Integrand size = 20, antiderivative size = 194 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=-\frac {a^2 b^3 x}{\left (a^2-b^2\right )^3}-\frac {a^2 b x}{2 \left (a^2-b^2\right )^2}+\frac {b x}{8 \left (a^2-b^2\right )}+\frac {a^3 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {a^2 b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}+\frac {b \cosh (x) \sinh (x)}{8 \left (a^2-b^2\right )}-\frac {b \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )}-\frac {a b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {a \sinh ^4(x)}{4 \left (a^2-b^2\right )} \]
-a^2*b^3*x/(a^2-b^2)^3-1/2*a^2*b*x/(a^2-b^2)^2+1/8*b*x/(a^2-b^2)+a^3*b^2*l n(a*cosh(x)+b*sinh(x))/(a^2-b^2)^3+1/2*a^2*b*cosh(x)*sinh(x)/(a^2-b^2)^2+1 /8*b*cosh(x)*sinh(x)/(a^2-b^2)-1/4*b*cosh(x)^3*sinh(x)/(a^2-b^2)-1/2*a*b^2 *sinh(x)^2/(a^2-b^2)^2+1/4*a*sinh(x)^4/(a^2-b^2)
Time = 0.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.66 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {-4 a \left (a^4-b^4\right ) \cosh (2 x)+a \left (a^2-b^2\right )^2 \cosh (4 x)-b \left (4 \left (3 a^4 x+6 a^2 b^2 x-b^4 x-8 a^3 b \log (a \cosh (x)+b \sinh (x))\right )-8 a^2 \left (a^2-b^2\right ) \sinh (2 x)+\left (a^2-b^2\right )^2 \sinh (4 x)\right )}{32 (a-b)^3 (a+b)^3} \]
(-4*a*(a^4 - b^4)*Cosh[2*x] + a*(a^2 - b^2)^2*Cosh[4*x] - b*(4*(3*a^4*x + 6*a^2*b^2*x - b^4*x - 8*a^3*b*Log[a*Cosh[x] + b*Sinh[x]]) - 8*a^2*(a^2 - b ^2)*Sinh[2*x] + (a^2 - b^2)^2*Sinh[4*x]))/(32*(a - b)^3*(a + b)^3)
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.02, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3042, 26, 3588, 25, 26, 3042, 25, 26, 3044, 15, 3048, 3042, 3115, 24, 3588, 25, 26, 3042, 25, 26, 3044, 15, 3115, 24, 3576, 26, 3042, 3612}
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.
| \(\displaystyle \int \frac {\sinh ^3(x) \cosh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3 \cos (i x)^2}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i x)^2 \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (-\frac {i b \int -\cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {a \int -i \cosh (x) \sinh ^3(x)dx}{a^2-b^2}+\frac {i a b \int -\frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i b \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {a \int -i \cosh (x) \sinh ^3(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i b \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a \int \cosh (x) \sinh ^3(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i b \int -\cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {i a \int i \cos (i x) \sin (i x)^3dx}{a^2-b^2}-\frac {i a b \int -\frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {i a \int i \cos (i x) \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (-\frac {i a \int -i \sinh ^3(x)d(i \sinh (x))}{a^2-b^2}-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (-\frac {i b \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \left (\frac {1}{4} \int \cosh ^2(x)dx-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \left (-\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {1}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\cos (i x) \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {a \int -\sinh ^2(x)dx}{a^2-b^2}-\frac {i b \int i \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {a \int \sinh ^2(x)dx}{a^2-b^2}-\frac {i b \int i \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {i \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {a \int \sinh ^2(x)dx}{a^2-b^2}+\frac {b \int \cosh (x) \sinh (x)dx}{a^2-b^2}-\frac {a b \int \frac {\sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {a \int -\sin (i x)^2dx}{a^2-b^2}+\frac {b \int -i \cos (i x) \sin (i x)dx}{a^2-b^2}-\frac {a b \int -\frac {i \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}+\frac {b \int -i \cos (i x) \sin (i x)dx}{a^2-b^2}-\frac {a b \int -\frac {i \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}-\frac {i b \int \cos (i x) \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}-\frac {b \int i \sinh (x)d(i \sinh (x))}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {a \int \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}+\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \int \frac {\sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3576 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (-\frac {a \int -\frac {i (b \cosh (x)+a \sinh (x))}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i b x}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (\frac {i a \int \frac {b \cosh (x)+a \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {i b x}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a b \left (\frac {i a \int \frac {b \cos (i x)-i a \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b x}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle i \left (-\frac {i a \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {i b \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}+\frac {i a b \left (\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}+\frac {a \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}+\frac {i a b \left (\frac {i a \log (a \cosh (x)+b \sinh (x))}{a^2-b^2}-\frac {i b x}{a^2-b^2}\right )}{a^2-b^2}\right )}{a^2-b^2}\right )\) |
I*(((-1/4*I)*a*Sinh[x]^4)/(a^2 - b^2) + (I*a*b*((I*a*b*(((-I)*b*x)/(a^2 - b^2) + (I*a*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)))/(a^2 - b^2) + (b*Sin h[x]^2)/(2*(a^2 - b^2)) + (a*(x/2 - (Cosh[x]*Sinh[x])/2))/(a^2 - b^2)))/(a ^2 - b^2) - (I*b*(-1/4*(Cosh[x]^3*Sinh[x]) + (x/2 + (Cosh[x]*Sinh[x])/2)/4 ))/(a^2 - b^2))
3.8.11.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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] - Simp[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]
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]
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)*(Log[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]
Time = 10.94 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(-\frac {3 a x b}{8 \left (a +b \right )^{3}}-\frac {b^{2} x}{8 \left (a +b \right )^{3}}+\frac {{\mathrm e}^{4 x}}{64 a +64 b}-\frac {{\mathrm e}^{2 x} a}{16 \left (a +b \right )^{2}}-\frac {{\mathrm e}^{-2 x} a}{16 \left (a -b \right )^{2}}+\frac {{\mathrm e}^{-4 x}}{64 a -64 b}-\frac {2 a^{3} b^{2} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {a^{3} b^{2} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}\) | \(160\) |
| default | \(\frac {4}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{4} \left (16 a +16 b \right )}+\frac {16}{\left (32 a +32 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {-a -3 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {a -b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \left (3 a +b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}+\frac {4}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{4} \left (16 a -16 b \right )}-\frac {16}{\left (32 a -32 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a -b}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {-a +3 b}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {b \left (3 a -b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 \left (a -b \right )^{3}}+\frac {a^{3} b^{2} \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(249\) |
-3/8*a*x/(a+b)^3*b-1/8*b^2*x/(a+b)^3+1/64/(a+b)*exp(4*x)-1/16/(a+b)^2*exp( 2*x)*a-1/16/(a-b)^2*exp(-2*x)*a+1/64/(a-b)*exp(-4*x)-2*a^3*b^2/(a^6-3*a^4* b^2+3*a^2*b^4-b^6)*x+a^3*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*ln(exp(2*x)+(a- b)/(a+b))
Leaf count of result is larger than twice the leaf count of optimal. 1158 vs. \(2 (180) = 360\).
Time = 0.27 (sec) , antiderivative size = 1158, normalized size of antiderivative = 5.97 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Too large to display} \]
1/64*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^8 + 8*(a ^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)*sinh(x)^7 + (a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*sinh(x)^8 - 4*(a^5 - 2*a^4 *b + 2*a^2*b^3 - a*b^4)*cosh(x)^6 - 4*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4 - 7*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^2)*sinh(x)^ 6 - 8*(3*a^4*b + 8*a^3*b^2 + 6*a^2*b^3 - b^5)*x*cosh(x)^4 + 8*(7*(a^5 - a^ 4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^3 - 3*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4)*cosh(x))*sinh(x)^5 + a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^ 3 + a*b^4 + b^5 + 2*(35*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 )*cosh(x)^4 - 30*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4)*cosh(x)^2 - 4*(3*a^4* b + 8*a^3*b^2 + 6*a^2*b^3 - b^5)*x)*sinh(x)^4 + 8*(7*(a^5 - a^4*b - 2*a^3* b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^5 - 10*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4)*cosh(x)^3 - 4*(3*a^4*b + 8*a^3*b^2 + 6*a^2*b^3 - b^5)*x*cosh(x))*s inh(x)^3 - 4*(a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4)*cosh(x)^2 + 4*(7*(a^5 - a ^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^6 - a^5 - 2*a^4*b + 2* a^2*b^3 + a*b^4 - 15*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4)*cosh(x)^4 - 12*(3 *a^4*b + 8*a^3*b^2 + 6*a^2*b^3 - b^5)*x*cosh(x)^2)*sinh(x)^2 + 64*(a^3*b^2 *cosh(x)^4 + 4*a^3*b^2*cosh(x)^3*sinh(x) + 6*a^3*b^2*cosh(x)^2*sinh(x)^2 + 4*a^3*b^2*cosh(x)*sinh(x)^3 + a^3*b^2*sinh(x)^4)*log(2*(a*cosh(x) + b*sin h(x))/(cosh(x) - sinh(x))) + 8*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + ...
Timed out. \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.79 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {a^{3} b^{2} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {{\left (3 \, a b + b^{2}\right )} x}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (4 \, a e^{\left (-2 \, x\right )} - a - b\right )} e^{\left (4 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {4 \, a e^{\left (-2 \, x\right )} - {\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} \]
a^3*b^2*log(-(a - b)*e^(-2*x) - a - b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - 1/8*(3*a*b + b^2)*x/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/64*(4*a*e^(-2*x ) - a - b)*e^(4*x)/(a^2 + 2*a*b + b^2) - 1/64*(4*a*e^(-2*x) - (a - b)*e^(- 4*x))/(a^2 - 2*a*b + b^2)
Time = 0.25 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {a^{3} b^{2} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {{\left (3 \, a b - b^{2}\right )} x}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (18 \, a b e^{\left (4 \, x\right )} - 6 \, b^{2} e^{\left (4 \, x\right )} - 4 \, a^{2} e^{\left (2 \, x\right )} + 4 \, a b e^{\left (2 \, x\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 4 \, a e^{\left (2 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} \]
a^3*b^2*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^6 - 3*a^4*b^2 + 3*a^2*b ^4 - b^6) - 1/8*(3*a*b - b^2)*x/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/64*(18 *a*b*e^(4*x) - 6*b^2*e^(4*x) - 4*a^2*e^(2*x) + 4*a*b*e^(2*x) + a^2 - 2*a*b + b^2)*e^(-4*x)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/64*(a*e^(4*x) + b*e^( 4*x) - 4*a*e^(2*x))/(a^2 + 2*a*b + b^2)
Time = 2.70 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.65 \[ \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-4\,x}}{64\,a-64\,b}+\frac {{\mathrm {e}}^{4\,x}}{64\,a+64\,b}-\frac {x\,\left (3\,a\,b-b^2\right )}{8\,{\left (a-b\right )}^3}-\frac {a\,{\mathrm {e}}^{2\,x}}{16\,{\left (a+b\right )}^2}-\frac {a\,{\mathrm {e}}^{-2\,x}}{16\,{\left (a-b\right )}^2}+\frac {a^3\,b^2\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6} \]