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*ln(a*cosh(x)+b*sinh(x))/(a^2-b^2)^2+b^2*ln(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 = 0.20, antiderivative size = 93, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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 (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*}

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Mathematica [A]  time = 0.24, 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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fricas [B]  time = 0.61, size = 376, normalized size = 4.04 \[ -\frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \relax (x)^{2} + 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \cosh \relax (x) \sinh \relax (x) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x \sinh \relax (x)^{2} + 2 \, a^{2} b - 2 \, a b^{2} + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} x - {\left (a^{3} - a^{2} b + a b^{2} - b^{3} + {\left (a^{3} + a^{2} b + a b^{2} + b^{3}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{3} + a^{2} b + a b^{2} + b^{3}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{3} + a^{2} b + a b^{2} + b^{3}\right )} \sinh \relax (x)^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \sinh \relax (x)^{2}} \]

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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giac [A]  time = 0.13, size = 128, normalized size = 1.38 \[ \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {x}{a^{2} - 2 \, a b + b^{2}} - \frac {a^{2} e^{\left (2 \, x\right )} + b^{2} e^{\left (2 \, x\right )} + a^{2} - b^{2}}{{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \]

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))

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maple [A]  time = 0.25, size = 181, normalized size = 1.95 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{2}}+\frac {2 a^{2} \tanh \left (\frac {x}{2}\right ) b}{\left (a -b \right )^{2} \left (a +b \right )^{2} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}-\frac {2 b^{3} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}+\frac {\ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right ) a^{2}}{\left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {\ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right ) b^{2}}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{\left (a -b \right )^{2}} \]

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 = 0.45, size = 107, normalized size = 1.15 \[ \frac {2 \, a b}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {x}{a^{2} + 2 \, a b + b^{2}} \]

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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mupad [B]  time = 1.93, size = 98, normalized size = 1.05 \[ \ln \left (a\,\mathrm {cosh}\relax (x)+b\,\mathrm {sinh}\relax (x)\right )\,\left (\frac {1}{2\,{\left (a+b\right )}^2}+\frac {1}{2\,{\left (a-b\right )}^2}\right )-\frac {\frac {a\,\mathrm {cosh}\relax (x)}{a^2-b^2}+\frac {2\,a^2\,b\,x\,\mathrm {cosh}\relax (x)}{{\left (a^2-b^2\right )}^2}+\frac {2\,a\,b^2\,x\,\mathrm {sinh}\relax (x)}{{\left (a^2-b^2\right )}^2}}{a\,\mathrm {cosh}\relax (x)+b\,\mathrm {sinh}\relax (x)} \]

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]

log(a*cosh(x) + b*sinh(x))*(1/(2*(a + b)^2) + 1/(2*(a - b)^2)) - ((a*cosh(x))/(a^2 - b^2) + (2*a^2*b*x*cosh(x)
)/(a^2 - b^2)^2 + (2*a*b^2*x*sinh(x))/(a^2 - b^2)^2)/(a*cosh(x) + b*sinh(x))

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sympy [A]  time = 1.39, size = 962, normalized size = 10.34 \[ \text {result too large to display} \]

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)), (-2*x*sinh(x)**2/(8*b**2*sinh(x)**2 - 16*b**2*sinh(x)*cosh(
x) + 8*b**2*cosh(x)**2) + 4*x*sinh(x)*cosh(x)/(8*b**2*sinh(x)**2 - 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2
) - 2*x*cosh(x)**2/(8*b**2*sinh(x)**2 - 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) + sinh(x)**2/(8*b**2*sinh
(x)**2 - 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) + cosh(x)**2/(8*b**2*sinh(x)**2 - 16*b**2*sinh(x)*cosh(x
) + 8*b**2*cosh(x)**2), Eq(a, -b)), (2*x*sinh(x)**2/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh
(x)**2) + 4*x*sinh(x)*cosh(x)/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) + 2*x*cosh(x)*
*2/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) + sinh(x)**2/(8*b**2*sinh(x)**2 + 16*b**2
*sinh(x)*cosh(x) + 8*b**2*cosh(x)**2) + cosh(x)**2/(8*b**2*sinh(x)**2 + 16*b**2*sinh(x)*cosh(x) + 8*b**2*cosh(
x)**2), Eq(a, b)), (log(sinh(x))/b**2, Eq(a, 0)), (a**3*log(cosh(x) + b*sinh(x)/a)*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(cosh(x) + b*sinh(x)/a)*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(cosh(x) + b*sinh(x)/a)*co
sh(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*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(cosh(x) + b*sinh(x)/a)*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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