Integrand size = 24, antiderivative size = 101 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=-\frac {e^{a-5 d-4 b x}}{128 b}-\frac {e^{a-3 d-2 b x}}{64 b}-\frac {e^{a+d+2 b x}}{32 b}+\frac {e^{a+3 d+4 b x}}{128 b}+\frac {e^{a+5 d+6 b x}}{192 b}-\frac {1}{16} e^{a-d} x \] Output:
-1/128*exp(-4*b*x+a-5*d)/b-1/64*exp(-2*b*x+a-3*d)/b-1/32*exp(2*b*x+a+d)/b+ 1/128*exp(4*b*x+a+3*d)/b+1/192*exp(6*b*x+a+5*d)/b-1/16*exp(a-d)*x
Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.14 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=\frac {e^a \left (-12 \left (\left (e^{2 b x}+2 b x\right ) \cosh (d)+\left (e^{2 b x}-2 b x\right ) \sinh (d)\right )+3 e^{-2 b x} \left (\left (-2+e^{6 b x}\right ) \cosh (3 d)+\left (2+e^{6 b x}\right ) \sinh (3 d)\right )+e^{-4 b x} \left (\left (-3+2 e^{10 b x}\right ) \cosh (5 d)+\left (3+2 e^{10 b x}\right ) \sinh (5 d)\right )\right )}{384 b} \] Input:
Integrate[E^(a + b*x)*Cosh[d + b*x]^3*Sinh[d + b*x]^2,x]
Output:
(E^a*(-12*((E^(2*b*x) + 2*b*x)*Cosh[d] + (E^(2*b*x) - 2*b*x)*Sinh[d]) + (3 *((-2 + E^(6*b*x))*Cosh[3*d] + (2 + E^(6*b*x))*Sinh[3*d]))/E^(2*b*x) + ((- 3 + 2*E^(10*b*x))*Cosh[5*d] + (3 + 2*E^(10*b*x))*Sinh[5*d])/E^(4*b*x)))/(3 84*b)
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2720, 27, 354, 99, 2009}
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.
\(\displaystyle \int e^{a+b x} \sinh ^2(b x+d) \cosh ^3(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {1}{32} e^{a-5 b x} \left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )^3de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^a \int e^{-5 b x} \left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )^3de^{b x}}{32 b}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {e^a \int e^{-3 b x} \left (1-e^{2 b x}\right )^2 \left (1+e^{2 b x}\right )^3de^{2 b x}}{64 b}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {e^a \int \left (-2+e^{-3 b x}+e^{-2 b x}-2 e^{-b x}+2 e^{2 b x}\right )de^{2 b x}}{64 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^a \left (-\frac {1}{2} e^{-2 b x}-e^{-b x}-\frac {3}{2} e^{2 b x}+\frac {1}{3} e^{3 b x}-2 \log \left (e^{2 b x}\right )\right )}{64 b}\) |
Input:
Int[E^(a + b*x)*Cosh[d + b*x]^3*Sinh[d + b*x]^2,x]
Output:
(E^a*(-1/2*1/E^(2*b*x) - E^(-(b*x)) - (3*E^(2*b*x))/2 + E^(3*b*x)/3 - 2*Lo g[E^(2*b*x)]))/(64*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.64
\[-\frac {\cosh \left (a -d \right ) x}{16}-\frac {\sinh \left (-4 b x +a -5 d \right )}{128 b}-\frac {\sinh \left (-2 b x +a -3 d \right )}{64 b}-\frac {\sinh \left (2 b x +a +d \right )}{32 b}+\frac {\sinh \left (4 b x +a +3 d \right )}{128 b}+\frac {\sinh \left (6 b x +a +5 d \right )}{192 b}-\frac {\sinh \left (a -d \right ) x}{16}-\frac {\cosh \left (-4 b x +a -5 d \right )}{128 b}-\frac {\cosh \left (-2 b x +a -3 d \right )}{64 b}-\frac {\cosh \left (2 b x +a +d \right )}{32 b}+\frac {\cosh \left (4 b x +a +3 d \right )}{128 b}+\frac {\cosh \left (6 b x +a +5 d \right )}{192 b}\]
Input:
int(exp(b*x+a)*cosh(b*x+d)^3*sinh(b*x+d)^2,x)
Output:
-1/16*cosh(a-d)*x-1/128/b*sinh(-4*b*x+a-5*d)-1/64/b*sinh(-2*b*x+a-3*d)-1/3 2/b*sinh(2*b*x+a+d)+1/128/b*sinh(4*b*x+a+3*d)+1/192/b*sinh(6*b*x+a+5*d)-1/ 16*sinh(a-d)*x-1/128*cosh(-4*b*x+a-5*d)/b-1/64*cosh(-2*b*x+a-3*d)/b-1/32*c osh(2*b*x+a+d)/b+1/128*cosh(4*b*x+a+3*d)/b+1/192*cosh(6*b*x+a+5*d)/b
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (83) = 166\).
Time = 0.08 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.85 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=-\frac {\cosh \left (b x + d\right )^{5} \cosh \left (-a + d\right ) - 5 \, {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{5} + 5 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{4} + 3 \, \cosh \left (b x + d\right )^{3} \cosh \left (-a + d\right ) - {\left (50 \, \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) - {\left (50 \, \cosh \left (b x + d\right )^{2} + 9\right )} \sinh \left (-a + d\right ) + 9 \, \cosh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{3} + 12 \, {\left (2 \, b x + 1\right )} \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) + {\left (10 \, \cosh \left (b x + d\right )^{3} \cosh \left (-a + d\right ) + 9 \, \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - {\left (10 \, \cosh \left (b x + d\right )^{3} + 9 \, \cosh \left (b x + d\right )\right )} \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} - {\left (25 \, \cosh \left (b x + d\right )^{4} \cosh \left (-a + d\right ) + 27 \, \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) + 12 \, {\left (2 \, b x - 1\right )} \cosh \left (-a + d\right ) - {\left (25 \, \cosh \left (b x + d\right )^{4} + 24 \, b x + 27 \, \cosh \left (b x + d\right )^{2} - 12\right )} \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{5} + 3 \, \cosh \left (b x + d\right )^{3} + 12 \, {\left (2 \, b x + 1\right )} \cosh \left (b x + d\right )\right )} \sinh \left (-a + d\right )}{384 \, {\left (b \cosh \left (b x + d\right ) - b \sinh \left (b x + d\right )\right )}} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^3*sinh(b*x+d)^2,x, algorithm="fricas")
Output:
-1/384*(cosh(b*x + d)^5*cosh(-a + d) - 5*(cosh(-a + d) - sinh(-a + d))*sin h(b*x + d)^5 + 5*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d)) *sinh(b*x + d)^4 + 3*cosh(b*x + d)^3*cosh(-a + d) - (50*cosh(b*x + d)^2*co sh(-a + d) - (50*cosh(b*x + d)^2 + 9)*sinh(-a + d) + 9*cosh(-a + d))*sinh( b*x + d)^3 + 12*(2*b*x + 1)*cosh(b*x + d)*cosh(-a + d) + (10*cosh(b*x + d) ^3*cosh(-a + d) + 9*cosh(b*x + d)*cosh(-a + d) - (10*cosh(b*x + d)^3 + 9*c osh(b*x + d))*sinh(-a + d))*sinh(b*x + d)^2 - (25*cosh(b*x + d)^4*cosh(-a + d) + 27*cosh(b*x + d)^2*cosh(-a + d) + 12*(2*b*x - 1)*cosh(-a + d) - (25 *cosh(b*x + d)^4 + 24*b*x + 27*cosh(b*x + d)^2 - 12)*sinh(-a + d))*sinh(b* x + d) - (cosh(b*x + d)^5 + 3*cosh(b*x + d)^3 + 12*(2*b*x + 1)*cosh(b*x + d))*sinh(-a + d))/(b*cosh(b*x + d) - b*sinh(b*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (82) = 164\).
Time = 5.31 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.22 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=\begin {cases} \frac {x e^{a} e^{b x} \sinh ^{5}{\left (b x + d \right )}}{16} - \frac {x e^{a} e^{b x} \sinh ^{4}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{16} - \frac {x e^{a} e^{b x} \sinh ^{3}{\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{8} + \frac {x e^{a} e^{b x} \sinh ^{2}{\left (b x + d \right )} \cosh ^{3}{\left (b x + d \right )}}{8} + \frac {x e^{a} e^{b x} \sinh {\left (b x + d \right )} \cosh ^{4}{\left (b x + d \right )}}{16} - \frac {x e^{a} e^{b x} \cosh ^{5}{\left (b x + d \right )}}{16} - \frac {13 e^{a} e^{b x} \sinh ^{5}{\left (b x + d \right )}}{96 b} + \frac {7 e^{a} e^{b x} \sinh ^{4}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{96 b} + \frac {e^{a} e^{b x} \sinh ^{3}{\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{3 b} - \frac {e^{a} e^{b x} \sinh ^{2}{\left (b x + d \right )} \cosh ^{3}{\left (b x + d \right )}}{6 b} + \frac {e^{a} e^{b x} \sinh {\left (b x + d \right )} \cosh ^{4}{\left (b x + d \right )}}{96 b} + \frac {5 e^{a} e^{b x} \cosh ^{5}{\left (b x + d \right )}}{96 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{2}{\left (d \right )} \cosh ^{3}{\left (d \right )} & \text {otherwise} \end {cases} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)**3*sinh(b*x+d)**2,x)
Output:
Piecewise((x*exp(a)*exp(b*x)*sinh(b*x + d)**5/16 - x*exp(a)*exp(b*x)*sinh( b*x + d)**4*cosh(b*x + d)/16 - x*exp(a)*exp(b*x)*sinh(b*x + d)**3*cosh(b*x + d)**2/8 + x*exp(a)*exp(b*x)*sinh(b*x + d)**2*cosh(b*x + d)**3/8 + x*exp (a)*exp(b*x)*sinh(b*x + d)*cosh(b*x + d)**4/16 - x*exp(a)*exp(b*x)*cosh(b* x + d)**5/16 - 13*exp(a)*exp(b*x)*sinh(b*x + d)**5/(96*b) + 7*exp(a)*exp(b *x)*sinh(b*x + d)**4*cosh(b*x + d)/(96*b) + exp(a)*exp(b*x)*sinh(b*x + d)* *3*cosh(b*x + d)**2/(3*b) - exp(a)*exp(b*x)*sinh(b*x + d)**2*cosh(b*x + d) **3/(6*b) + exp(a)*exp(b*x)*sinh(b*x + d)*cosh(b*x + d)**4/(96*b) + 5*exp( a)*exp(b*x)*cosh(b*x + d)**5/(96*b), Ne(b, 0)), (x*exp(a)*sinh(d)**2*cosh( d)**3, True))
Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=-\frac {{\left (2 \, e^{\left (2 \, b x + 5 \, a + 2 \, d\right )} + e^{\left (5 \, a\right )}\right )} e^{\left (-4 \, b x - 4 \, a - 5 \, d\right )}}{128 \, b} + \frac {{\left (2 \, e^{\left (6 \, b x + 6 \, a + 5 \, d\right )} + 3 \, e^{\left (4 \, b x + 6 \, a + 3 \, d\right )} - 12 \, e^{\left (2 \, b x + 6 \, a + d\right )}\right )} e^{\left (-5 \, a\right )}}{384 \, b} - \frac {{\left (b x + a\right )} e^{\left (a - d\right )}}{16 \, b} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^3*sinh(b*x+d)^2,x, algorithm="maxima")
Output:
-1/128*(2*e^(2*b*x + 5*a + 2*d) + e^(5*a))*e^(-4*b*x - 4*a - 5*d)/b + 1/38 4*(2*e^(6*b*x + 6*a + 5*d) + 3*e^(4*b*x + 6*a + 3*d) - 12*e^(2*b*x + 6*a + d))*e^(-5*a)/b - 1/16*(b*x + a)*e^(a - d)/b
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=-\frac {{\left (24 \, b x e^{\left (a + 4 \, d\right )} - 3 \, {\left (6 \, e^{\left (4 \, b x + a + 4 \, d\right )} - 2 \, e^{\left (2 \, b x + a + 2 \, d\right )} - e^{a}\right )} e^{\left (-4 \, b x\right )} - 2 \, e^{\left (6 \, b x + a + 10 \, d\right )} - 3 \, e^{\left (4 \, b x + a + 8 \, d\right )} + 12 \, e^{\left (2 \, b x + a + 6 \, d\right )}\right )} e^{\left (-5 \, d\right )}}{384 \, b} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^3*sinh(b*x+d)^2,x, algorithm="giac")
Output:
-1/384*(24*b*x*e^(a + 4*d) - 3*(6*e^(4*b*x + a + 4*d) - 2*e^(2*b*x + a + 2 *d) - e^a)*e^(-4*b*x) - 2*e^(6*b*x + a + 10*d) - 3*e^(4*b*x + a + 8*d) + 1 2*e^(2*b*x + a + 6*d))*e^(-5*d)/b
Time = 0.65 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.35 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=\frac {{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {e}}^{a+b\,x}}{48\,b}-\frac {{\mathrm {cosh}\left (d+b\,x\right )}^5\,{\mathrm {e}}^{a+b\,x}}{24\,b}-\frac {x\,\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x}}{16}+\frac {x\,{\mathrm {e}}^{a+b\,x}\,\mathrm {sinh}\left (d+b\,x\right )}{16}-\frac {\mathrm {cosh}\left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x}}{16\,b}-\frac {{\mathrm {cosh}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{a+b\,x}\,\mathrm {sinh}\left (d+b\,x\right )}{16\,b}+\frac {5\,{\mathrm {cosh}\left (d+b\,x\right )}^4\,{\mathrm {e}}^{a+b\,x}\,\mathrm {sinh}\left (d+b\,x\right )}{24\,b} \] Input:
int(cosh(d + b*x)^3*exp(a + b*x)*sinh(d + b*x)^2,x)
Output:
(cosh(d + b*x)^3*exp(a + b*x))/(48*b) - (cosh(d + b*x)^5*exp(a + b*x))/(24 *b) - (x*cosh(d + b*x)*exp(a + b*x))/16 + (x*exp(a + b*x)*sinh(d + b*x))/1 6 - (cosh(d + b*x)*exp(a + b*x))/(16*b) - (cosh(d + b*x)^2*exp(a + b*x)*si nh(d + b*x))/(16*b) + (5*cosh(d + b*x)^4*exp(a + b*x)*sinh(d + b*x))/(24*b )
Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.83 \[ \int e^{a+b x} \cosh ^3(d+b x) \sinh ^2(d+b x) \, dx=\frac {e^{a} \left (2 e^{10 b x +10 d}+3 e^{8 b x +8 d}-12 e^{6 b x +6 d}-24 e^{4 b x +4 d} b x -6 e^{2 b x +2 d}-3\right )}{384 e^{4 b x +5 d} b} \] Input:
int(exp(b*x+a)*cosh(b*x+d)^3*sinh(b*x+d)^2,x)
Output:
(e**a*(2*e**(10*b*x + 10*d) + 3*e**(8*b*x + 8*d) - 12*e**(6*b*x + 6*d) - 2 4*e**(4*b*x + 4*d)*b*x - 6*e**(2*b*x + 2*d) - 3))/(384*e**(4*b*x + 5*d)*b)