Integrand size = 24, antiderivative size = 76 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=\frac {e^{2 (a-2 d)-2 b x}}{32 b}-\frac {e^{2 (a+d)+4 b x}}{32 b}+\frac {e^{2 (a+2 d)+6 b x}}{96 b}+\frac {1}{8} e^{2 a-2 d} x \] Output:
1/32*exp(-2*b*x+2*a-4*d)/b-1/32*exp(4*b*x+2*a+2*d)/b+1/96*exp(6*b*x+2*a+4* d)/b+1/8*exp(2*a-2*d)*x
Time = 0.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=\frac {e^{2 a} \left (-3 \left (\left (e^{4 b x}-4 b x\right ) \cosh (2 d)+\left (e^{4 b x}+4 b x\right ) \sinh (2 d)\right )+e^{-2 b x} \left (\left (3+e^{8 b x}\right ) \cosh (4 d)+\left (-3+e^{8 b x}\right ) \sinh (4 d)\right )\right )}{96 b} \] Input:
Integrate[E^(2*(a + b*x))*Cosh[d + b*x]*Sinh[d + b*x]^3,x]
Output:
(E^(2*a)*(-3*((E^(4*b*x) - 4*b*x)*Cosh[2*d] + (E^(4*b*x) + 4*b*x)*Sinh[2*d ]) + ((3 + E^(8*b*x))*Cosh[4*d] + (-3 + E^(8*b*x))*Sinh[4*d])/E^(2*b*x)))/ (96*b)
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.61, 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, 84, 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.
\(\displaystyle \int e^{2 (a+b x)} \sinh ^3(b x+d) \cosh (b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {1}{16} e^{2 a-3 b x} \left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {e^{2 a} \int e^{-3 b x} \left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )de^{b x}}{16 b}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {e^{2 a} \int e^{-2 b x} \left (1-e^{2 b x}\right )^3 \left (1+e^{2 b x}\right )de^{2 b x}}{32 b}\) |
\(\Big \downarrow \) 84 |
\(\displaystyle -\frac {e^{2 a} \int \left (e^{-2 b x}-2 e^{-b x}+e^{2 b x}\right )de^{2 b x}}{32 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^{2 a} \left (-e^{-b x}+e^{2 b x}-\frac {1}{3} e^{3 b x}-2 \log \left (e^{2 b x}\right )\right )}{32 b}\) |
Input:
Int[E^(2*(a + b*x))*Cosh[d + b*x]*Sinh[d + b*x]^3,x]
Output:
-1/32*(E^(2*a)*(-E^(-(b*x)) + E^(2*b*x) - E^(3*b*x)/3 - 2*Log[E^(2*b*x)])) /b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
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 = 267.81 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {{\mathrm e}^{-2 b x +2 a -4 d}}{32 b}-\frac {{\mathrm e}^{4 b x +2 a +2 d}}{32 b}+\frac {{\mathrm e}^{6 b x +2 a +4 d}}{96 b}+\frac {{\mathrm e}^{2 a -2 d} x}{8}\) | \(64\) |
default | \(\frac {x \cosh \left (2 a -2 d \right )}{8}+\frac {\sinh \left (-2 b x +2 a -4 d \right )}{32 b}-\frac {\sinh \left (4 b x +2 a +2 d \right )}{32 b}+\frac {\sinh \left (6 b x +2 a +4 d \right )}{96 b}+\frac {x \sinh \left (2 a -2 d \right )}{8}+\frac {\cosh \left (-2 b x +2 a -4 d \right )}{32 b}-\frac {\cosh \left (4 b x +2 a +2 d \right )}{32 b}+\frac {\cosh \left (6 b x +2 a +4 d \right )}{96 b}\) | \(126\) |
orering | \(\frac {\left (12 b x -1\right ) {\mathrm e}^{2 b x +2 a} \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3}}{12 b}+\frac {\left (b x +2\right ) \left (2 \,{\mathrm e}^{2 b x +2 a} b \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3}+{\mathrm e}^{2 b x +2 a} b \sinh \left (b x +d \right )^{4}+3 \,{\mathrm e}^{2 b x +2 a} \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2} b \right )}{12 b^{2}}-\frac {\left (8 b x +1\right ) \left (14 \,{\mathrm e}^{2 b x +2 a} b^{2} \cosh \left (b x +d \right ) \sinh \left (b x +d \right )^{3}+4 \,{\mathrm e}^{2 b x +2 a} b^{2} \sinh \left (b x +d \right )^{4}+12 \,{\mathrm e}^{2 b x +2 a} b^{2} \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2}+6 \,{\mathrm e}^{2 b x +2 a} \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right ) b^{2}\right )}{48 b^{3}}+\frac {x \left (68 \,{\mathrm e}^{2 b x +2 a} \sinh \left (b x +d \right )^{3} \cosh \left (b x +d \right ) b^{3}+22 \,{\mathrm e}^{2 b x +2 a} \sinh \left (b x +d \right )^{4} b^{3}+84 \,{\mathrm e}^{2 b x +2 a} b^{3} \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right )^{2}+36 \,{\mathrm e}^{2 b x +2 a} \sinh \left (b x +d \right ) \cosh \left (b x +d \right )^{3} b^{3}+6 \,{\mathrm e}^{2 b x +2 a} \cosh \left (b x +d \right )^{4} b^{3}\right )}{48 b^{3}}\) | \(377\) |
Input:
int(exp(2*b*x+2*a)*cosh(b*x+d)*sinh(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
1/32*exp(-2*b*x+2*a-4*d)/b-1/32*exp(4*b*x+2*a+2*d)/b+1/96*exp(6*b*x+2*a+4* d)/b+1/8*exp(2*a-2*d)*x
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (63) = 126\).
Time = 0.10 (sec) , antiderivative size = 345, normalized size of antiderivative = 4.54 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=\frac {4 \, \cosh \left (b x + d\right )^{4} \cosh \left (-2 \, a + 2 \, d\right ) + 4 \, {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{4} + 3 \, {\left (4 \, b x - 1\right )} \cosh \left (b x + d\right )^{2} \cosh \left (-2 \, a + 2 \, d\right ) - 8 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) - \cosh \left (b x + d\right ) \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{3} + 3 \, {\left (8 \, \cosh \left (b x + d\right )^{2} \cosh \left (-2 \, a + 2 \, d\right ) + {\left (4 \, b x - 1\right )} \cosh \left (-2 \, a + 2 \, d\right ) - {\left (4 \, b x + 8 \, \cosh \left (b x + d\right )^{2} - 1\right )} \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{2} - 2 \, {\left (4 \, \cosh \left (b x + d\right )^{3} \cosh \left (-2 \, a + 2 \, d\right ) + 3 \, {\left (4 \, b x + 1\right )} \cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) - {\left (4 \, \cosh \left (b x + d\right )^{3} + 3 \, {\left (4 \, b x + 1\right )} \cosh \left (b x + d\right )\right )} \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right ) - {\left (4 \, \cosh \left (b x + d\right )^{4} + 3 \, {\left (4 \, b x - 1\right )} \cosh \left (b x + d\right )^{2}\right )} \sinh \left (-2 \, a + 2 \, d\right )}{96 \, {\left (b \cosh \left (b x + d\right )^{2} - 2 \, b \cosh \left (b x + d\right ) \sinh \left (b x + d\right ) + b \sinh \left (b x + d\right )^{2}\right )}} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)*sinh(b*x+d)^3,x, algorithm="fricas")
Output:
1/96*(4*cosh(b*x + d)^4*cosh(-2*a + 2*d) + 4*(cosh(-2*a + 2*d) - sinh(-2*a + 2*d))*sinh(b*x + d)^4 + 3*(4*b*x - 1)*cosh(b*x + d)^2*cosh(-2*a + 2*d) - 8*(cosh(b*x + d)*cosh(-2*a + 2*d) - cosh(b*x + d)*sinh(-2*a + 2*d))*sinh (b*x + d)^3 + 3*(8*cosh(b*x + d)^2*cosh(-2*a + 2*d) + (4*b*x - 1)*cosh(-2* a + 2*d) - (4*b*x + 8*cosh(b*x + d)^2 - 1)*sinh(-2*a + 2*d))*sinh(b*x + d) ^2 - 2*(4*cosh(b*x + d)^3*cosh(-2*a + 2*d) + 3*(4*b*x + 1)*cosh(b*x + d)*c osh(-2*a + 2*d) - (4*cosh(b*x + d)^3 + 3*(4*b*x + 1)*cosh(b*x + d))*sinh(- 2*a + 2*d))*sinh(b*x + d) - (4*cosh(b*x + d)^4 + 3*(4*b*x - 1)*cosh(b*x + d)^2)*sinh(-2*a + 2*d))/(b*cosh(b*x + d)^2 - 2*b*cosh(b*x + d)*sinh(b*x + d) + b*sinh(b*x + d)^2)
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (61) = 122\).
Time = 2.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.09 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=\begin {cases} - \frac {x e^{2 a} e^{2 b x} \sinh ^{4}{\left (b x + d \right )}}{8} + \frac {x e^{2 a} e^{2 b x} \sinh ^{3}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{4} - \frac {x e^{2 a} e^{2 b x} \sinh {\left (b x + d \right )} \cosh ^{3}{\left (b x + d \right )}}{4} + \frac {x e^{2 a} e^{2 b x} \cosh ^{4}{\left (b x + d \right )}}{8} + \frac {7 e^{2 a} e^{2 b x} \sinh ^{4}{\left (b x + d \right )}}{48 b} - \frac {e^{2 a} e^{2 b x} \sinh ^{3}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{6 b} + \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{4 b} - \frac {e^{2 a} e^{2 b x} \cosh ^{4}{\left (b x + d \right )}}{16 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{3}{\left (d \right )} \cosh {\left (d \right )} & \text {otherwise} \end {cases} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)*sinh(b*x+d)**3,x)
Output:
Piecewise((-x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/8 + x*exp(2*a)*exp(2*b* x)*sinh(b*x + d)**3*cosh(b*x + d)/4 - x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)* cosh(b*x + d)**3/4 + x*exp(2*a)*exp(2*b*x)*cosh(b*x + d)**4/8 + 7*exp(2*a) *exp(2*b*x)*sinh(b*x + d)**4/(48*b) - exp(2*a)*exp(2*b*x)*sinh(b*x + d)**3 *cosh(b*x + d)/(6*b) + exp(2*a)*exp(2*b*x)*sinh(b*x + d)**2*cosh(b*x + d)* *2/(4*b) - exp(2*a)*exp(2*b*x)*cosh(b*x + d)**4/(16*b), Ne(b, 0)), (x*exp( 2*a)*sinh(d)**3*cosh(d), True))
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=-\frac {{\left (3 \, e^{\left (-2 \, b x - 2 \, d\right )} - 1\right )} e^{\left (6 \, b x + 2 \, a + 4 \, d\right )}}{96 \, b} + \frac {{\left (b x + d\right )} e^{\left (2 \, a - 2 \, d\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x + 2 \, a - 4 \, d\right )}}{32 \, b} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)*sinh(b*x+d)^3,x, algorithm="maxima")
Output:
-1/96*(3*e^(-2*b*x - 2*d) - 1)*e^(6*b*x + 2*a + 4*d)/b + 1/8*(b*x + d)*e^( 2*a - 2*d)/b + 1/32*e^(-2*b*x + 2*a - 4*d)/b
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.07 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=-\frac {{\left (3 \, {\left (2 \, e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, b x - 2 \, d\right )} - 12 \, {\left (b x + d\right )} e^{\left (2 \, a\right )} - e^{\left (6 \, b x + 2 \, a + 6 \, d\right )} + 3 \, e^{\left (4 \, b x + 2 \, a + 4 \, d\right )}\right )} e^{\left (-2 \, d\right )}}{96 \, b} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)*sinh(b*x+d)^3,x, algorithm="giac")
Output:
-1/96*(3*(2*e^(2*b*x + 2*a + 2*d) - e^(2*a))*e^(-2*b*x - 2*d) - 12*(b*x + d)*e^(2*a) - e^(6*b*x + 2*a + 6*d) + 3*e^(4*b*x + 2*a + 4*d))*e^(-2*d)/b
Time = 3.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (2\,\mathrm {cosh}\left (2\,d+2\,b\,x\right )+2\,\mathrm {cosh}\left (4\,d+4\,b\,x\right )-5\,\mathrm {sinh}\left (2\,d+2\,b\,x\right )-\mathrm {sinh}\left (4\,d+4\,b\,x\right )-6\,b\,x\,\mathrm {sinh}\left (2\,d+2\,b\,x\right )+6\,b\,x\,\mathrm {cosh}\left (2\,d+2\,b\,x\right )\right )}{48\,b} \] Input:
int(cosh(d + b*x)*exp(2*a + 2*b*x)*sinh(d + b*x)^3,x)
Output:
(exp(2*a + 2*b*x)*(2*cosh(2*d + 2*b*x) + 2*cosh(4*d + 4*b*x) - 5*sinh(2*d + 2*b*x) - sinh(4*d + 4*b*x) - 6*b*x*sinh(2*d + 2*b*x) + 6*b*x*cosh(2*d + 2*b*x)))/(48*b)
Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int e^{2 (a+b x)} \cosh (d+b x) \sinh ^3(d+b x) \, dx=\frac {e^{2 a} \left (e^{8 b x +8 d}-3 e^{6 b x +6 d}+12 e^{2 b x +2 d} b x +3\right )}{96 e^{2 b x +4 d} b} \] Input:
int(exp(2*b*x+2*a)*cosh(b*x+d)*sinh(b*x+d)^3,x)
Output:
(e**(2*a)*(e**(8*b*x + 8*d) - 3*e**(6*b*x + 6*d) + 12*e**(2*b*x + 2*d)*b*x + 3))/(96*e**(2*b*x + 4*d)*b)