Integrand size = 20, antiderivative size = 194 \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {a^3 b^2 x}{\left (a^2-b^2\right )^3}-\frac {a b^2 x}{2 \left (a^2-b^2\right )^2}-\frac {a x}{8 \left (a^2-b^2\right )}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a^2 b^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {a b^2 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}-\frac {a \cosh (x) \sinh (x)}{8 \left (a^2-b^2\right )}+\frac {a \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )}+\frac {a^2 b \sinh ^2(x)}{2 \left (a^2-b^2\right )^2} \] Output:
a^3*b^2*x/(a^2-b^2)^3-1/2*a*b^2*x/(a^2-b^2)^2-a*x/(8*a^2-8*b^2)-b*cosh(x)^ 4/(4*a^2-4*b^2)-a^2*b^3*ln(a*cosh(x)+b*sinh(x))/(a^2-b^2)^3-1/2*a*b^2*cosh (x)*sinh(x)/(a^2-b^2)^2-a*cosh(x)*sinh(x)/(8*a^2-8*b^2)+a*cosh(x)^3*sinh(x )/(4*a^2-4*b^2)+1/2*a^2*b*sinh(x)^2/(a^2-b^2)^2
Time = 0.40 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.65 \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {4 b \left (a^4-b^4\right ) \cosh (2 x)-b \left (a^2-b^2\right )^2 \cosh (4 x)+a \left (-4 \left (\left (a^4-6 a^2 b^2-3 b^4\right ) x+8 a b^3 \log (a \cosh (x)+b \sinh (x))\right )+8 b^2 \left (-a^2+b^2\right ) \sinh (2 x)+\left (a^2-b^2\right )^2 \sinh (4 x)\right )}{32 (a-b)^3 (a+b)^3} \] Input:
Integrate[(Cosh[x]^3*Sinh[x]^2)/(a*Cosh[x] + b*Sinh[x]),x]
Output:
(4*b*(a^4 - b^4)*Cosh[2*x] - b*(a^2 - b^2)^2*Cosh[4*x] + a*(-4*((a^4 - 6*a ^2*b^2 - 3*b^4)*x + 8*a*b^3*Log[a*Cosh[x] + b*Sinh[x]]) + 8*b^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 = 1.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.98, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {3042, 25, 3588, 25, 26, 3042, 25, 26, 3045, 15, 3048, 3042, 3115, 24, 3588, 26, 3042, 26, 3044, 15, 3115, 24, 3577, 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.
| \(\displaystyle \int \frac {\sinh ^2(x) \cosh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i x)^2 \cos (i x)^3}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos (i x)^3 \sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle \frac {i b \int i \cosh ^3(x) \sinh (x)dx}{a^2-b^2}-\frac {a \int -\cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a b \int \frac {i \cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i b \int i \cosh ^3(x) \sinh (x)dx}{a^2-b^2}+\frac {a \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a b \int \frac {i \cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \cosh ^3(x) \sinh (x)dx}{a^2-b^2}+\frac {a \int \cosh ^2(x) \sinh ^2(x)dx}{a^2-b^2}+\frac {a b \int \frac {\cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int -i \cos (i x)^3 \sin (i x)dx}{a^2-b^2}+\frac {a \int -\cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {a b \int -\frac {i \cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int -i \cos (i x)^3 \sin (i x)dx}{a^2-b^2}-\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}+\frac {a b \int -\frac {i \cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i b \int \cos (i x)^3 \sin (i x)dx}{a^2-b^2}-\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle -\frac {b \int \cosh ^3(x)d\cosh (x)}{a^2-b^2}-\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {a \int \cos (i x)^2 \sin (i x)^2dx}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle -\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \left (\frac {1}{4} \int \cosh ^2(x)dx-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \left (-\frac {1}{4} \sinh (x) \cosh ^3(x)+\frac {1}{4} \int \sin \left (i x+\frac {\pi }{2}\right )^2dx\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \left (\frac {1}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {i a b \int \frac {\cos (i x)^2 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -\frac {i a b \left (-\frac {i b \int \cosh ^2(x)dx}{a^2-b^2}+\frac {a \int i \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i a b \left (-\frac {i b \int \cosh ^2(x)dx}{a^2-b^2}+\frac {i a \int \cosh (x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {i a \int -i \cos (i x) \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {a \int \cos (i x) \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}-\frac {i a \int i \sinh (x)d(i \sinh (x))}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {i a b \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^2dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {i a b \left (\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {i b \left (\frac {\int 1dx}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}+\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {i a b \left (\frac {i a b \int \frac {\cos (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac {i b \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle -\frac {i a b \left (\frac {i a b \left (\frac {a x}{a^2-b^2}-\frac {i b \int -\frac {i (b \cosh (x)+a \sinh (x))}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac {i b \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i a b \left (\frac {i a b \left (\frac {a x}{a^2-b^2}-\frac {b \int \frac {b \cosh (x)+a \sinh (x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac {i b \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i a b \left (\frac {i a b \left (\frac {a x}{a^2-b^2}-\frac {b \int \frac {b \cos (i x)-i a \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )}{a^2-b^2}+\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac {i b \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}\right )}{a^2-b^2}-\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle -\frac {b \cosh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {a \left (\frac {1}{4} \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )-\frac {1}{4} \sinh (x) \cosh ^3(x)\right )}{a^2-b^2}-\frac {i a b \left (\frac {i a \sinh ^2(x)}{2 \left (a^2-b^2\right )}-\frac {i b \left (\frac {x}{2}+\frac {1}{2} \sinh (x) \cosh (x)\right )}{a^2-b^2}+\frac {i a b \left (\frac {a x}{a^2-b^2}-\frac {b \log (a \cosh (x)+b \sinh (x))}{a^2-b^2}\right )}{a^2-b^2}\right )}{a^2-b^2}\) |
Input:
Int[(Cosh[x]^3*Sinh[x]^2)/(a*Cosh[x] + b*Sinh[x]),x]
Output:
-1/4*(b*Cosh[x]^4)/(a^2 - b^2) - (a*(-1/4*(Cosh[x]^3*Sinh[x]) + (x/2 + (Co sh[x]*Sinh[x])/2)/4))/(a^2 - b^2) - (I*a*b*((I*a*b*((a*x)/(a^2 - b^2) - (b *Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)))/(a^2 - b^2) + ((I/2)*a*Sinh[x]^ 2)/(a^2 - b^2) - (I*b*(x/2 + (Cosh[x]*Sinh[x])/2))/(a^2 - b^2)))/(a^2 - b^ 2)
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_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_. ) + (d_.)*(x_)]), x_Symbol] :> Simp[a*(x/(a^2 + b^2)), x] + Simp[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]
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 = 10.66 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(-\frac {a^{2} x}{8 \left (a +b \right )^{3}}-\frac {3 b x a}{8 \left (a +b \right )^{3}}+\frac {{\mathrm e}^{4 x}}{64 a +64 b}+\frac {b \,{\mathrm e}^{2 x}}{16 \left (a +b \right )^{2}}+\frac {b \,{\mathrm e}^{-2 x}}{16 \left (a -b \right )^{2}}-\frac {{\mathrm e}^{-4 x}}{64 \left (a -b \right )}+\frac {2 a^{2} b^{3} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}-\frac {a^{2} b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}\) | \(161\) |
| default | \(\frac {2}{\left (8 a +8 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {8}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {-3 a -5 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-a -3 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (a +3 b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {a^{2} b^{3} \ln \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \tanh \left (\frac {x}{2}\right )^{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {2}{\left (8 a -8 b \right ) \left (1+\tanh \left (\frac {x}{2}\right )\right )^{4}}+\frac {8}{\left (1+\tanh \left (\frac {x}{2}\right )\right )^{3} \left (16 a -16 b \right )}-\frac {-a +3 b}{8 \left (a -b \right )^{2} \left (1+\tanh \left (\frac {x}{2}\right )\right )}-\frac {3 a -5 b}{8 \left (a -b \right )^{2} \left (1+\tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {a \left (a -3 b \right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{8 \left (a -b \right )^{3}}\) | \(250\) |
Input:
int(cosh(x)^3*sinh(x)^2/(a*cosh(x)+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
-1/8*a^2*x/(a+b)^3-3/8*b*x/(a+b)^3*a+1/64/(a+b)*exp(4*x)+1/16/(a+b)^2*b*ex p(2*x)+1/16*b/(a-b)^2*exp(-2*x)-1/64/(a-b)*exp(-4*x)+2*a^2*b^3/(a^6-3*a^4* b^2+3*a^2*b^4-b^6)*x-a^2*b^3/(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. 1162 vs. \(2 (180) = 360\).
Time = 0.10 (sec) , antiderivative size = 1162, normalized size of antiderivative = 5.99 \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Too large to display} \] Input:
integrate(cosh(x)^3*sinh(x)^2/(a*cosh(x)+b*sinh(x)),x, algorithm="fricas")
Output:
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^4*b - 2*a ^3*b^2 + 2*a*b^4 - b^5)*cosh(x)^6 + 4*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + 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*(a^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*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^4*b - 2*a^3*b^ 2 + 2*a*b^4 - b^5)*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^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5)*cosh(x)^2 - 4*(a^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*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^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5)*cosh(x)^3 - 4*(a^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*x*cosh(x))*s inh(x)^3 + 4*(a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*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^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5 + 15*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5)*cosh(x)^4 - 12*(a ^5 - 6*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4)*x*cosh(x)^2)*sinh(x)^2 - 64*(a^2*b^3 *cosh(x)^4 + 4*a^2*b^3*cosh(x)^3*sinh(x) + 6*a^2*b^3*cosh(x)^2*sinh(x)^2 + 4*a^2*b^3*cosh(x)*sinh(x)^3 + a^2*b^3*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. \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(x)**3*sinh(x)**2/(a*cosh(x)+b*sinh(x)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.77 \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=-\frac {a^{2} b^{3} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {{\left (a^{2} + 3 \, a b\right )} x}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (4 \, b e^{\left (-2 \, x\right )} + a + b\right )} e^{\left (4 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {4 \, b e^{\left (-2 \, x\right )} - {\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} \] Input:
integrate(cosh(x)^3*sinh(x)^2/(a*cosh(x)+b*sinh(x)),x, algorithm="maxima")
Output:
-a^2*b^3*log(-(a - b)*e^(-2*x) - a - b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 ) - 1/8*(a^2 + 3*a*b)*x/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/64*(4*b*e^(-2* x) + a + b)*e^(4*x)/(a^2 + 2*a*b + b^2) + 1/64*(4*b*e^(-2*x) - (a - b)*e^( -4*x))/(a^2 - 2*a*b + b^2)
Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.04 \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=-\frac {a^{2} b^{3} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {{\left (a^{2} - 3 \, a b\right )} x}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (6 \, a^{2} e^{\left (4 \, x\right )} - 18 \, a b e^{\left (4 \, x\right )} + 4 \, a b e^{\left (2 \, x\right )} - 4 \, b^{2} e^{\left (2 \, x\right )} - a^{2} + 2 \, a b - b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 4 \, b e^{\left (2 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} \] Input:
integrate(cosh(x)^3*sinh(x)^2/(a*cosh(x)+b*sinh(x)),x, algorithm="giac")
Output:
-a^2*b^3*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*(a^2 - 3*a*b)*x/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/64*(6 *a^2*e^(4*x) - 18*a*b*e^(4*x) + 4*a*b*e^(2*x) - 4*b^2*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*b*e^(2*x))/(a^2 + 2*a*b + b^2)
Time = 1.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.66 \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{4\,x}}{64\,a+64\,b}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a-64\,b}+\frac {x\,\left (3\,a\,b-a^2\right )}{8\,{\left (a-b\right )}^3}+\frac {b\,{\mathrm {e}}^{2\,x}}{16\,{\left (a+b\right )}^2}+\frac {b\,{\mathrm {e}}^{-2\,x}}{16\,{\left (a-b\right )}^2}-\frac {a^2\,b^3\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6} \] Input:
int((cosh(x)^3*sinh(x)^2)/(a*cosh(x) + b*sinh(x)),x)
Output:
exp(4*x)/(64*a + 64*b) - exp(-4*x)/(64*a - 64*b) + (x*(3*a*b - a^2))/(8*(a - b)^3) + (b*exp(2*x))/(16*(a + b)^2) + (b*exp(-2*x))/(16*(a - b)^2) - (a ^2*b^3*log(a - b + a*exp(2*x) + b*exp(2*x)))/(a^6 - b^6 + 3*a^2*b^4 - 3*a^ 4*b^2)
Time = 0.23 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.62 \[ \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {-2 e^{8 x} a^{3} b^{2}-8 e^{4 x} a^{5} x -4 e^{6 x} b^{5}-a^{4} b -a \,b^{4}-4 e^{2 x} b^{5}+2 a^{3} b^{2}+2 a^{2} b^{3}+8 e^{2 x} a^{3} b^{2}-8 e^{2 x} a \,b^{4}+4 e^{6 x} a^{4} b -8 e^{6 x} a^{3} b^{2}+8 e^{6 x} a \,b^{4}+4 e^{2 x} a^{4} b +2 e^{8 x} a^{2} b^{3}-e^{8 x} a^{4} b +e^{8 x} a \,b^{4}+48 e^{4 x} a^{3} b^{2} x +64 e^{4 x} a^{2} b^{3} x -64 e^{4 x} \mathrm {log}\left (e^{2 x} a +e^{2 x} b +a -b \right ) a^{2} b^{3}+24 e^{4 x} a \,b^{4} x -b^{5}-a^{5}+e^{8 x} a^{5}-e^{8 x} b^{5}}{64 e^{4 x} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )} \] Input:
int(cosh(x)^3*sinh(x)^2/(a*cosh(x)+b*sinh(x)),x)
Output:
(e**(8*x)*a**5 - e**(8*x)*a**4*b - 2*e**(8*x)*a**3*b**2 + 2*e**(8*x)*a**2* b**3 + e**(8*x)*a*b**4 - e**(8*x)*b**5 + 4*e**(6*x)*a**4*b - 8*e**(6*x)*a* *3*b**2 + 8*e**(6*x)*a*b**4 - 4*e**(6*x)*b**5 - 64*e**(4*x)*log(e**(2*x)*a + e**(2*x)*b + a - b)*a**2*b**3 - 8*e**(4*x)*a**5*x + 48*e**(4*x)*a**3*b* *2*x + 64*e**(4*x)*a**2*b**3*x + 24*e**(4*x)*a*b**4*x + 4*e**(2*x)*a**4*b + 8*e**(2*x)*a**3*b**2 - 8*e**(2*x)*a*b**4 - 4*e**(2*x)*b**5 - a**5 - a**4 *b + 2*a**3*b**2 + 2*a**2*b**3 - a*b**4 - b**5)/(64*e**(4*x)*(a**6 - 3*a** 4*b**2 + 3*a**2*b**4 - b**6))