347737, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2754, 12, 2659, 208} \[ \frac{\left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}-\frac{\sinh (x) \left (11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right )}{6 \left (a^2-b^2\right )^3 (a+b \cosh (x))}-\frac{\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}-\frac{\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(a + b*Cosh[x])^4,x]

[Out]

((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b
)^(7/2)) - ((A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sinh[x])/(
6*(a^2 - b^2)^2*(a + b*Cosh[x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sinh[x])/(6*(a^2 - b^2)^3*
(a + b*Cosh[x]))

Rule 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 12

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

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 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{A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx &=-\frac{(A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}-\frac{\int \frac{-3 (a A-b B)+2 (A b-a B) \cosh (x)}{(a+b \cosh (x))^3} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac{(A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}-\frac{\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}+\frac{\int \frac{2 \left (3 a^2 A+2 A b^2-5 a b B\right )-\left (5 a A b-2 a^2 B-3 b^2 B\right ) \cosh (x)}{(a+b \cosh (x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=-\frac{(A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}-\frac{\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}-\frac{\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^3 (a+b \cosh (x))}-\frac{\int -\frac{3 \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right )}{a+b \cosh (x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=-\frac{(A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}-\frac{\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}-\frac{\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^3 (a+b \cosh (x))}+\frac{\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \int \frac{1}{a+b \cosh (x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=-\frac{(A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}-\frac{\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}-\frac{\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^3 (a+b \cosh (x))}+\frac{\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^3}\\ &=\frac{\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}-\frac{(A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}-\frac{\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}-\frac{\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^3 (a+b \cosh (x))}\\ \end{align*}

Mathematica [A]  time = 0.753572, size = 196, normalized size = 0.99 \[ \frac{1}{6} \left (\frac{6 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}+\frac{\sinh (x) \left (-11 a^2 A b+2 a^3 B+13 a b^2 B-4 A b^3\right )}{(a-b)^3 (a+b)^3 (a+b \cosh (x))}+\frac{\sinh (x) \left (2 a^2 B-5 a A b+3 b^2 B\right )}{(a-b)^2 (a+b)^2 (a+b \cosh (x))^2}+\frac{2 \sinh (x) (a B-A b)}{(a-b) (a+b) (a+b \cosh (x))^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2)
+ (2*(-(A*b) + a*B)*Sinh[x])/((a - b)*(a + b)*(a + b*Cosh[x])^3) + ((-5*a*A*b + 2*a^2*B + 3*b^2*B)*Sinh[x])/(
(a - b)^2*(a + b)^2*(a + b*Cosh[x])^2) + ((-11*a^2*A*b - 4*A*b^3 + 2*a^3*B + 13*a*b^2*B)*Sinh[x])/((a - b)^3*(
a + b)^3*(a + b*Cosh[x])))/6

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Maple [A]  time = 0.025, size = 342, normalized size = 1.7 \begin{align*} -2\,{\frac{1}{ \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b-a-b \right ) ^{3}} \left ( -1/2\,{\frac{ \left ( 6\,A{a}^{2}b+3\,Aa{b}^{2}+2\,A{b}^{3}-2\,{a}^{3}B-2\,B{a}^{2}b-6\,Ba{b}^{2}-B{b}^{3} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{5}}{ \left ( a-b \right ) \left ({a}^{3}+3\,{a}^{2}b+3\,a{b}^{2}+{b}^{3} \right ) }}+2/3\,{\frac{ \left ( 9\,A{a}^{2}b+A{b}^{3}-3\,{a}^{3}B-7\,Ba{b}^{2} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{3}}{ \left ({a}^{2}+2\,ab+{b}^{2} \right ) \left ({a}^{2}-2\,ab+{b}^{2} \right ) }}-1/2\,{\frac{ \left ( 6\,A{a}^{2}b-3\,Aa{b}^{2}+2\,A{b}^{3}-2\,{a}^{3}B+2\,B{a}^{2}b-6\,Ba{b}^{2}+B{b}^{3} \right ) \tanh \left ( x/2 \right ) }{ \left ( a+b \right ) \left ({a}^{3}-3\,{a}^{2}b+3\,a{b}^{2}-{b}^{3} \right ) }} \right ) }+{\frac{2\,A{a}^{3}+3\,Aa{b}^{2}-4\,B{a}^{2}b-B{b}^{3}}{{a}^{6}-3\,{a}^{4}{b}^{2}+3\,{a}^{2}{b}^{4}-{b}^{6}}{\it Artanh} \left ({(a-b)\tanh \left ({\frac{x}{2}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-1/2*(6*A*a^2*b+3*A*a*b^2+2*A*b^3-2*B*a^3-2*B*a^2*b-6*B*a*b^2-B*b^3)/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(
1/2*x)^5+2/3*(9*A*a^2*b+A*b^3-3*B*a^3-7*B*a*b^2)/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tanh(1/2*x)^3-1/2*(6*A*a^2*b-
3*A*a*b^2+2*A*b^3-2*B*a^3+2*B*a^2*b-6*B*a*b^2+B*b^3)/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*x))/(a*tanh(1/2*
x)^2-tanh(1/2*x)^2*b-a-b)^3+(2*A*a^3+3*A*a*b^2-4*B*a^2*b-B*b^3)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1
/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))

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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((A+B*cosh(x))/(a+b*cosh(x))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.39518, size = 17173, normalized size = 87.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(4*B*a^5*b^3 - 22*A*a^4*b^4 + 22*B*a^3*b^5 + 14*A*a^2*b^6 - 26*B*a*b^7 + 8*A*b^8 - 6*(2*A*a^5*b^3 - 4*B*
a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(x)^5 - 6*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 +
3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*sinh(x)^5 - 30*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2
*b^6 + B*a*b^7)*cosh(x)^4 - 30*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7 +
(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(x))*sinh(x)^4 + 4*(4*B*a^8 - 22
*A*a^7*b + 28*B*a^6*b^2 - 19*A*a^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^7)*cosh(x)^3 + 4
*(4*B*a^8 - 22*A*a^7*b + 28*B*a^6*b^2 - 19*A*a^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^7
- 15*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(x)^2 - 30*(2*A*a^6*b^2 - 4
*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(x))*sinh(x)^3 + 12*(4*B*a^7*b - 17*A*a^6*b^
2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8)*cosh(x)^2 + 12*(4*B*a^7*b
- 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8 - 5*(2*A*a^5*b
^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(x)^3 - 15*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*
a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(x)^2 + (4*B*a^8 - 22*A*a^7*b + 28*B*a^6*b^2 - 19*A*a^5*b^3
+ 7*B*a^4*b^4 + 29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^7)*cosh(x))*sinh(x)^2 - 3*(2*A*a^3*b^4 - 4*B*a^2*b^5 +
3*A*a*b^6 - B*b^7 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 +
3*A*a*b^6 - B*b^7)*sinh(x)^6 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^5 + 6*(2*A*a^4*b^
3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x))*sinh(x)^5 +
3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^4 + 3*(8*A*a^5*b^2 -
16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b
^7)*cosh(x)^2 + 10*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(x))*sinh(x)^4 + 4*(4*A*a^6*b - 8*B
*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(x)^3 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*
A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)
^3 + 15*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^2 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a
^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x))*sinh(x)^3 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 -
8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^2 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*
a*b^6 - B*b^7 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^4 + 20*(2*A*a^4*b^3 - 4*B*a^3*b^4 +
3*A*a^2*b^5 - B*a*b^6)*cosh(x)^3 + 6*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*
b^7)*cosh(x)^2 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(x))*
sinh(x)^2 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(x) + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A
*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^5 + 5*(2*A*a^4*b^3 - 4*B*a^3*b^4
+ 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^4 + 2*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 -
B*b^7)*cosh(x)^3 + 2*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(x)
^2 + (8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x))*sinh(x))*sqrt(a^2
- b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sq
rt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x
) + b)) + 6*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4 + 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B*b^8)*cosh
(x) + 6*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4 + 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B*b^8 - 5*(2*A*
a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(x)^4 - 20*(2*A*a^6*b^2 - 4*B*a^5*b^3
+ A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(x)^3 + 2*(4*B*a^8 - 22*A*a^7*b + 28*B*a^6*b^2 - 19*A*
a^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^7)*cosh(x)^2 + 4*(4*B*a^7*b - 17*A*a^6*b^2 + 13
*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8)*cosh(x))*sinh(x))/(a^8*b^4 - 4*a
^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(x)^6 + (a^
8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*sinh(x)^6 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9
+ a*b^11)*cosh(x)^5 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b
^8 - 4*a^2*b^10 + b^12)*cosh(x))*sinh(x)^5 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*cosh
(x)^4 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12 + 5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a
^2*b^10 + b^12)*cosh(x)^2 + 10*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x))*sinh(x)^4 + 4*(
2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*cosh(x)^3 + 4*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 -
10*a^3*b^9 + 3*a*b^11 + 5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(x)^3 + 15*(a^9*b^3 - 4*a^
7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x)^2 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^1
2)*cosh(x))*sinh(x)^3 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*cosh(x)^2 + 3*(4*a^10*b^2
- 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12 + 5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(
x)^4 + 20*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x)^3 + 6*(4*a^10*b^2 - 15*a^8*b^4 + 20*a
^6*b^6 - 10*a^4*b^8 + b^12)*cosh(x)^2 + 4*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*cosh(x))
*sinh(x)^2 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x) + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5
*b^7 - 4*a^3*b^9 + a*b^11 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(x)^5 + 5*(a^9*b^3 - 4*a
^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x)^4 + 2*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^
12)*cosh(x)^3 + 2*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*cosh(x)^2 + (4*a^10*b^2 - 15*a^8
*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*cosh(x))*sinh(x)), -1/3*(2*B*a^5*b^3 - 11*A*a^4*b^4 + 11*B*a^3*b^5 + 7*
A*a^2*b^6 - 13*B*a*b^7 + 4*A*b^8 - 3*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)
*cosh(x)^5 - 3*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*sinh(x)^5 - 15*(2*A*a
^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(x)^4 - 15*(2*A*a^6*b^2 - 4*B*a^5*
b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7 + (2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 -
3*A*a*b^7 + B*b^8)*cosh(x))*sinh(x)^4 + 2*(4*B*a^8 - 22*A*a^7*b + 28*B*a^6*b^2 - 19*A*a^5*b^3 + 7*B*a^4*b^4 +
29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^7)*cosh(x)^3 + 2*(4*B*a^8 - 22*A*a^7*b + 28*B*a^6*b^2 - 19*A*a^5*b^3 +
7*B*a^4*b^4 + 29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^7 - 15*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*
b^6 - 3*A*a*b^7 + B*b^8)*cosh(x)^2 - 30*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B
*a*b^7)*cosh(x))*sinh(x)^3 + 6*(4*B*a^7*b - 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^
2*b^6 - 4*B*a*b^7 + 2*A*b^8)*cosh(x)^2 + 6*(4*B*a^7*b - 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*
b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8 - 5*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 +
B*b^8)*cosh(x)^3 - 15*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(x)^2
+ (4*B*a^8 - 22*A*a^7*b + 28*B*a^6*b^2 - 19*A*a^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^
7)*cosh(x))*sinh(x)^2 + 3*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*
b^6 - B*b^7)*cosh(x)^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*sinh(x)^6 + 6*(2*A*a^4*b^3 - 4*B*a^3*
b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^5 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 -
4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x))*sinh(x)^5 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*
b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^4 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 -
B*b^7 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^2 + 10*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*
b^5 - B*a*b^6)*cosh(x))*sinh(x)^4 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3
*B*a*b^6)*cosh(x)^3 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6 + 5*(
2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^3 + 15*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b
^6)*cosh(x)^2 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x))*sinh(
x)^3 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^2 + 3*(8*A*a^5*
b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6
- B*b^7)*cosh(x)^4 + 20*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^3 + 6*(8*A*a^5*b^2 - 16*B
*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^2 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b
^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(x))*sinh(x)^2 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 -
B*a*b^6)*cosh(x) + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*
b^6 - B*b^7)*cosh(x)^5 + 5*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^4 + 2*(8*A*a^5*b^2 - 16
*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x)^3 + 2*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4
*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(x)^2 + (8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^
2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(x))*sinh(x))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x)
+ a)/(a^2 - b^2)) + 3*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4 + 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B
*b^8)*cosh(x) + 3*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4 + 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B*b^8
- 5*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(x)^4 - 20*(2*A*a^6*b^2 - 4
*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(x)^3 + 2*(4*B*a^8 - 22*A*a^7*b + 28*B*a^6*b
^2 - 19*A*a^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 - 39*B*a^2*b^6 + 12*A*a*b^7)*cosh(x)^2 + 4*(4*B*a^7*b - 17*A*a^
6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8)*cosh(x))*sinh(x))/(a^8
*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(
x)^6 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*sinh(x)^6 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 -
4*a^3*b^9 + a*b^11)*cosh(x)^5 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11 + (a^8*b^4 - 4*a^6*b^6
+ 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(x))*sinh(x)^5 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 +
b^12)*cosh(x)^4 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12 + 5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4
*b^8 - 4*a^2*b^10 + b^12)*cosh(x)^2 + 10*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x))*sinh(
x)^4 + 4*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*cosh(x)^3 + 4*(2*a^11*b - 5*a^9*b^3 + 10*
a^5*b^7 - 10*a^3*b^9 + 3*a*b^11 + 5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(x)^3 + 15*(a^9*
b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x)^2 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4
*b^8 + b^12)*cosh(x))*sinh(x)^3 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*cosh(x)^2 + 3*(
4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12 + 5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b
^12)*cosh(x)^4 + 20*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x)^3 + 6*(4*a^10*b^2 - 15*a^8*
b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*cosh(x)^2 + 4*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11
)*cosh(x))*sinh(x)^2 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x) + 6*(a^9*b^3 - 4*a^7*b
^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cosh(x)^5 + 5*(a^9
*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cosh(x)^4 + 2*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^
4*b^8 + b^12)*cosh(x)^3 + 2*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*cosh(x)^2 + (4*a^10*b^
2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*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((A+B*cosh(x))/(a+b*cosh(x))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.19912, size = 612, normalized size = 3.11 \begin{align*} \frac{{\left (2 \, A a^{3} - 4 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt{-a^{2} + b^{2}}} + \frac{6 \, A a^{3} b^{3} e^{\left (5 \, x\right )} - 12 \, B a^{2} b^{4} e^{\left (5 \, x\right )} + 9 \, A a b^{5} e^{\left (5 \, x\right )} - 3 \, B b^{6} e^{\left (5 \, x\right )} + 30 \, A a^{4} b^{2} e^{\left (4 \, x\right )} - 60 \, B a^{3} b^{3} e^{\left (4 \, x\right )} + 45 \, A a^{2} b^{4} e^{\left (4 \, x\right )} - 15 \, B a b^{5} e^{\left (4 \, x\right )} - 8 \, B a^{6} e^{\left (3 \, x\right )} + 44 \, A a^{5} b e^{\left (3 \, x\right )} - 64 \, B a^{4} b^{2} e^{\left (3 \, x\right )} + 82 \, A a^{3} b^{3} e^{\left (3 \, x\right )} - 78 \, B a^{2} b^{4} e^{\left (3 \, x\right )} + 24 \, A a b^{5} e^{\left (3 \, x\right )} - 24 \, B a^{5} b e^{\left (2 \, x\right )} + 102 \, A a^{4} b^{2} e^{\left (2 \, x\right )} - 102 \, B a^{3} b^{3} e^{\left (2 \, x\right )} + 36 \, A a^{2} b^{4} e^{\left (2 \, x\right )} - 24 \, B a b^{5} e^{\left (2 \, x\right )} + 12 \, A b^{6} e^{\left (2 \, x\right )} - 12 \, B a^{4} b^{2} e^{x} + 60 \, A a^{3} b^{3} e^{x} - 66 \, B a^{2} b^{4} e^{x} + 15 \, A a b^{5} e^{x} + 3 \, B b^{6} e^{x} - 2 \, B a^{3} b^{3} + 11 \, A a^{2} b^{4} - 13 \, B a b^{5} + 4 \, A b^{6}}{3 \,{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )}{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*A*a^3 - 4*B*a^2*b + 3*A*a*b^2 - B*b^3)*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 -
b^6)*sqrt(-a^2 + b^2)) + 1/3*(6*A*a^3*b^3*e^(5*x) - 12*B*a^2*b^4*e^(5*x) + 9*A*a*b^5*e^(5*x) - 3*B*b^6*e^(5*x
) + 30*A*a^4*b^2*e^(4*x) - 60*B*a^3*b^3*e^(4*x) + 45*A*a^2*b^4*e^(4*x) - 15*B*a*b^5*e^(4*x) - 8*B*a^6*e^(3*x)
+ 44*A*a^5*b*e^(3*x) - 64*B*a^4*b^2*e^(3*x) + 82*A*a^3*b^3*e^(3*x) - 78*B*a^2*b^4*e^(3*x) + 24*A*a*b^5*e^(3*x)
- 24*B*a^5*b*e^(2*x) + 102*A*a^4*b^2*e^(2*x) - 102*B*a^3*b^3*e^(2*x) + 36*A*a^2*b^4*e^(2*x) - 24*B*a*b^5*e^(2
*x) + 12*A*b^6*e^(2*x) - 12*B*a^4*b^2*e^x + 60*A*a^3*b^3*e^x - 66*B*a^2*b^4*e^x + 15*A*a*b^5*e^x + 3*B*b^6*e^x
- 2*B*a^3*b^3 + 11*A*a^2*b^4 - 13*B*a*b^5 + 4*A*b^6)/((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*(b*e^(2*x) + 2*a*
e^x + b)^3)