Integrand size = 24, antiderivative size = 195 \[ \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}-\frac {b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac {5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )} \]
1/8*d*exp(d*x+c)*cosh(b*x+a)/(b^2-d^2)-1/16*d*exp(d*x+c)*cosh(3*b*x+3*a)/( 9*b^2-d^2)-1/16*d*exp(d*x+c)*cosh(5*b*x+5*a)/(25*b^2-d^2)-1/8*b*exp(d*x+c) *sinh(b*x+a)/(b^2-d^2)+3/16*b*exp(d*x+c)*sinh(3*b*x+3*a)/(9*b^2-d^2)+5/16* b*exp(d*x+c)*sinh(5*b*x+5*a)/(25*b^2-d^2)
Time = 0.81 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.61 \[ \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {1}{16} e^{c+d x} \left (\frac {2 d \cosh (a+b x)-2 b \sinh (a+b x)}{(b-d) (b+d)}+\frac {-d \cosh (3 (a+b x))+3 b \sinh (3 (a+b x))}{9 b^2-d^2}+\frac {-d \cosh (5 (a+b x))+5 b \sinh (5 (a+b x))}{25 b^2-d^2}\right ) \]
(E^(c + d*x)*((2*d*Cosh[a + b*x] - 2*b*Sinh[a + b*x])/((b - d)*(b + d)) + (-(d*Cosh[3*(a + b*x)]) + 3*b*Sinh[3*(a + b*x)])/(9*b^2 - d^2) + (-(d*Cosh [5*(a + b*x)]) + 5*b*Sinh[5*(a + b*x)])/(25*b^2 - d^2)))/16
Time = 0.37 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6035, 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.
\(\displaystyle \int e^{c+d x} \sinh ^2(a+b x) \cosh ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 6035 |
\(\displaystyle \int \left (-\frac {1}{8} e^{c+d x} \cosh (a+b x)+\frac {1}{16} e^{c+d x} \cosh (3 a+3 b x)+\frac {1}{16} e^{c+d x} \cosh (5 a+5 b x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b e^{c+d x} \sinh (a+b x)}{8 \left (b^2-d^2\right )}+\frac {3 b e^{c+d x} \sinh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}+\frac {5 b e^{c+d x} \sinh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}+\frac {d e^{c+d x} \cosh (a+b x)}{8 \left (b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (3 a+3 b x)}{16 \left (9 b^2-d^2\right )}-\frac {d e^{c+d x} \cosh (5 a+5 b x)}{16 \left (25 b^2-d^2\right )}\) |
(d*E^(c + d*x)*Cosh[a + b*x])/(8*(b^2 - d^2)) - (d*E^(c + d*x)*Cosh[3*a + 3*b*x])/(16*(9*b^2 - d^2)) - (d*E^(c + d*x)*Cosh[5*a + 5*b*x])/(16*(25*b^2 - d^2)) - (b*E^(c + d*x)*Sinh[a + b*x])/(8*(b^2 - d^2)) + (3*b*E^(c + d*x )*Sinh[3*a + 3*b*x])/(16*(9*b^2 - d^2)) + (5*b*E^(c + d*x)*Sinh[5*a + 5*b* x])/(16*(25*b^2 - d^2))
3.10.60.3.1 Defintions of rubi rules used
Int[Cosh[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[( d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[F^(c*(a + b*x)) , Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g} , x] && IGtQ[m, 0] && IGtQ[n, 0]
Time = 0.02 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.43
\[-\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{16 \left (b -d \right )}-\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{16 \left (b +d \right )}+\frac {\sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{96 b -32 d}+\frac {\sinh \left (3 a +c +\left (3 b +d \right ) x \right )}{96 b +32 d}+\frac {\sinh \left (\left (5 b -d \right ) x +5 a -c \right )}{160 b -32 d}+\frac {\sinh \left (\left (5 b +d \right ) x +5 a +c \right )}{160 b +32 d}+\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{16 b -16 d}-\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{16 \left (b +d \right )}-\frac {\cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {\cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{96 b +32 d}-\frac {\cosh \left (\left (5 b -d \right ) x +5 a -c \right )}{32 \left (5 b -d \right )}+\frac {\cosh \left (\left (5 b +d \right ) x +5 a +c \right )}{160 b +32 d}\]
-1/16*sinh(a-c+(b-d)*x)/(b-d)-1/16*sinh(a+c+(b+d)*x)/(b+d)+1/32*sinh(3*a-c +(3*b-d)*x)/(3*b-d)+1/32*sinh(3*a+c+(3*b+d)*x)/(3*b+d)+1/32/(5*b-d)*sinh(( 5*b-d)*x+5*a-c)+1/32/(5*b+d)*sinh((5*b+d)*x+5*a+c)+1/16*cosh(a-c+(b-d)*x)/ (b-d)-1/16*cosh(a+c+(b+d)*x)/(b+d)-1/32*cosh(3*a-c+(3*b-d)*x)/(3*b-d)+1/32 *cosh(3*a+c+(3*b+d)*x)/(3*b+d)-1/32*cosh((5*b-d)*x+5*a-c)/(5*b-d)+1/32*cos h((5*b+d)*x+5*a+c)/(5*b+d)
Leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (177) = 354\).
Time = 0.26 (sec) , antiderivative size = 917, normalized size of antiderivative = 4.70 \[ \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\text {Too large to display} \]
-1/16*(5*(9*b^4*d - 10*b^2*d^3 + d^5)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^4 - 5*(9*b^5 - 10*b^3*d^2 + b*d^4)*cosh(d*x + c)*sinh(b*x + a)^5 - ( 75*b^5 - 78*b^3*d^2 + 3*b*d^4 + 50*(9*b^5 - 10*b^3*d^2 + b*d^4)*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a)^3 + (10*(9*b^4*d - 10*b^2*d^3 + d^5)*co sh(b*x + a)^3 + 3*(25*b^4*d - 26*b^2*d^3 + d^5)*cosh(b*x + a))*cosh(d*x + c)*sinh(b*x + a)^2 + (450*b^5 - 68*b^3*d^2 + 2*b*d^4 - 25*(9*b^5 - 10*b^3* d^2 + b*d^4)*cosh(b*x + a)^4 - 9*(25*b^5 - 26*b^3*d^2 + b*d^4)*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a) + ((9*b^4*d - 10*b^2*d^3 + d^5)*cosh(b*x + a)^5 + (25*b^4*d - 26*b^2*d^3 + d^5)*cosh(b*x + a)^3 - 2*(225*b^4*d - 3 4*b^2*d^3 + d^5)*cosh(b*x + a))*cosh(d*x + c) + ((9*b^4*d - 10*b^2*d^3 + d ^5)*cosh(b*x + a)^5 + 5*(9*b^4*d - 10*b^2*d^3 + d^5)*cosh(b*x + a)*sinh(b* x + a)^4 - 5*(9*b^5 - 10*b^3*d^2 + b*d^4)*sinh(b*x + a)^5 + (25*b^4*d - 26 *b^2*d^3 + d^5)*cosh(b*x + a)^3 - (75*b^5 - 78*b^3*d^2 + 3*b*d^4 + 50*(9*b ^5 - 10*b^3*d^2 + b*d^4)*cosh(b*x + a)^2)*sinh(b*x + a)^3 + (10*(9*b^4*d - 10*b^2*d^3 + d^5)*cosh(b*x + a)^3 + 3*(25*b^4*d - 26*b^2*d^3 + d^5)*cosh( b*x + a))*sinh(b*x + a)^2 - 2*(225*b^4*d - 34*b^2*d^3 + d^5)*cosh(b*x + a) + (450*b^5 - 68*b^3*d^2 + 2*b*d^4 - 25*(9*b^5 - 10*b^3*d^2 + b*d^4)*cosh( b*x + a)^4 - 9*(25*b^5 - 26*b^3*d^2 + b*d^4)*cosh(b*x + a)^2)*sinh(b*x + a ))*sinh(d*x + c))/((225*b^6 - 259*b^4*d^2 + 35*b^2*d^4 - d^6)*cosh(b*x + a )^6 - 3*(225*b^6 - 259*b^4*d^2 + 35*b^2*d^4 - d^6)*cosh(b*x + a)^4*sinh...
Leaf count of result is larger than twice the leaf count of optimal. 2761 vs. \(2 (168) = 336\).
Time = 25.24 (sec) , antiderivative size = 2761, normalized size of antiderivative = 14.16 \[ \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\text {Too large to display} \]
Piecewise((x*exp(c)*sinh(a)**2*cosh(a)**3, Eq(b, 0) & Eq(d, 0)), (-x*exp(c )*exp(d*x)*sinh(a - d*x)**5/16 - x*exp(c)*exp(d*x)*sinh(a - d*x)**4*cosh(a - d*x)/16 + x*exp(c)*exp(d*x)*sinh(a - d*x)**3*cosh(a - d*x)**2/8 + x*exp (c)*exp(d*x)*sinh(a - d*x)**2*cosh(a - d*x)**3/8 - x*exp(c)*exp(d*x)*sinh( a - d*x)*cosh(a - d*x)**4/16 - x*exp(c)*exp(d*x)*cosh(a - d*x)**5/16 + 13* exp(c)*exp(d*x)*sinh(a - d*x)**5/(96*d) + 7*exp(c)*exp(d*x)*sinh(a - d*x)* *4*cosh(a - d*x)/(96*d) - exp(c)*exp(d*x)*sinh(a - d*x)**3*cosh(a - d*x)** 2/(3*d) - exp(c)*exp(d*x)*sinh(a - d*x)**2*cosh(a - d*x)**3/(6*d) - exp(c) *exp(d*x)*sinh(a - d*x)*cosh(a - d*x)**4/(96*d) + 5*exp(c)*exp(d*x)*cosh(a - d*x)**5/(96*d), Eq(b, -d)), (-x*exp(c)*exp(d*x)*sinh(a - d*x/3)**5/32 - 3*x*exp(c)*exp(d*x)*sinh(a - d*x/3)**4*cosh(a - d*x/3)/32 - x*exp(c)*exp( d*x)*sinh(a - d*x/3)**3*cosh(a - d*x/3)**2/16 + x*exp(c)*exp(d*x)*sinh(a - d*x/3)**2*cosh(a - d*x/3)**3/16 + 3*x*exp(c)*exp(d*x)*sinh(a - d*x/3)*cos h(a - d*x/3)**4/32 + x*exp(c)*exp(d*x)*cosh(a - d*x/3)**5/32 - 23*exp(c)*e xp(d*x)*sinh(a - d*x/3)**5/(64*d) - 75*exp(c)*exp(d*x)*sinh(a - d*x/3)**4* cosh(a - d*x/3)/(64*d) - exp(c)*exp(d*x)*sinh(a - d*x/3)**3*cosh(a - d*x/3 )**2/d + exp(c)*exp(d*x)*sinh(a - d*x/3)**2*cosh(a - d*x/3)**3/(2*d) + 27* exp(c)*exp(d*x)*sinh(a - d*x/3)*cosh(a - d*x/3)**4/(64*d) + 7*exp(c)*exp(d *x)*cosh(a - d*x/3)**5/(64*d), Eq(b, -d/3)), (x*exp(c)*exp(d*x)*sinh(a - d *x/5)**5/32 + 5*x*exp(c)*exp(d*x)*sinh(a - d*x/5)**4*cosh(a - d*x/5)/32...
Exception generated. \[ \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(-d/b>0)', see `assume?` for more details)I
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.68 \[ \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {e^{\left (5 \, b x + d x + 5 \, a + c\right )}}{32 \, {\left (5 \, b + d\right )}} + \frac {e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{32 \, {\left (3 \, b + d\right )}} - \frac {e^{\left (b x + d x + a + c\right )}}{16 \, {\left (b + d\right )}} + \frac {e^{\left (-b x + d x - a + c\right )}}{16 \, {\left (b - d\right )}} - \frac {e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{32 \, {\left (3 \, b - d\right )}} - \frac {e^{\left (-5 \, b x + d x - 5 \, a + c\right )}}{32 \, {\left (5 \, b - d\right )}} \]
1/32*e^(5*b*x + d*x + 5*a + c)/(5*b + d) + 1/32*e^(3*b*x + d*x + 3*a + c)/ (3*b + d) - 1/16*e^(b*x + d*x + a + c)/(b + d) + 1/16*e^(-b*x + d*x - a + c)/(b - d) - 1/32*e^(-3*b*x + d*x - 3*a + c)/(3*b - d) - 1/32*e^(-5*b*x + d*x - 5*a + c)/(5*b - d)
Time = 3.20 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.02 \[ \int e^{c+d x} \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {{\mathrm {cosh}\left (a+b\,x\right )}^5\,{\mathrm {e}}^{c+d\,x}\,\left (26\,b^4\,d-2\,b^2\,d^3\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}+\frac {3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (25\,b^5-10\,b^3\,d^2+b\,d^4\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}+\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b\,d^4-13\,b^3\,d^2\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}-\frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (65\,b^4\,d-18\,b^2\,d^3+d^5\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}-\frac {6\,b^3\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^5\,\left (5\,b^2-d^2\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6}+\frac {6\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^4\,\left (5\,b^2-d^2\right )}{225\,b^6-259\,b^4\,d^2+35\,b^2\,d^4-d^6} \]
(cosh(a + b*x)^5*exp(c + d*x)*(26*b^4*d - 2*b^2*d^3))/(225*b^6 - d^6 + 35* b^2*d^4 - 259*b^4*d^2) + (3*cosh(a + b*x)^2*exp(c + d*x)*sinh(a + b*x)^3*( b*d^4 + 25*b^5 - 10*b^3*d^2))/(225*b^6 - d^6 + 35*b^2*d^4 - 259*b^4*d^2) + (2*cosh(a + b*x)^4*exp(c + d*x)*sinh(a + b*x)*(b*d^4 - 13*b^3*d^2))/(225* b^6 - d^6 + 35*b^2*d^4 - 259*b^4*d^2) - (cosh(a + b*x)^3*exp(c + d*x)*sinh (a + b*x)^2*(65*b^4*d + d^5 - 18*b^2*d^3))/(225*b^6 - d^6 + 35*b^2*d^4 - 2 59*b^4*d^2) - (6*b^3*exp(c + d*x)*sinh(a + b*x)^5*(5*b^2 - d^2))/(225*b^6 - d^6 + 35*b^2*d^4 - 259*b^4*d^2) + (6*b^2*d*cosh(a + b*x)*exp(c + d*x)*si nh(a + b*x)^4*(5*b^2 - d^2))/(225*b^6 - d^6 + 35*b^2*d^4 - 259*b^4*d^2)