Integrand size = 16, antiderivative size = 72 \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {a b \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {a \cosh (x)}{a^2-b^2}-\frac {b \sinh (x)}{a^2-b^2} \] Output:
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)
Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {2 a b \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}+\frac {a \cosh (x)}{a^2-b^2}+\frac {b \sinh (x)}{-a^2+b^2} \] Input:
Integrate[(Cosh[x]*Sinh[x])/(a*Cosh[x] + b*Sinh[x]),x]
Output:
(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)
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3042, 26, 3588, 26, 3042, 26, 3117, 3118, 3553, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sinh (x) \cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x) \cos (i x)}{a \cos (i x)-i b \sin (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos (i x) \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx\) |
\(\Big \downarrow \) 3588 |
\(\displaystyle -i \left (\frac {a \int i \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {i a \int \sinh (x)dx}{a^2-b^2}-\frac {i b \int \cosh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {i a \int -i \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {a \int \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle -i \left (\frac {a \int \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle -i \left (\frac {i a b \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}+\frac {i a \cosh (x)}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 3553 |
\(\displaystyle -i \left (-\frac {a b \int \frac {1}{a^2-b^2-(-i b \cosh (x)-i a \sinh (x))^2}d(-i b \cosh (x)-i a \sinh (x))}{a^2-b^2}-\frac {i b \sinh (x)}{a^2-b^2}+\frac {i a \cosh (x)}{a^2-b^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -i \left (-\frac {a b \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {i b \sinh (x)}{a^2-b^2}+\frac {i a \cosh (x)}{a^2-b^2}\right )\) |
Input:
Int[(Cosh[x]*Sinh[x])/(a*Cosh[x] + b*Sinh[x]),x]
Output:
(-I)*(-((a*b*ArcTanh[((-I)*b*Cosh[x] - I*a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) + (I*a*Cosh[x])/(a^2 - b^2) - (I*b*Sinh[x])/(a^2 - b^2))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-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]
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[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]
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28
method | result | size |
default | \(\frac {2 a b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {a^{2}-b^{2}}}-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {4}{\left (4 a -4 b \right ) \left (1+\tanh \left (\frac {x}{2}\right )\right )}\) | \(92\) |
risch | \(\frac {{\mathrm e}^{x}}{2 a +2 b}+\frac {{\mathrm e}^{-x}}{2 a -2 b}-\frac {a b \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}+\frac {a b \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right )}\) | \(120\) |
Input:
int(cosh(x)*sinh(x)/(a*cosh(x)+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
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)/(1+tanh(1/2*x))
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (68) = 136\).
Time = 0.12 (sec) , antiderivative size = 427, normalized size of antiderivative = 5.93 \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\left [\frac {a^{3} + a^{2} b - a b^{2} - b^{3} + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}, \frac {a^{3} + a^{2} b - a b^{2} - b^{3} + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{2} - 4 \, {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right )}{2 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )\right )}}\right ] \] Input:
integrate(cosh(x)*sinh(x)/(a*cosh(x)+b*sinh(x)),x, algorithm="fricas")
Output:
[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)*s inh(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*sinh(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))]
Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (58) = 116\).
Time = 68.10 (sec) , antiderivative size = 678, normalized size of antiderivative = 9.42 \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx =\text {Too large to display} \] Input:
integrate(cosh(x)*sinh(x)/(a*cosh(x)+b*sinh(x)),x)
Output:
Piecewise((zoo*sinh(x), Eq(a, 0) & Eq(b, 0)), (sinh(x)/b, Eq(a, 0)), (-sin h(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)*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 )/(a**2*sqrt(-a**2 + b**2)*tanh(x/2)**2 - a**2*sqrt(-a**2 + b**2) - b**2*s qrt(-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)*ta nh(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)) - 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))
Exception generated. \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(x)*sinh(x)/(a*cosh(x)+b*sinh(x)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {2 \, a b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} + \frac {e^{\left (-x\right )}}{2 \, {\left (a - b\right )}} + \frac {e^{x}}{2 \, {\left (a + b\right )}} \] Input:
integrate(cosh(x)*sinh(x)/(a*cosh(x)+b*sinh(x)),x, algorithm="giac")
Output:
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)
Time = 0.94 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.18 \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^x}{2\,a+2\,b}+\frac {{\mathrm {e}}^{-x}}{2\,a-2\,b}+\frac {2\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^x\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{a^3\,\sqrt {a^2\,b^2}+b^3\,\sqrt {a^2\,b^2}-a\,b^2\,\sqrt {a^2\,b^2}-a^2\,b\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}} \] Input:
int((cosh(x)*sinh(x))/(a*cosh(x) + b*sinh(x)),x)
Output:
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)
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.75 \[ \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {4 e^{x} \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {e^{x} a +e^{x} b}{\sqrt {a^{2}-b^{2}}}\right ) a b +e^{2 x} a^{3}-e^{2 x} a^{2} b -e^{2 x} a \,b^{2}+e^{2 x} b^{3}+a^{3}+a^{2} b -a \,b^{2}-b^{3}}{2 e^{x} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )} \] Input:
int(cosh(x)*sinh(x)/(a*cosh(x)+b*sinh(x)),x)
Output:
(4*e**x*sqrt(a**2 - b**2)*atan((e**x*a + e**x*b)/sqrt(a**2 - b**2))*a*b + e**(2*x)*a**3 - e**(2*x)*a**2*b - e**(2*x)*a*b**2 + e**(2*x)*b**3 + a**3 + a**2*b - a*b**2 - b**3)/(2*e**x*(a**4 - 2*a**2*b**2 + b**4))