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

Optimal. Leaf size=194 \[ -\frac{a x}{8 \left (a^2-b^2\right )}-\frac{a b^2 x}{2 \left (a^2-b^2\right )^2}+\frac{a^3 b^2 x}{\left (a^2-b^2\right )^3}+\frac{a^2 b \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac{b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac{a \sinh (x) \cosh ^3(x)}{4 \left (a^2-b^2\right )}-\frac{a \sinh (x) \cosh (x)}{8 \left (a^2-b^2\right )}-\frac{a b^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}-\frac{a^2 b^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]

[Out]

(a^3*b^2*x)/(a^2 - b^2)^3 - (a*b^2*x)/(2*(a^2 - b^2)^2) - (a*x)/(8*(a^2 - b^2)) - (b*Cosh[x]^4)/(4*(a^2 - b^2)
) - (a^2*b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (a*b^2*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) - (a*Cosh[x
]*Sinh[x])/(8*(a^2 - b^2)) + (a*Cosh[x]^3*Sinh[x])/(4*(a^2 - b^2)) + (a^2*b*Sinh[x]^2)/(2*(a^2 - b^2)^2)

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Rubi [A]  time = 0.33815, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3109, 2565, 30, 2568, 2635, 8, 2564, 3098, 3133} \[ -\frac{a x}{8 \left (a^2-b^2\right )}-\frac{a b^2 x}{2 \left (a^2-b^2\right )^2}+\frac{a^3 b^2 x}{\left (a^2-b^2\right )^3}+\frac{a^2 b \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac{b \cosh ^4(x)}{4 \left (a^2-b^2\right )}+\frac{a \sinh (x) \cosh ^3(x)}{4 \left (a^2-b^2\right )}-\frac{a \sinh (x) \cosh (x)}{8 \left (a^2-b^2\right )}-\frac{a b^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}-\frac{a^2 b^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^3*Sinh[x]^2)/(a*Cosh[x] + b*Sinh[x]),x]

[Out]

(a^3*b^2*x)/(a^2 - b^2)^3 - (a*b^2*x)/(2*(a^2 - b^2)^2) - (a*x)/(8*(a^2 - b^2)) - (b*Cosh[x]^4)/(4*(a^2 - b^2)
) - (a^2*b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (a*b^2*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) - (a*Cosh[x
]*Sinh[x])/(8*(a^2 - b^2)) + (a*Cosh[x]^3*Sinh[x])/(4*(a^2 - b^2)) + (a^2*b*Sinh[x]^2)/(2*(a^2 - b^2)^2)

Rule 3109

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Dist[
a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Dist[(a*b)/(a^2 + b^2), Int[(Cos[c + d*x]^(m
- 1)*Sin[c + d*x]^(n - 1))/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 3098

Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(a*x)/(a^2 + b^2), x] + Dist[b/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c +
d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3133

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rubi steps

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

Mathematica [A]  time = 0.569968, size = 126, normalized size = 0.65 \[ \frac{-b \left (a^2-b^2\right )^2 \cosh (4 x)+4 b \left (a^4-b^4\right ) \cosh (2 x)+a \left (8 b^2 \left (b^2-a^2\right ) \sinh (2 x)+\left (a^2-b^2\right )^2 \sinh (4 x)-4 \left (x \left (-6 a^2 b^2+a^4-3 b^4\right )+8 a b^3 \log (a \cosh (x)+b \sinh (x))\right )\right )}{32 (a-b)^3 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]^3*Sinh[x]^2)/(a*Cosh[x] + b*Sinh[x]),x]

[Out]

(4*b*(a^4 - b^4)*Cosh[2*x] - b*(a^2 - b^2)^2*Cosh[4*x] + a*(-4*((a^4 - 6*a^2*b^2 - 3*b^4)*x + 8*a*b^3*Log[a*Co
sh[x] + b*Sinh[x]]) + 8*b^2*(-a^2 + b^2)*Sinh[2*x] + (a^2 - b^2)^2*Sinh[4*x]))/(32*(a - b)^3*(a + b)^3)

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Maple [A]  time = 0.051, size = 322, normalized size = 1.7 \begin{align*} -2\,{\frac{1}{ \left ( 8\,a-8\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{4}}}+8\,{\frac{1}{ \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{3} \left ( 16\,a-16\,b \right ) }}+{\frac{a}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{3\,b}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{3\,a}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{5\,b}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{{a}^{2}}{8\, \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{3\,ab}{8\, \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{{a}^{2}{b}^{3}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }+2\,{\frac{1}{ \left ( 8\,a+8\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{4}}}+8\,{\frac{1}{ \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{3} \left ( 16\,a+16\,b \right ) }}+{\frac{3\,a}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{5\,b}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{a}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3\,b}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{{a}^{2}}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }+{\frac{3\,ab}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(8*a-8*b)/(tanh(1/2*x)+1)^4+8/(tanh(1/2*x)+1)^3/(16*a-16*b)+1/8*a/(a-b)^2/(tanh(1/2*x)+1)-3/8*b/(a-b)^2/(ta
nh(1/2*x)+1)-3/8/(a-b)^2/(tanh(1/2*x)+1)^2*a+5/8/(a-b)^2/(tanh(1/2*x)+1)^2*b-1/8*a^2/(a-b)^3*ln(tanh(1/2*x)+1)
+3/8*b/(a-b)^3*ln(tanh(1/2*x)+1)*a-a^2*b^3/(a-b)^3/(a+b)^3*ln(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+2/(8*a+8*b)/(
tanh(1/2*x)-1)^4+8/(tanh(1/2*x)-1)^3/(16*a+16*b)+3/8/(a+b)^2/(tanh(1/2*x)-1)^2*a+5/8/(a+b)^2/(tanh(1/2*x)-1)^2
*b+1/8*a/(a+b)^2/(tanh(1/2*x)-1)+3/8*b/(a+b)^2/(tanh(1/2*x)-1)+1/8*a^2/(a+b)^3*ln(tanh(1/2*x)-1)+3/8*b/(a+b)^3
*ln(tanh(1/2*x)-1)*a

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Maxima [A]  time = 1.3266, size = 203, normalized size = 1.05 \begin{align*} -\frac{a^{2} b^{3} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac{{\left (a^{2} + 3 \, a b\right )} x}{8 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac{{\left (4 \, b e^{\left (-2 \, x\right )} + a + b\right )} e^{\left (4 \, x\right )}}{64 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac{4 \, b e^{\left (-2 \, x\right )} -{\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*b^3*log(-(a - b)*e^(-2*x) - a - b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - 1/8*(a^2 + 3*a*b)*x/(a^3 + 3*a^2
*b + 3*a*b^2 + b^3) + 1/64*(4*b*e^(-2*x) + a + b)*e^(4*x)/(a^2 + 2*a*b + b^2) + 1/64*(4*b*e^(-2*x) - (a - b)*e
^(-4*x))/(a^2 - 2*a*b + b^2)

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Fricas [B]  time = 2.2313, size = 2612, normalized size = 13.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^8 + 8*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 +
a*b^4 - b^5)*cosh(x)*sinh(x)^7 + (a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*sinh(x)^8 + 4*(a^4*b - 2
*a^3*b^2 + 2*a*b^4 - b^5)*cosh(x)^6 + 4*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + 7*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^
2*b^3 + a*b^4 - b^5)*cosh(x)^2)*sinh(x)^6 - 8*(a^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*x*cosh(x)^4 + 8*(7*(a^5
- a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^3 + 3*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5)*cosh(x))*sinh
(x)^5 - a^5 - a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - a*b^4 - b^5 + 2*(35*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4
- b^5)*cosh(x)^4 + 30*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5)*cosh(x)^2 - 4*(a^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^
4)*x)*sinh(x)^4 + 8*(7*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^5 + 10*(a^4*b - 2*a^3*b^2 +
2*a*b^4 - b^5)*cosh(x)^3 - 4*(a^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*x*cosh(x))*sinh(x)^3 + 4*(a^4*b + 2*a^3*
b^2 - 2*a*b^4 - b^5)*cosh(x)^2 + 4*(7*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^6 + a^4*b +
2*a^3*b^2 - 2*a*b^4 - b^5 + 15*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5)*cosh(x)^4 - 12*(a^5 - 6*a^3*b^2 - 8*a^2*b^3
- 3*a*b^4)*x*cosh(x)^2)*sinh(x)^2 - 64*(a^2*b^3*cosh(x)^4 + 4*a^2*b^3*cosh(x)^3*sinh(x) + 6*a^2*b^3*cosh(x)^2
*sinh(x)^2 + 4*a^2*b^3*cosh(x)*sinh(x)^3 + a^2*b^3*sinh(x)^4)*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x)
)) + 8*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^7 + 3*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5)*
cosh(x)^5 - 4*(a^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*x*cosh(x)^3 + (a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*cosh(x
))*sinh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3*s
inh(x) + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2*sinh(x)^2 + 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos
h(x)*sinh(x)^3 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^4)

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

[Out]

Timed out

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Giac [A]  time = 1.15873, size = 273, normalized size = 1.41 \begin{align*} -\frac{a^{2} b^{3} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac{{\left (a^{2} - 3 \, a b\right )} x}{8 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{{\left (6 \, a^{2} e^{\left (4 \, x\right )} - 18 \, a b e^{\left (4 \, x\right )} + 4 \, a b e^{\left (2 \, x\right )} - 4 \, b^{2} e^{\left (2 \, x\right )} - a^{2} + 2 \, a b - b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 4 \, b e^{\left (2 \, x\right )}}{64 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*b^3*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - 1/8*(a^2 - 3*a*b)*x/(a^
3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/64*(6*a^2*e^(4*x) - 18*a*b*e^(4*x) + 4*a*b*e^(2*x) - 4*b^2*e^(2*x) - a^2 + 2*
a*b - b^2)*e^(-4*x)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/64*(a*e^(4*x) + b*e^(4*x) + 4*b*e^(2*x))/(a^2 + 2*a*b
+ b^2)