Integrand size = 16, antiderivative size = 65 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=-\frac {e^{a-3 d-2 b x}}{16 b}+\frac {3 e^{a+d+2 b x}}{16 b}+\frac {e^{a+3 d+4 b x}}{32 b}+\frac {3}{8} e^{a-d} x \] Output:
-1/16*exp(-2*b*x+a-3*d)/b+3/16*exp(2*b*x+a+d)/b+1/32*exp(4*b*x+a+3*d)/b+3/ 8*exp(a-d)*x
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.37 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=\frac {e^{a-2 b x} \left (6 e^{2 b x} \left (e^{2 b x}+2 b x\right ) \cosh (d)+\left (-2+e^{6 b x}\right ) \cosh (3 d)+6 e^{4 b x} \sinh (d)-12 b e^{2 b x} x \sinh (d)+2 \sinh (3 d)+e^{6 b x} \sinh (3 d)\right )}{32 b} \] Input:
Integrate[E^(a + b*x)*Cosh[d + b*x]^3,x]
Output:
(E^(a - 2*b*x)*(6*E^(2*b*x)*(E^(2*b*x) + 2*b*x)*Cosh[d] + (-2 + E^(6*b*x)) *Cosh[3*d] + 6*E^(4*b*x)*Sinh[d] - 12*b*E^(2*b*x)*x*Sinh[d] + 2*Sinh[3*d] + E^(6*b*x)*Sinh[3*d]))/(32*b)
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, 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.
\(\displaystyle \int e^{a+b x} \cosh ^3(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {1}{8} e^{a-3 b x} \left (1+e^{2 b x}\right )^3de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^a \int e^{-3 b x} \left (1+e^{2 b x}\right )^3de^{b x}}{8 b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {e^a \int e^{-2 b x} \left (1+e^{2 b x}\right )^3de^{2 b x}}{16 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {e^a \int \left (3+e^{-2 b x}+3 e^{-b x}+e^{2 b x}\right )de^{2 b x}}{16 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^a \left (-e^{-b x}+\frac {7}{2} e^{2 b x}+3 \log \left (e^{2 b x}\right )\right )}{16 b}\) |
Input:
Int[E^(a + b*x)*Cosh[d + b*x]^3,x]
Output:
(E^a*(-E^(-(b*x)) + (7*E^(2*b*x))/2 + 3*Log[E^(2*b*x)]))/(16*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 = 0.77 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {{\mathrm e}^{-2 b x +a -3 d}}{16 b}+\frac {3 \,{\mathrm e}^{2 b x +a +d}}{16 b}+\frac {{\mathrm e}^{4 b x +a +3 d}}{32 b}+\frac {3 \,{\mathrm e}^{a -d} x}{8}\) | \(54\) |
parallelrisch | \(\frac {{\mathrm e}^{b x +a} \left (12 b x \cosh \left (b x +d \right )-12 x b \sinh \left (b x +d \right )+\cosh \left (b x +d \right )+11 \sinh \left (b x +d \right )-\cosh \left (3 b x +3 d \right )+3 \sinh \left (3 b x +3 d \right )\right )}{32 b}\) | \(69\) |
default | \(\frac {3 \cosh \left (a -d \right ) x}{8}-\frac {\sinh \left (-2 b x +a -3 d \right )}{16 b}+\frac {3 \sinh \left (2 b x +a +d \right )}{16 b}+\frac {\sinh \left (4 b x +a +3 d \right )}{32 b}+\frac {3 \sinh \left (a -d \right ) x}{8}-\frac {\cosh \left (-2 b x +a -3 d \right )}{16 b}+\frac {3 \cosh \left (2 b x +a +d \right )}{16 b}+\frac {\cosh \left (4 b x +a +3 d \right )}{32 b}\) | \(106\) |
orering | \(\frac {\left (4 b x +1\right ) {\mathrm e}^{b x +a} \cosh \left (b x +d \right )^{3}}{4 b}-\frac {\left (b x -1\right ) \left ({\mathrm e}^{b x +a} \cosh \left (b x +d \right )^{3} b +3 \,{\mathrm e}^{b x +a} \cosh \left (b x +d \right )^{2} \sinh \left (b x +d \right ) b \right )}{4 b^{2}}-\frac {\left (4 b x +1\right ) \left (4 \,{\mathrm e}^{b x +a} \cosh \left (b x +d \right )^{3} b^{2}+6 \,{\mathrm e}^{b x +a} \sinh \left (b x +d \right ) \cosh \left (b x +d \right )^{2} b^{2}+6 \,{\mathrm e}^{b x +a} \sinh \left (b x +d \right )^{2} \cosh \left (b x +d \right ) b^{2}\right )}{16 b^{3}}+\frac {x \left (10 \,{\mathrm e}^{b x +a} \cosh \left (b x +d \right )^{3} b^{3}+30 \,{\mathrm e}^{b x +a} \sinh \left (b x +d \right ) \cosh \left (b x +d \right )^{2} b^{3}+18 \,{\mathrm e}^{b x +a} \sinh \left (b x +d \right )^{2} \cosh \left (b x +d \right ) b^{3}+6 \,{\mathrm e}^{b x +a} \sinh \left (b x +d \right )^{3} b^{3}\right )}{16 b^{3}}\) | \(253\) |
Input:
int(exp(b*x+a)*cosh(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/16*exp(-2*b*x+a-3*d)/b+3/16*exp(2*b*x+a+d)/b+1/32*exp(4*b*x+a+3*d)/b+3/ 8*exp(a-d)*x
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (53) = 106\).
Time = 0.08 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.29 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=-\frac {\cosh \left (b x + d\right )^{3} \cosh \left (-a + d\right ) - 3 \, {\left (\cosh \left (-a + d\right ) - \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{3} - 6 \, {\left (2 \, b x + 1\right )} \cosh \left (b x + d\right ) \cosh \left (-a + d\right ) + 3 \, {\left (\cosh \left (b x + d\right ) \cosh \left (-a + d\right ) - \cosh \left (b x + d\right ) \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right )^{2} - 3 \, {\left (3 \, \cosh \left (b x + d\right )^{2} \cosh \left (-a + d\right ) - 2 \, {\left (2 \, b x - 1\right )} \cosh \left (-a + d\right ) + {\left (4 \, b x - 3 \, \cosh \left (b x + d\right )^{2} - 2\right )} \sinh \left (-a + d\right )\right )} \sinh \left (b x + d\right ) - {\left (\cosh \left (b x + d\right )^{3} - 6 \, {\left (2 \, b x + 1\right )} \cosh \left (b x + d\right )\right )} \sinh \left (-a + d\right )}{32 \, {\left (b \cosh \left (b x + d\right ) - b \sinh \left (b x + d\right )\right )}} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^3,x, algorithm="fricas")
Output:
-1/32*(cosh(b*x + d)^3*cosh(-a + d) - 3*(cosh(-a + d) - sinh(-a + d))*sinh (b*x + d)^3 - 6*(2*b*x + 1)*cosh(b*x + d)*cosh(-a + d) + 3*(cosh(b*x + d)* cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d)^2 - 3*(3*cosh(b*x + d)^2*cosh(-a + d) - 2*(2*b*x - 1)*cosh(-a + d) + (4*b*x - 3*cosh(b*x + d)^2 - 2)*sinh(-a + d))*sinh(b*x + d) - (cosh(b*x + d)^3 - 6*(2*b*x + 1)*c osh(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. 207 vs. \(2 (54) = 108\).
Time = 0.94 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.18 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=\begin {cases} \frac {3 x e^{a} e^{b x} \sinh ^{3}{\left (b x + d \right )}}{8} - \frac {3 x e^{a} e^{b x} \sinh ^{2}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{8} - \frac {3 x e^{a} e^{b x} \sinh {\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{8} + \frac {3 x e^{a} e^{b x} \cosh ^{3}{\left (b x + d \right )}}{8} - \frac {5 e^{a} e^{b x} \sinh ^{3}{\left (b x + d \right )}}{8 b} + \frac {e^{a} e^{b x} \sinh ^{2}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{4 b} + \frac {e^{a} e^{b x} \sinh {\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{b} - \frac {3 e^{a} e^{b x} \cosh ^{3}{\left (b x + d \right )}}{8 b} & \text {for}\: b \neq 0 \\x e^{a} \cosh ^{3}{\left (d \right )} & \text {otherwise} \end {cases} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)**3,x)
Output:
Piecewise((3*x*exp(a)*exp(b*x)*sinh(b*x + d)**3/8 - 3*x*exp(a)*exp(b*x)*si nh(b*x + d)**2*cosh(b*x + d)/8 - 3*x*exp(a)*exp(b*x)*sinh(b*x + d)*cosh(b* x + d)**2/8 + 3*x*exp(a)*exp(b*x)*cosh(b*x + d)**3/8 - 5*exp(a)*exp(b*x)*s inh(b*x + d)**3/(8*b) + exp(a)*exp(b*x)*sinh(b*x + d)**2*cosh(b*x + d)/(4* b) + exp(a)*exp(b*x)*sinh(b*x + d)*cosh(b*x + d)**2/b - 3*exp(a)*exp(b*x)* cosh(b*x + d)**3/(8*b), Ne(b, 0)), (x*exp(a)*cosh(d)**3, True))
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=\frac {3 \, {\left (b x + a\right )} e^{\left (a - d\right )}}{8 \, b} + \frac {e^{\left (4 \, b x + a + 3 \, d\right )}}{32 \, b} + \frac {3 \, e^{\left (2 \, b x + a + d\right )}}{16 \, b} - \frac {e^{\left (-2 \, b x + a - 3 \, d\right )}}{16 \, b} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^3,x, algorithm="maxima")
Output:
3/8*(b*x + a)*e^(a - d)/b + 1/32*e^(4*b*x + a + 3*d)/b + 3/16*e^(2*b*x + a + d)/b - 1/16*e^(-2*b*x + a - 3*d)/b
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=\frac {{\left (12 \, b x e^{\left (a + 2 \, d\right )} - 2 \, {\left (3 \, e^{\left (2 \, b x + a + 2 \, d\right )} + e^{a}\right )} e^{\left (-2 \, b x\right )} + e^{\left (4 \, b x + a + 6 \, d\right )} + 6 \, e^{\left (2 \, b x + a + 4 \, d\right )}\right )} e^{\left (-3 \, d\right )}}{32 \, b} \] Input:
integrate(exp(b*x+a)*cosh(b*x+d)^3,x, algorithm="giac")
Output:
1/32*(12*b*x*e^(a + 2*d) - 2*(3*e^(2*b*x + a + 2*d) + e^a)*e^(-2*b*x) + e^ (4*b*x + a + 6*d) + 6*e^(2*b*x + a + 4*d))*e^(-3*d)/b
Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=\frac {{\mathrm {e}}^{a+b\,x}\,\left (3\,\mathrm {cosh}\left (d+b\,x\right )+3\,{\mathrm {cosh}\left (d+b\,x\right )}^2\,\mathrm {sinh}\left (d+b\,x\right )-{\mathrm {cosh}\left (d+b\,x\right )}^3+3\,b\,x\,\mathrm {cosh}\left (d+b\,x\right )-3\,b\,x\,\mathrm {sinh}\left (d+b\,x\right )\right )}{8\,b} \] Input:
int(cosh(d + b*x)^3*exp(a + b*x),x)
Output:
(exp(a + b*x)*(3*cosh(d + b*x) + 3*cosh(d + b*x)^2*sinh(d + b*x) - cosh(d + b*x)^3 + 3*b*x*cosh(d + b*x) - 3*b*x*sinh(d + b*x)))/(8*b)
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int e^{a+b x} \cosh ^3(d+b x) \, dx=\frac {e^{a} \left (e^{6 b x +6 d}+6 e^{4 b x +4 d}+12 e^{2 b x +2 d} b x -2\right )}{32 e^{2 b x +3 d} b} \] Input:
int(exp(b*x+a)*cosh(b*x+d)^3,x)
Output:
(e**a*(e**(6*b*x + 6*d) + 6*e**(4*b*x + 4*d) + 12*e**(2*b*x + 2*d)*b*x - 2 ))/(32*e**(2*b*x + 3*d)*b)