3.8.15 \(\int \frac {\cosh (x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx\) [715]

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

[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 = 0.15, 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 = {3190, 3177, 3212, 3176, 3154} \begin {gather*} -\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} \end {gather*}

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 3154

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 3176

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 3177

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 3190

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 3212

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 = 0.16, size = 60, normalized size = 0.65 \begin {gather*} \frac {-2 a b x+\left (a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))+\frac {(a-b) b (a+b) \sinh (x)}{a \cosh (x)+b \sinh (x)}}{(a-b)^2 (a+b)^2} \end {gather*}

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 = 1.36, size = 108, normalized size = 1.16

method	result	size
default	\(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{\left (a -b \right )^{2}}+\frac {\frac {2 b \left (a^{2}-b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}+\left (a^{2}+b^{2}\right ) \ln \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{2}}\)	\(108\)
risch	\(\frac {x}{a^{2}+2 a b +b^{2}}-\frac {2 x \,a^{2}}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {2 x \,b^{2}}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {2 a b}{\left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{2}}{a^{4}-2 a^{2} b^{2}+b^{4}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) b^{2}}{a^{4}-2 a^{2} b^{2}+b^{4}}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(x)/(a*cosh(x)+b*sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 107, normalized size = 1.15 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (93) = 186\).
time = 0.39, size = 376, normalized size = 4.04 \begin {gather*} -\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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 962 vs. \(2 (83) = 166\).
time = 0.66, size = 962, normalized size = 10.34 \begin {gather*} \begin {cases} \tilde {\infty } \log {\left (\sinh {\left (x \right )} \right )} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x \sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {4 x \sinh {\left (x \right )} \cosh {\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} - \frac {2 x \cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {\sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {\cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} - 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} & \text {for}\: a = - b \\\frac {2 x \sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {4 x \sinh {\left (x \right )} \cosh {\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {2 x \cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {\sinh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} + \frac {\cosh ^{2}{\left (x \right )}}{8 b^{2} \sinh ^{2}{\left (x \right )} + 16 b^{2} \sinh {\left (x \right )} \cosh {\left (x \right )} + 8 b^{2} \cosh ^{2}{\left (x \right )}} & \text {for}\: a = b \\\frac {\log {\left (\sinh {\left (x \right )} \right )}}{b^{2}} & \text {for}\: a = 0 \\\frac {a^{3} \log {\left (\cosh {\left (x \right )} + \frac {b \sinh {\left (x \right )}}{a} \right )} \cosh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} - \frac {a^{3} \cosh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} - \frac {2 a^{2} b x \cosh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} + \frac {a^{2} b \log {\left (\cosh {\left (x \right )} + \frac {b \sinh {\left (x \right )}}{a} \right )} \sinh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} - \frac {2 a b^{2} x \sinh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} + \frac {a b^{2} \log {\left (\cosh {\left (x \right )} + \frac {b \sinh {\left (x \right )}}{a} \right )} \cosh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} + \frac {a b^{2} \cosh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} + \frac {b^{3} \log {\left (\cosh {\left (x \right )} + \frac {b \sinh {\left (x \right )}}{a} \right )} \sinh {\left (x \right )}}{a^{5} \cosh {\left (x \right )} + a^{4} b \sinh {\left (x \right )} - 2 a^{3} b^{2} \cosh {\left (x \right )} - 2 a^{2} b^{3} \sinh {\left (x \right )} + a b^{4} \cosh {\left (x \right )} + b^{5} \sinh {\left (x \right )}} & \text {otherwise} \end {cases} \end {gather*}

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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Giac [A]
time = 0.41, size = 128, normalized size = 1.38 \begin {gather*} \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 {gather*}

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

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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