3.8.6 \(\int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx\) [706]

Optimal. Leaf size=72 \[ \frac {a b \text {ArcTan}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {a \cosh (x)}{a^2-b^2}-\frac {b \sinh (x)}{a^2-b^2} \]

[Out]

a*b*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(3/2)+a*cosh(x)/(a^2-b^2)-b*sinh(x)/(a^2-b^2)

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Rubi [A]
time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3188, 2717, 2718, 3153, 212} \begin {gather*} \frac {a b \text {ArcTan}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {b \sinh (x)}{a^2-b^2}+\frac {a \cosh (x)}{a^2-b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a*Cosh[x])/(a^2 - b^2) - (b*Sinh[x]
)/(a^2 - b^2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3153

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

Rule 3188

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]

Rubi steps

\begin {align*} \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx &=\frac {a \int \sinh (x) \, dx}{a^2-b^2}-\frac {b \int \cosh (x) \, dx}{a^2-b^2}+\frac {(a b) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}\\ &=\frac {a \cosh (x)}{a^2-b^2}-\frac {b \sinh (x)}{a^2-b^2}+\frac {(i a b) \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{a^2-b^2}\\ &=\frac {a b \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {a \cosh (x)}{a^2-b^2}-\frac {b \sinh (x)}{a^2-b^2}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 79, normalized size = 1.10 \begin {gather*} \frac {2 a b \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}+\frac {a \cosh (x)}{a^2-b^2}+\frac {b \sinh (x)}{-a^2+b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(3/2)*(a + b)^(3/2)) + (a*Cosh[x])/(a^2 -
b^2) + (b*Sinh[x])/(-a^2 + b^2)

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Maple [A]
time = 0.91, size = 92, normalized size = 1.28

method	result	size
default	\(\frac {2 a b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {a^{2}-b^{2}}}-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {4}{\left (4 a -4 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )}\)	\(92\)
risch	\(\frac {{\mathrm e}^{x}}{2 b +2 a}+\frac {{\mathrm e}^{-x}}{2 a -2 b}-\frac {b a \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}+\frac {b a \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*b/(a+b)/(a-b)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))-4/(4*a+4*b)/(tanh(1/2*x)-1
)+4/(4*a-4*b)/(tanh(1/2*x)+1)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (68) = 136\).
time = 0.39, size = 427, normalized size = 5.93 \begin {gather*} \left [\frac {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} + 2 \, {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}, \frac {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} - 4 \, {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}\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)),x, algorithm="fricas")

[Out]

[1/2*(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 + 2*(a*b*cosh(x) + a*b*sinh(x))*sqrt(-a^2 + b^2)*log(((a + b
)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)/
((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)))/((a^4 - 2*a^2*b^2 + b^4)*cosh(x)
+ (a^4 - 2*a^2*b^2 + b^4)*sinh(x)), 1/2*(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 - 4*(a*b*cosh(x) + a*b*s
inh(x))*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))))/((a^4 - 2*a^2*b^2 + b^4)*
cosh(x) + (a^4 - 2*a^2*b^2 + b^4)*sinh(x))]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (58) = 116\).
time = 124.06, size = 678, normalized size = 9.42 \begin {gather*} \begin {cases} \tilde {\infty } \sinh {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\sinh {\left (x \right )}}{b} & \text {for}\: a = 0 \\- \frac {\sinh ^{2}{\left (x \right )}}{- 3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} + \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{- 3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} - \frac {\cosh ^{2}{\left (x \right )}}{- 3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} & \text {for}\: a = - b \\\frac {\sinh ^{2}{\left (x \right )}}{3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} + \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} + \frac {\cosh ^{2}{\left (x \right )}}{3 b \sinh {\left (x \right )} + 3 b \cosh {\left (x \right )}} & \text {for}\: a = b \\\frac {a b \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} - \frac {a b \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} - \frac {a b \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} + \frac {a b \log {\left (\tanh {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} - \frac {2 a \sqrt {- a^{2} + b^{2}}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} + \frac {2 b \sqrt {- a^{2} + b^{2}} \tanh {\left (\frac {x}{2} \right )}}{a^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} - a^{2} \sqrt {- a^{2} + b^{2}} - b^{2} \sqrt {- a^{2} + b^{2}} \tanh ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {- a^{2} + b^{2}}} & \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)),x)

[Out]

Piecewise((zoo*sinh(x), Eq(a, 0) & Eq(b, 0)), (sinh(x)/b, Eq(a, 0)), (-sinh(x)**2/(-3*b*sinh(x) + 3*b*cosh(x))
+ sinh(x)*cosh(x)/(-3*b*sinh(x) + 3*b*cosh(x)) - cosh(x)**2/(-3*b*sinh(x) + 3*b*cosh(x)), Eq(a, -b)), (sinh(x
)**2/(3*b*sinh(x) + 3*b*cosh(x)) + sinh(x)*cosh(x)/(3*b*sinh(x) + 3*b*cosh(x)) + cosh(x)**2/(3*b*sinh(x) + 3*b
*cosh(x)), Eq(a, b)), (a*b*log(tanh(x/2) + b/a - sqrt(-a**2 + b**2)/a)*tanh(x/2)**2/(a**2*sqrt(-a**2 + b**2)*t
anh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) - a*b*
log(tanh(x/2) + b/a - sqrt(-a**2 + b**2)/a)/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) -
b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) - a*b*log(tanh(x/2) + b/a + sqrt(-a**2 + b**2)
/a)*tanh(x/2)**2/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*tan
h(x/2)**2 + b**2*sqrt(-a**2 + b**2)) + a*b*log(tanh(x/2) + b/a + sqrt(-a**2 + b**2)/a)/(a**2*sqrt(-a**2 + b**2
)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) - 2
*a*sqrt(-a**2 + b**2)/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2
)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)) + 2*b*sqrt(-a**2 + b**2)*tanh(x/2)/(a**2*sqrt(-a**2 + b**2)*tanh(x/2
)**2 - a**2*sqrt(-a**2 + b**2) - b**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 + b**2*sqrt(-a**2 + b**2)), True))

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Giac [A]
time = 0.40, size = 60, normalized size = 0.83 \begin {gather*} \frac {2 \, a b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} + \frac {e^{\left (-x\right )}}{2 \, {\left (a - b\right )}} + \frac {e^{x}}{2 \, {\left (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)),x, algorithm="giac")

[Out]

2*a*b*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/(a^2 - b^2)^(3/2) + 1/2*e^(-x)/(a - b) + 1/2*e^x/(a + b)

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Mupad [B]
time = 1.69, size = 157, normalized size = 2.18 \begin {gather*} \frac {{\mathrm {e}}^x}{2\,a+2\,b}+\frac {{\mathrm {e}}^{-x}}{2\,a-2\,b}+\frac {2\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^x\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{a^3\,\sqrt {a^2\,b^2}+b^3\,\sqrt {a^2\,b^2}-a\,b^2\,\sqrt {a^2\,b^2}-a^2\,b\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)/(2*a + 2*b) + exp(-x)/(2*a - 2*b) + (2*atan((a*b*exp(x)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)^(1/2))/(a^3
*(a^2*b^2)^(1/2) + b^3*(a^2*b^2)^(1/2) - a*b^2*(a^2*b^2)^(1/2) - a^2*b*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(a^6
- b^6 + 3*a^2*b^4 - 3*a^4*b^2)^(1/2)

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