Integrand size = 20, antiderivative size = 144 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=\frac {3 e^{\frac {5 (a-d)}{3}+\frac {1}{3} (-d-b x)}}{4 b}-\frac {3 e^{\frac {5 (a-d)}{3}-\frac {7}{3} (d+b x)}}{112 b}+\frac {9 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{40 b}-\frac {3 e^{\frac {5 (a-d)}{3}+\frac {11}{3} (d+b x)}}{44 b}+\frac {3 e^{\frac {5 (a-d)}{3}+\frac {17}{3} (d+b x)}}{272 b} \] Output:
3/4*exp(5/3*a-2*d-1/3*b*x)/b-3/112*exp(5/3*a-4*d-7/3*b*x)/b+9/40*exp(5/3*b *x+5/3*a)/b-3/44*exp(5/3*a+2*d+11/3*b*x)/b+3/272*exp(5/3*a+4*d+17/3*b*x)/b
Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.75 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=\frac {3 e^{\frac {5 a}{3}-\frac {7 b x}{3}} \left (7854 e^{4 b x}-2380 e^{2 b x} \left (-11+e^{4 b x}\right ) \cosh (2 d)+55 \left (-17+7 e^{8 b x}\right ) \cosh (4 d)-26180 e^{2 b x} \sinh (2 d)-2380 e^{6 b x} \sinh (2 d)+935 \sinh (4 d)+385 e^{8 b x} \sinh (4 d)\right )}{104720 b} \] Input:
Integrate[E^((5*(a + b*x))/3)*Sinh[d + b*x]^4,x]
Output:
(3*E^((5*a)/3 - (7*b*x)/3)*(7854*E^(4*b*x) - 2380*E^(2*b*x)*(-11 + E^(4*b* x))*Cosh[2*d] + 55*(-17 + 7*E^(8*b*x))*Cosh[4*d] - 26180*E^(2*b*x)*Sinh[2* d] - 2380*E^(6*b*x)*Sinh[2*d] + 935*Sinh[4*d] + 385*E^(8*b*x)*Sinh[4*d]))/ (104720*b)
Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.51, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2720, 27, 802, 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.
\(\displaystyle \int e^{\frac {5}{3} (a+b x)} \sinh ^4(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {3 \int \frac {1}{16} e^{\frac {5 a}{3}-\frac {8 b x}{3}} \left (1-e^{2 b x}\right )^4de^{\frac {b x}{3}}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 e^{5 a/3} \int e^{-\frac {8 b x}{3}} \left (1-e^{2 b x}\right )^4de^{\frac {b x}{3}}}{16 b}\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \frac {3 e^{5 a/3} \int \left (e^{-\frac {8 b x}{3}}-4 e^{-\frac {2 b x}{3}}+6 e^{\frac {4 b x}{3}}-4 e^{\frac {10 b x}{3}}+e^{\frac {16 b x}{3}}\right )de^{\frac {b x}{3}}}{16 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 e^{5 a/3} \left (-\frac {1}{7} e^{-\frac {7 b x}{3}}+4 e^{-\frac {b x}{3}}+\frac {6}{5} e^{\frac {5 b x}{3}}-\frac {4}{11} e^{\frac {11 b x}{3}}+\frac {1}{17} e^{\frac {17 b x}{3}}\right )}{16 b}\) |
Input:
Int[E^((5*(a + b*x))/3)*Sinh[d + b*x]^4,x]
Output:
(3*E^((5*a)/3)*(-1/7*1/E^((7*b*x)/3) + 4/E^((b*x)/3) + (6*E^((5*b*x)/3))/5 - (4*E^((11*b*x)/3))/11 + E^((17*b*x)/3)/17))/(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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
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 = 1.40 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.42
method | result | size |
parallelrisch | \(-\frac {3 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \left (-11900 \cosh \left (2 b x +2 d \right )+275 \cosh \left (4 b x +4 d \right )+14280 \sinh \left (2 b x +2 d \right )-660 \sinh \left (4 b x +4 d \right )-3927\right )}{52360 b}\) | \(61\) |
risch | \(\frac {3 \,{\mathrm e}^{\frac {5 a}{3}-2 d -\frac {b x}{3}}}{4 b}-\frac {3 \,{\mathrm e}^{\frac {5 a}{3}-4 d -\frac {7 b x}{3}}}{112 b}+\frac {9 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}}}{40 b}-\frac {3 \,{\mathrm e}^{\frac {5 a}{3}+2 d +\frac {11 b x}{3}}}{44 b}+\frac {3 \,{\mathrm e}^{\frac {5 a}{3}+4 d +\frac {17 b x}{3}}}{272 b}\) | \(84\) |
default | \(\frac {9 \sinh \left (\frac {5 b x}{3}+\frac {5 a}{3}\right )}{40 b}-\frac {3 \sinh \left (\frac {5 a}{3}-4 d -\frac {7 b x}{3}\right )}{112 b}+\frac {3 \sinh \left (\frac {5 a}{3}-2 d -\frac {b x}{3}\right )}{4 b}-\frac {3 \sinh \left (\frac {5 a}{3}+2 d +\frac {11 b x}{3}\right )}{44 b}+\frac {3 \sinh \left (\frac {5 a}{3}+4 d +\frac {17 b x}{3}\right )}{272 b}+\frac {9 \cosh \left (\frac {5 b x}{3}+\frac {5 a}{3}\right )}{40 b}-\frac {3 \cosh \left (\frac {5 a}{3}-4 d -\frac {7 b x}{3}\right )}{112 b}+\frac {3 \cosh \left (\frac {5 a}{3}-2 d -\frac {b x}{3}\right )}{4 b}-\frac {3 \cosh \left (\frac {5 a}{3}+2 d +\frac {11 b x}{3}\right )}{44 b}+\frac {3 \cosh \left (\frac {5 a}{3}+4 d +\frac {17 b x}{3}\right )}{272 b}\) | \(166\) |
orering | \(-\frac {15573 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{4}}{6545 b}+\frac {\frac {4350 b \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{4}}{1309}+\frac {10440 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{3} b \cosh \left (b x +d \right )}{1309}}{b^{2}}+\frac {\frac {366 b^{2} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{4}}{187}+\frac {720 b^{2} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{3} \cosh \left (b x +d \right )}{187}+\frac {648 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{2} b^{2} \cosh \left (b x +d \right )^{2}}{187}}{b^{3}}-\frac {405 \left (\frac {665 b^{3} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{4}}{27}+\frac {220 b^{3} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{3} \cosh \left (b x +d \right )}{3}+60 b^{3} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{2} \cosh \left (b x +d \right )^{2}+24 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right ) b^{3} \cosh \left (b x +d \right )^{3}\right )}{1309 b^{4}}+\frac {\frac {327 b^{4} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{4}}{77}+\frac {16560 b^{4} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{3} \cosh \left (b x +d \right )}{1309}+\frac {13608 b^{4} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right )^{2} \cosh \left (b x +d \right )^{2}}{935}+\frac {7776 b^{4} {\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} \sinh \left (b x +d \right ) \cosh \left (b x +d \right )^{3}}{1309}+\frac {5832 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}} b^{4} \cosh \left (b x +d \right )^{4}}{6545}}{b^{5}}\) | \(412\) |
Input:
int(exp(5/3*b*x+5/3*a)*sinh(b*x+d)^4,x,method=_RETURNVERBOSE)
Output:
-3/52360*exp(5/3*b*x+5/3*a)*(-11900*cosh(2*b*x+2*d)+275*cosh(4*b*x+4*d)+14 280*sinh(2*b*x+2*d)-660*sinh(4*b*x+4*d)-3927)/b
Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (83) = 166\).
Time = 0.09 (sec) , antiderivative size = 1051, normalized size of antiderivative = 7.30 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(5/3*b*x+5/3*a)*sinh(b*x+d)^4,x, algorithm="fricas")
Output:
-3/52360*(275*cosh(1/3*b*x + 1/3*d)^12*cosh(-5/3*a + 5/3*d) + 275*(cosh(-5 /3*a + 5/3*d) - sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^12 - 7920*(cos h(1/3*b*x + 1/3*d)*cosh(-5/3*a + 5/3*d) - cosh(1/3*b*x + 1/3*d)*sinh(-5/3* a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^11 + 18150*(cosh(1/3*b*x + 1/3*d)^2*cosh (-5/3*a + 5/3*d) - cosh(1/3*b*x + 1/3*d)^2*sinh(-5/3*a + 5/3*d))*sinh(1/3* b*x + 1/3*d)^10 - 145200*(cosh(1/3*b*x + 1/3*d)^3*cosh(-5/3*a + 5/3*d) - c osh(1/3*b*x + 1/3*d)^3*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^9 + 136 125*(cosh(1/3*b*x + 1/3*d)^4*cosh(-5/3*a + 5/3*d) - cosh(1/3*b*x + 1/3*d)^ 4*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^8 - 522720*(cosh(1/3*b*x + 1 /3*d)^5*cosh(-5/3*a + 5/3*d) - cosh(1/3*b*x + 1/3*d)^5*sinh(-5/3*a + 5/3*d ))*sinh(1/3*b*x + 1/3*d)^7 - 11900*cosh(1/3*b*x + 1/3*d)^6*cosh(-5/3*a + 5 /3*d) + 700*(363*cosh(1/3*b*x + 1/3*d)^6*cosh(-5/3*a + 5/3*d) - (363*cosh( 1/3*b*x + 1/3*d)^6 - 17)*sinh(-5/3*a + 5/3*d) - 17*cosh(-5/3*a + 5/3*d))*s inh(1/3*b*x + 1/3*d)^6 - 720*(726*cosh(1/3*b*x + 1/3*d)^7*cosh(-5/3*a + 5/ 3*d) - 119*cosh(1/3*b*x + 1/3*d)*cosh(-5/3*a + 5/3*d) - (726*cosh(1/3*b*x + 1/3*d)^7 - 119*cosh(1/3*b*x + 1/3*d))*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3*d)^5 + 375*(363*cosh(1/3*b*x + 1/3*d)^8*cosh(-5/3*a + 5/3*d) - 476* cosh(1/3*b*x + 1/3*d)^2*cosh(-5/3*a + 5/3*d) - (363*cosh(1/3*b*x + 1/3*d)^ 8 - 476*cosh(1/3*b*x + 1/3*d)^2)*sinh(-5/3*a + 5/3*d))*sinh(1/3*b*x + 1/3* d)^4 - 1200*(121*cosh(1/3*b*x + 1/3*d)^9*cosh(-5/3*a + 5/3*d) - 238*cos...
Time = 2.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.24 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=\begin {cases} - \frac {3093 e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{4}{\left (b x + d \right )}}{6545 b} + \frac {2340 e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{3}{\left (b x + d \right )} \cosh {\left (b x + d \right )}}{1309 b} - \frac {324 e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh ^{2}{\left (b x + d \right )} \cosh ^{2}{\left (b x + d \right )}}{595 b} - \frac {1944 e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \sinh {\left (b x + d \right )} \cosh ^{3}{\left (b x + d \right )}}{1309 b} + \frac {5832 e^{\frac {5 a}{3}} e^{\frac {5 b x}{3}} \cosh ^{4}{\left (b x + d \right )}}{6545 b} & \text {for}\: b \neq 0 \\x e^{\frac {5 a}{3}} \sinh ^{4}{\left (d \right )} & \text {otherwise} \end {cases} \] Input:
integrate(exp(5/3*b*x+5/3*a)*sinh(b*x+d)**4,x)
Output:
Piecewise((-3093*exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**4/(6545*b) + 2340* exp(5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**3*cosh(b*x + d)/(1309*b) - 324*exp( 5*a/3)*exp(5*b*x/3)*sinh(b*x + d)**2*cosh(b*x + d)**2/(595*b) - 1944*exp(5 *a/3)*exp(5*b*x/3)*sinh(b*x + d)*cosh(b*x + d)**3/(1309*b) + 5832*exp(5*a/ 3)*exp(5*b*x/3)*cosh(b*x + d)**4/(6545*b), Ne(b, 0)), (x*exp(5*a/3)*sinh(d )**4, True))
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.54 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=-\frac {3 \, {\left (340 \, e^{\left (-2 \, b x - 2 \, d\right )} - 1122 \, e^{\left (-4 \, b x - 4 \, d\right )} - 55\right )} e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )}}{14960 \, b} + \frac {3 \, {\left (28 \, e^{\left (-\frac {1}{3} \, b x - \frac {1}{3} \, d\right )} - e^{\left (-\frac {7}{3} \, b x - \frac {7}{3} \, d\right )}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} \, d\right )}}{112 \, b} \] Input:
integrate(exp(5/3*b*x+5/3*a)*sinh(b*x+d)^4,x, algorithm="maxima")
Output:
-3/14960*(340*e^(-2*b*x - 2*d) - 1122*e^(-4*b*x - 4*d) - 55)*e^(17/3*b*x + 5/3*a + 4*d)/b + 3/112*(28*e^(-1/3*b*x - 1/3*d) - e^(-7/3*b*x - 7/3*d))*e ^(5/3*a - 5/3*d)/b
Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.56 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=\frac {3 \, {\left (935 \, {\left (28 \, e^{\left (2 \, b x + \frac {5}{3} \, a + 2 \, d\right )} - e^{\left (\frac {5}{3} \, a\right )}\right )} e^{\left (-\frac {7}{3} \, b x\right )} + 385 \, e^{\left (\frac {17}{3} \, b x + \frac {5}{3} \, a + 8 \, d\right )} - 2380 \, e^{\left (\frac {11}{3} \, b x + \frac {5}{3} \, a + 6 \, d\right )} + 7854 \, e^{\left (\frac {5}{3} \, b x + \frac {5}{3} \, a + 4 \, d\right )}\right )} e^{\left (-4 \, d\right )}}{104720 \, b} \] Input:
integrate(exp(5/3*b*x+5/3*a)*sinh(b*x+d)^4,x, algorithm="giac")
Output:
3/104720*(935*(28*e^(2*b*x + 5/3*a + 2*d) - e^(5/3*a))*e^(-7/3*b*x) + 385* e^(17/3*b*x + 5/3*a + 8*d) - 2380*e^(11/3*b*x + 5/3*a + 6*d) + 7854*e^(5/3 *b*x + 5/3*a + 4*d))*e^(-4*d)/b
Time = 3.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}-2\,d-\frac {b\,x}{3}}}{4\,b}-\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}-4\,d-\frac {7\,b\,x}{3}}}{112\,b}-\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}+2\,d+\frac {11\,b\,x}{3}}}{44\,b}+\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}+4\,d+\frac {17\,b\,x}{3}}}{272\,b}+\frac {9\,{\mathrm {e}}^{\frac {5\,a}{3}+\frac {5\,b\,x}{3}}}{40\,b} \] Input:
int(exp((5*a)/3 + (5*b*x)/3)*sinh(d + b*x)^4,x)
Output:
(3*exp((5*a)/3 - 2*d - (b*x)/3))/(4*b) - (3*exp((5*a)/3 - 4*d - (7*b*x)/3) )/(112*b) - (3*exp((5*a)/3 + 2*d + (11*b*x)/3))/(44*b) + (3*exp((5*a)/3 + 4*d + (17*b*x)/3))/(272*b) + (9*exp((5*a)/3 + (5*b*x)/3))/(40*b)
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.53 \[ \int e^{\frac {5}{3} (a+b x)} \sinh ^4(d+b x) \, dx=\frac {3 e^{\frac {5 b x}{3}+\frac {5 a}{3}} \left (385 e^{8 b x +8 d}-2380 e^{6 b x +6 d}+7854 e^{4 b x +4 d}+26180 e^{2 b x +2 d}-935\right )}{104720 e^{4 b x +4 d} b} \] Input:
int(exp(5/3*b*x+5/3*a)*sinh(b*x+d)^4,x)
Output:
(3*e**((5*a + 5*b*x)/3)*(385*e**(8*b*x + 8*d) - 2380*e**(6*b*x + 6*d) + 78 54*e**(4*b*x + 4*d) + 26180*e**(2*b*x + 2*d) - 935))/(104720*e**(4*b*x + 4 *d)*b)