Integrand size = 18, antiderivative size = 93 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=-\frac {e^{2 (a-2 d)-2 b x}}{32 b}+\frac {3 e^{2 a+2 b x}}{16 b}+\frac {e^{2 (a+d)+4 b x}}{16 b}+\frac {e^{2 (a+2 d)+6 b x}}{96 b}+\frac {1}{4} e^{2 a-2 d} x \] Output:
-1/32*exp(-2*b*x+2*a-4*d)/b+3/16*exp(2*b*x+2*a)/b+1/16*exp(4*b*x+2*a+2*d)/ b+1/96*exp(6*b*x+2*a+4*d)/b+1/4*exp(2*a-2*d)*x
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=\frac {e^{2 a-2 b x} \left (18 e^{4 b x}+6 e^{2 b x} \left (e^{4 b x}+4 b x\right ) \cosh (2 d)+\left (-3+e^{8 b x}\right ) \cosh (4 d)+6 e^{6 b x} \sinh (2 d)-24 b e^{2 b x} x \sinh (2 d)+3 \sinh (4 d)+e^{8 b x} \sinh (4 d)\right )}{96 b} \] Input:
Integrate[E^(2*(a + b*x))*Cosh[d + b*x]^4,x]
Output:
(E^(2*a - 2*b*x)*(18*E^(4*b*x) + 6*E^(2*b*x)*(E^(4*b*x) + 4*b*x)*Cosh[2*d] + (-3 + E^(8*b*x))*Cosh[4*d] + 6*E^(6*b*x)*Sinh[2*d] - 24*b*E^(2*b*x)*x*S inh[2*d] + 3*Sinh[4*d] + E^(8*b*x)*Sinh[4*d]))/(96*b)
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.52, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2720, 27, 243, 49, 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.
\(\displaystyle \int e^{2 (a+b x)} \cosh ^4(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {1}{16} e^{2 a-3 b x} \left (1+e^{2 b x}\right )^4de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^{2 a} \int e^{-3 b x} \left (1+e^{2 b x}\right )^4de^{b x}}{16 b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {e^{2 a} \int e^{-2 b x} \left (1+e^{2 b x}\right )^4de^{2 b x}}{32 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {e^{2 a} \int \left (6+e^{-2 b x}+4 e^{-b x}+5 e^{2 b x}\right )de^{2 b x}}{32 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{2 a} \left (-e^{-b x}+8 e^{2 b x}+\frac {1}{3} e^{3 b x}+4 \log \left (e^{2 b x}\right )\right )}{32 b}\) |
Input:
Int[E^(2*(a + b*x))*Cosh[d + b*x]^4,x]
Output:
(E^(2*a)*(-E^(-(b*x)) + 8*E^(2*b*x) + E^(3*b*x)/3 + 4*Log[E^(2*b*x)]))/(32 *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_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(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 = 1.72 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {{\mathrm e}^{-2 b x +2 a -4 d}}{32 b}+\frac {3 \,{\mathrm e}^{2 b x +2 a}}{16 b}+\frac {{\mathrm e}^{4 b x +2 a +2 d}}{16 b}+\frac {{\mathrm e}^{6 b x +2 a +4 d}}{96 b}+\frac {{\mathrm e}^{2 a -2 d} x}{4}\) | \(78\) |
parallelrisch | \(\frac {{\mathrm e}^{2 b x +2 a} \left (-12 b x \sinh \left (2 b x +2 d \right )+12 b x \cosh \left (2 b x +2 d \right )+2 \sinh \left (4 b x +4 d \right )+14 \sinh \left (2 b x +2 d \right )-\cosh \left (4 b x +4 d \right )-8 \cosh \left (2 b x +2 d \right )+9\right )}{48 b}\) | \(87\) |
default | \(\frac {x \cosh \left (2 a -2 d \right )}{4}+\frac {3 \sinh \left (2 b x +2 a \right )}{16 b}-\frac {\sinh \left (-2 b x +2 a -4 d \right )}{32 b}+\frac {\sinh \left (4 b x +2 a +2 d \right )}{16 b}+\frac {\sinh \left (6 b x +2 a +4 d \right )}{96 b}+\frac {x \sinh \left (2 a -2 d \right )}{4}+\frac {3 \cosh \left (2 b x +2 a \right )}{16 b}-\frac {\cosh \left (-2 b x +2 a -4 d \right )}{32 b}+\frac {\cosh \left (4 b x +2 a +2 d \right )}{16 b}+\frac {\cosh \left (6 b x +2 a +4 d \right )}{96 b}\) | \(154\) |
orering | \(\frac {\left (12 b x +5\right ) {\mathrm e}^{2 b x +2 a} \cosh \left (b x +d \right )^{4}}{12 b}-\frac {5 \left (2 b x -1\right ) \left (2 \,{\mathrm e}^{2 b x +2 a} b \cosh \left (b x +d \right )^{4}+4 \,{\mathrm e}^{2 b x +2 a} \cosh \left (b x +d \right )^{3} b \sinh \left (b x +d \right )\right )}{24 b^{2}}-\frac {5 \left (2 b x +1\right ) \left (8 \,{\mathrm e}^{2 b x +2 a} b^{2} \cosh \left (b x +d \right )^{4}+16 \,{\mathrm e}^{2 b x +2 a} b^{2} \cosh \left (b x +d \right )^{3} \sinh \left (b x +d \right )+12 \,{\mathrm e}^{2 b x +2 a} \sinh \left (b x +d \right )^{2} b^{2} \cosh \left (b x +d \right )^{2}\right )}{48 b^{3}}+\frac {\left (10 b x +1\right ) \left (32 \,{\mathrm e}^{2 b x +2 a} \cosh \left (b x +d \right )^{4} b^{3}+88 \sinh \left (b x +d \right ) \cosh \left (b x +d \right )^{3} {\mathrm e}^{2 b x +2 a} b^{3}+72 \,{\mathrm e}^{2 b x +2 a} b^{3} \sinh \left (b x +d \right )^{2} \cosh \left (b x +d \right )^{2}+24 \sinh \left (b x +d \right )^{3} \cosh \left (b x +d \right ) {\mathrm e}^{2 b x +2 a} b^{3}\right )}{96 b^{4}}-\frac {x \left (152 \cosh \left (b x +d \right )^{4} {\mathrm e}^{2 b x +2 a} b^{4}+448 \sinh \left (b x +d \right ) \cosh \left (b x +d \right )^{3} {\mathrm e}^{2 b x +2 a} b^{4}+480 \sinh \left (b x +d \right )^{2} \cosh \left (b x +d \right )^{2} {\mathrm e}^{2 b x +2 a} b^{4}+192 \sinh \left (b x +d \right )^{3} \cosh \left (b x +d \right ) {\mathrm e}^{2 b x +2 a} b^{4}+24 \sinh \left (b x +d \right )^{4} {\mathrm e}^{2 b x +2 a} b^{4}\right )}{96 b^{4}}\) | \(437\) |
Input:
int(exp(2*b*x+2*a)*cosh(b*x+d)^4,x,method=_RETURNVERBOSE)
Output:
-1/32*exp(-2*b*x+2*a-4*d)/b+3/16*exp(2*b*x+2*a)/b+1/16*exp(4*b*x+2*a+2*d)/ b+1/96*exp(6*b*x+2*a+4*d)/b+1/4*exp(2*a-2*d)*x
Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (77) = 154\).
Time = 0.10 (sec) , antiderivative size = 352, normalized size of antiderivative = 3.78 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=-\frac {\cosh \left (b x + d\right )^{4} \cosh \left (-2 \, a + 2 \, d\right ) + {\left (\cosh \left (-2 \, a + 2 \, d\right ) - \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{4} - 3 \, {\left (4 \, b x + 1\right )} \cosh \left (b x + d\right )^{2} \cosh \left (-2 \, a + 2 \, d\right ) - 8 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) - \cosh \left (b x + d\right ) \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{3} + 3 \, {\left (2 \, \cosh \left (b x + d\right )^{2} \cosh \left (-2 \, a + 2 \, d\right ) - {\left (4 \, b x + 1\right )} \cosh \left (-2 \, a + 2 \, d\right ) + {\left (4 \, b x - 2 \, \cosh \left (b x + d\right )^{2} + 1\right )} \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right )^{2} - 2 \, {\left (4 \, \cosh \left (b x + d\right )^{3} \cosh \left (-2 \, a + 2 \, d\right ) - 3 \, {\left (4 \, b x - 1\right )} \cosh \left (b x + d\right ) \cosh \left (-2 \, a + 2 \, d\right ) - {\left (4 \, \cosh \left (b x + d\right )^{3} - 3 \, {\left (4 \, b x - 1\right )} \cosh \left (b x + d\right )\right )} \sinh \left (-2 \, a + 2 \, d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{4} - 3 \, {\left (4 \, b x + 1\right )} \cosh \left (b x + d\right )^{2} - 9\right )} \sinh \left (-2 \, a + 2 \, d\right ) - 9 \, \cosh \left (-2 \, a + 2 \, d\right )}{48 \, {\left (b \cosh \left (b x + d\right )^{2} - 2 \, b \cosh \left (b x + d\right ) \sinh \left (b x + d\right ) + b \sinh \left (b x + d\right )^{2}\right )}} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)^4,x, algorithm="fricas")
Output:
-1/48*(cosh(b*x + d)^4*cosh(-2*a + 2*d) + (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*(2*cosh(b*x + d)^2*cosh(-2*a + 2*d) - (4*b*x + 1)*cosh(-2*a + 2*d) + (4*b*x - 2*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)*cosh (-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) - (cosh(b*x + d)^4 - 3*(4*b*x + 1)*cosh(b*x + d)^2 - 9)*sinh(-2*a + 2*d) - 9*cosh(-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. 265 vs. \(2 (76) = 152\).
Time = 2.35 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.85 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=\begin {cases} - \frac {x e^{2 a} e^{2 b x} \sinh ^{4}{\left (b x + d \right )}}{4} + \frac {x e^{2 a} e^{2 b x} \sinh ^{3}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{2} - \frac {x e^{2 a} e^{2 b x} \sinh {\left (b x + d \right )} \cosh ^{3}{\left (b x + d \right )}}{2} + \frac {x e^{2 a} e^{2 b x} \cosh ^{4}{\left (b x + d \right )}}{4} + \frac {13 e^{2 a} e^{2 b x} \sinh ^{4}{\left (b x + d \right )}}{48 b} - \frac {7 e^{2 a} e^{2 b x} \sinh ^{3}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{24 b} - \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{2 b} + \frac {5 e^{2 a} e^{2 b x} \sinh {\left (b x + d \right )} \cosh ^{3}{\left (b x + d \right )}}{8 b} + \frac {e^{2 a} e^{2 b x} \cosh ^{4}{\left (b x + d \right )}}{16 b} & \text {for}\: b \neq 0 \\x e^{2 a} \cosh ^{4}{\left (d \right )} & \text {otherwise} \end {cases} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)**4,x)
Output:
Piecewise((-x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)**4/4 + x*exp(2*a)*exp(2*b* x)*sinh(b*x + d)**3*cosh(b*x + d)/2 - x*exp(2*a)*exp(2*b*x)*sinh(b*x + d)* cosh(b*x + d)**3/2 + x*exp(2*a)*exp(2*b*x)*cosh(b*x + d)**4/4 + 13*exp(2*a )*exp(2*b*x)*sinh(b*x + d)**4/(48*b) - 7*exp(2*a)*exp(2*b*x)*sinh(b*x + d) **3*cosh(b*x + d)/(24*b) - exp(2*a)*exp(2*b*x)*sinh(b*x + d)**2*cosh(b*x + d)**2/(2*b) + 5*exp(2*a)*exp(2*b*x)*sinh(b*x + d)*cosh(b*x + d)**3/(8*b) + exp(2*a)*exp(2*b*x)*cosh(b*x + d)**4/(16*b), Ne(b, 0)), (x*exp(2*a)*cosh (d)**4, True))
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=\frac {{\left (6 \, e^{\left (-2 \, b x - 2 \, d\right )} + 18 \, e^{\left (-4 \, b x - 4 \, d\right )} + 1\right )} e^{\left (6 \, b x + 2 \, a + 4 \, d\right )}}{96 \, b} + \frac {{\left (b x + d\right )} e^{\left (2 \, a - 2 \, d\right )}}{4 \, b} - \frac {e^{\left (-2 \, b x + 2 \, a - 4 \, d\right )}}{32 \, b} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)^4,x, algorithm="maxima")
Output:
1/96*(6*e^(-2*b*x - 2*d) + 18*e^(-4*b*x - 4*d) + 1)*e^(6*b*x + 2*a + 4*d)/ b + 1/4*(b*x + d)*e^(2*a - 2*d)/b - 1/32*e^(-2*b*x + 2*a - 4*d)/b
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=-\frac {{\left (3 \, {\left (4 \, e^{\left (2 \, b x + 2 \, a + 2 \, d\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, b x - 2 \, d\right )} - 24 \, {\left (b x + d\right )} e^{\left (2 \, a\right )} - e^{\left (6 \, b x + 2 \, a + 6 \, d\right )} - 6 \, e^{\left (4 \, b x + 2 \, a + 4 \, d\right )} - 18 \, e^{\left (2 \, b x + 2 \, a + 2 \, d\right )}\right )} e^{\left (-2 \, d\right )}}{96 \, b} \] Input:
integrate(exp(2*b*x+2*a)*cosh(b*x+d)^4,x, algorithm="giac")
Output:
-1/96*(3*(4*e^(2*b*x + 2*a + 2*d) + e^(2*a))*e^(-2*b*x - 2*d) - 24*(b*x + d)*e^(2*a) - e^(6*b*x + 2*a + 6*d) - 6*e^(4*b*x + 2*a + 4*d) - 18*e^(2*b*x + 2*a + 2*d))*e^(-2*d)/b
Time = 3.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.60 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=\frac {3\,{\mathrm {e}}^{2\,a+2\,b\,x}}{16\,b}+\frac {x\,{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {cosh}\left (2\,d+2\,b\,x\right )}{4}-\frac {x\,{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {sinh}\left (2\,d+2\,b\,x\right )}{4}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {cosh}\left (2\,d+2\,b\,x\right )}{6\,b}-\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {cosh}\left (4\,d+4\,b\,x\right )}{48\,b}-\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {sinh}\left (2\,d+2\,b\,x\right )}{24\,b}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\mathrm {sinh}\left (4\,d+4\,b\,x\right )}{24\,b} \] Input:
int(cosh(d + b*x)^4*exp(2*a + 2*b*x),x)
Output:
(3*exp(2*a + 2*b*x))/(16*b) + (x*exp(2*a + 2*b*x)*cosh(2*d + 2*b*x))/4 - ( x*exp(2*a + 2*b*x)*sinh(2*d + 2*b*x))/4 + (exp(2*a + 2*b*x)*cosh(2*d + 2*b *x))/(6*b) - (exp(2*a + 2*b*x)*cosh(4*d + 4*b*x))/(48*b) - (exp(2*a + 2*b* x)*sinh(2*d + 2*b*x))/(24*b) + (exp(2*a + 2*b*x)*sinh(4*d + 4*b*x))/(24*b)
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int e^{2 (a+b x)} \cosh ^4(d+b x) \, dx=\frac {e^{2 a} \left (e^{8 b x +8 d}+6 e^{6 b x +6 d}+18 e^{4 b x +4 d}+24 e^{2 b x +2 d} b x -3\right )}{96 e^{2 b x +4 d} b} \] Input:
int(exp(2*b*x+2*a)*cosh(b*x+d)^4,x)
Output:
(e**(2*a)*(e**(8*b*x + 8*d) + 6*e**(6*b*x + 6*d) + 18*e**(4*b*x + 4*d) + 2 4*e**(2*b*x + 2*d)*b*x - 3))/(96*e**(2*b*x + 4*d)*b)