### 3.715 $$\int \frac{\cosh (x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx$$

Optimal. Leaf size=93 $-\frac{2 a b x}{\left (a^2-b^2\right )^2}+\frac{b \sinh (x)}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{a^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}+\frac{b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}$

[Out]

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

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Rubi [A]  time = 0.204338, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.312, Rules used = {3111, 3098, 3133, 3097, 3075} $-\frac{2 a b x}{\left (a^2-b^2\right )^2}+\frac{b \sinh (x)}{\left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{a^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}+\frac{b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3111

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

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]

Rule 3097

Int[sin[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(b*x)/(a^2 + b^2), x] - Dist[a/(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 3075

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

Rubi steps

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

Mathematica [A]  time = 0.183565, size = 60, normalized size = 0.65 $\frac{\left (a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))-2 a b x+\frac{b (a-b) (a+b) \sinh (x)}{a \cosh (x)+b \sinh (x)}}{(a-b)^2 (a+b)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 181, normalized size = 2. \begin{align*} -{\frac{1}{ \left ( a-b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+2\,{\frac{{a}^{2}\tanh \left ( x/2 \right ) b}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{{b}^{3}\tanh \left ( x/2 \right ) }{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}+{\frac{{a}^{2}}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }+{\frac{{b}^{2}}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }-{\frac{1}{ \left ( a+b \right ) ^{2}}\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)*sinh(x)/(a*cosh(x)+b*sinh(x))^2,x)

[Out]

-1/(a-b)^2*ln(tanh(1/2*x)+1)+2*a^2/(a-b)^2/(a+b)^2/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)*tanh(1/2*x)*b-2/(a-b)^2
/(a+b)^2*b^3*tanh(1/2*x)/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+1/(a-b)^2/(a+b)^2*ln(a+2*tanh(1/2*x)*b+a*tanh(1/2
*x)^2)*a^2+1/(a-b)^2/(a+b)^2*ln(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)*b^2-1/(a+b)^2*ln(tanh(1/2*x)-1)

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Maxima [A]  time = 1.23402, size = 144, normalized size = 1.55 \begin{align*} \frac{2 \, a b}{a^{4} - 2 \, a^{2} b^{2} + b^{4} +{\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )}} + \frac{{\left (a^{2} + b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{x}{a^{2} + 2 \, a b + b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*b/(a^4 - 2*a^2*b^2 + b^4 + (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*x)) + (a^2 + b^2)*log(-(a - b)*e^(-2*x) -
a - b)/(a^4 - 2*a^2*b^2 + b^4) + x/(a^2 + 2*a*b + b^2)

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Fricas [B]  time = 2.41042, size = 892, normalized size = 9.59 \begin{align*} -\frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \left (x\right )^{2} + 2 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{2} b - 2 \, a b^{2} +{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} x -{\left (a^{3} - a^{2} b + a b^{2} - b^{3} +{\left (a^{3} + a^{2} b + a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{3} + a^{2} b + a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{3} + a^{2} b + a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} +{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \sinh \left (x\right )^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*x*cosh(x)^2 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*x*cosh(x)*sinh(x) + (a^3 + 3
*a^2*b + 3*a*b^2 + b^3)*x*sinh(x)^2 + 2*a^2*b - 2*a*b^2 + (a^3 + a^2*b - a*b^2 - b^3)*x - (a^3 - a^2*b + a*b^2
- b^3 + (a^3 + a^2*b + a*b^2 + b^3)*cosh(x)^2 + 2*(a^3 + a^2*b + a*b^2 + b^3)*cosh(x)*sinh(x) + (a^3 + a^2*b
+ a*b^2 + b^3)*sinh(x)^2)*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))))/(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2
*b^3 + a*b^4 - b^5 + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*cosh(x)^2 + 2*(a^5 + a^4*b - 2*a^3*b^
2 - 2*a^2*b^3 + a*b^4 + b^5)*cosh(x)*sinh(x) + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*sinh(x)^2)

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Sympy [A]  time = 146.537, size = 1986, normalized size = 21.35 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*log(sinh(x)), Eq(a, 0) & Eq(b, 0)), (log(cosh(x))/a**2, Eq(b, 0)), (-x*sinh(x)**2/(4*b**2*sinh(
x)**2 - 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) + 2*x*sinh(x)*cosh(x)/(4*b**2*sinh(x)**2 - 8*b**2*sinh(x)*
cosh(x) + 4*b**2*cosh(x)**2) - x*cosh(x)**2/(4*b**2*sinh(x)**2 - 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) +
sinh(x)*cosh(x)/(4*b**2*sinh(x)**2 - 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2), Eq(a, -b)), (x*sinh(x)**2/(
4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) + 2*x*sinh(x)*cosh(x)/(4*b**2*sinh(x)**2 + 8*b
**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) + x*cosh(x)**2/(4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*c
osh(x)**2) - sinh(x)*cosh(x)/(4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2), Eq(a, b)), (-x*
exp(4*x)*sinh(x)**2/(4*b**2*exp(4*x)*sinh(x)**2 - 8*b**2*exp(4*x)*sinh(x)*cosh(x) + 4*b**2*exp(4*x)*cosh(x)**2
+ 8*b**2*exp(2*x)*sinh(x)**2 - 8*b**2*exp(2*x)*cosh(x)**2 + 4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b*
*2*cosh(x)**2) + 2*x*exp(4*x)*sinh(x)*cosh(x)/(4*b**2*exp(4*x)*sinh(x)**2 - 8*b**2*exp(4*x)*sinh(x)*cosh(x) +
4*b**2*exp(4*x)*cosh(x)**2 + 8*b**2*exp(2*x)*sinh(x)**2 - 8*b**2*exp(2*x)*cosh(x)**2 + 4*b**2*sinh(x)**2 + 8*b
**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) - x*exp(4*x)*cosh(x)**2/(4*b**2*exp(4*x)*sinh(x)**2 - 8*b**2*exp(4*x)
*sinh(x)*cosh(x) + 4*b**2*exp(4*x)*cosh(x)**2 + 8*b**2*exp(2*x)*sinh(x)**2 - 8*b**2*exp(2*x)*cosh(x)**2 + 4*b*
*2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) + x*sinh(x)**2/(4*b**2*exp(4*x)*sinh(x)**2 - 8*b**
2*exp(4*x)*sinh(x)*cosh(x) + 4*b**2*exp(4*x)*cosh(x)**2 + 8*b**2*exp(2*x)*sinh(x)**2 - 8*b**2*exp(2*x)*cosh(x)
**2 + 4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) + 2*x*sinh(x)*cosh(x)/(4*b**2*exp(4*x)*s
inh(x)**2 - 8*b**2*exp(4*x)*sinh(x)*cosh(x) + 4*b**2*exp(4*x)*cosh(x)**2 + 8*b**2*exp(2*x)*sinh(x)**2 - 8*b**2
*exp(2*x)*cosh(x)**2 + 4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) + x*cosh(x)**2/(4*b**2*
exp(4*x)*sinh(x)**2 - 8*b**2*exp(4*x)*sinh(x)*cosh(x) + 4*b**2*exp(4*x)*cosh(x)**2 + 8*b**2*exp(2*x)*sinh(x)**
2 - 8*b**2*exp(2*x)*cosh(x)**2 + 4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2) + exp(4*x)*si
nh(x)*cosh(x)/(4*b**2*exp(4*x)*sinh(x)**2 - 8*b**2*exp(4*x)*sinh(x)*cosh(x) + 4*b**2*exp(4*x)*cosh(x)**2 + 8*b
**2*exp(2*x)*sinh(x)**2 - 8*b**2*exp(2*x)*cosh(x)**2 + 4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*cosh(x) + 4*b**2*cos
h(x)**2) + 4*exp(2*x)*cosh(x)**2/(4*b**2*exp(4*x)*sinh(x)**2 - 8*b**2*exp(4*x)*sinh(x)*cosh(x) + 4*b**2*exp(4*
x)*cosh(x)**2 + 8*b**2*exp(2*x)*sinh(x)**2 - 8*b**2*exp(2*x)*cosh(x)**2 + 4*b**2*sinh(x)**2 + 8*b**2*sinh(x)*c
osh(x) + 4*b**2*cosh(x)**2) - sinh(x)*cosh(x)/(4*b**2*exp(4*x)*sinh(x)**2 - 8*b**2*exp(4*x)*sinh(x)*cosh(x) +
4*b**2*exp(4*x)*cosh(x)**2 + 8*b**2*exp(2*x)*sinh(x)**2 - 8*b**2*exp(2*x)*cosh(x)**2 + 4*b**2*sinh(x)**2 + 8*b
**2*sinh(x)*cosh(x) + 4*b**2*cosh(x)**2), Eq(a, -(b*exp(2*x) - b)/(exp(2*x) + 1))), (a**3*log(a*cosh(x)/b + si
nh(x))*cosh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**4*cosh(x) + b
**5*sinh(x)) - a**3*cosh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**
4*cosh(x) + b**5*sinh(x)) - 2*a**2*b*x*cosh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b
**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)) + a**2*b*log(a*cosh(x)/b + sinh(x))*sinh(x)/(a**5*cosh(x) + a**4*
b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)) - 2*a*b**2*x*sinh(x)/(a
**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)) + a*
b**2*log(a*cosh(x)/b + sinh(x))*cosh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sin
h(x) + a*b**4*cosh(x) + b**5*sinh(x)) + a*b**2*cosh(x)/(a**5*cosh(x) + a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) -
2*a**2*b**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)) + b**3*log(a*cosh(x)/b + sinh(x))*sinh(x)/(a**5*cosh(x) +
a**4*b*sinh(x) - 2*a**3*b**2*cosh(x) - 2*a**2*b**3*sinh(x) + a*b**4*cosh(x) + b**5*sinh(x)), True))

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Giac [A]  time = 1.14981, size = 173, normalized size = 1.86 \begin{align*} \frac{{\left (a^{2} + b^{2}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{x}{a^{2} - 2 \, a b + b^{2}} - \frac{a^{2} e^{\left (2 \, x\right )} + b^{2} e^{\left (2 \, x\right )} + a^{2} - b^{2}}{{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )}{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2 + b^2)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^4 - 2*a^2*b^2 + b^4) - x/(a^2 - 2*a*b + b^2) - (a^2*e^(
2*x) + b^2*e^(2*x) + a^2 - b^2)/((a^3 - a^2*b - a*b^2 + b^3)*(a*e^(2*x) + b*e^(2*x) + a - b))