Integrand size = 20, antiderivative size = 209 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {5^{-1-m} e^{5 a} x^m (-b x)^{-m} \Gamma (1+m,-5 b x)}{32 b}+\frac {3^{-1-m} e^{3 a} x^m (-b x)^{-m} \Gamma (1+m,-3 b x)}{32 b}-\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{16 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{16 b}-\frac {3^{-1-m} e^{-3 a} x^m (b x)^{-m} \Gamma (1+m,3 b x)}{32 b}-\frac {5^{-1-m} e^{-5 a} x^m (b x)^{-m} \Gamma (1+m,5 b x)}{32 b} \]
1/32*5^(-1-m)*exp(5*a)*x^m*GAMMA(1+m,-5*b*x)/b/((-b*x)^m)+1/32*3^(-1-m)*ex p(3*a)*x^m*GAMMA(1+m,-3*b*x)/b/((-b*x)^m)-1/16*exp(a)*x^m*GAMMA(1+m,-b*x)/ b/((-b*x)^m)+1/16*x^m*GAMMA(1+m,b*x)/b/exp(a)/((b*x)^m)-1/32*3^(-1-m)*x^m* GAMMA(1+m,3*b*x)/b/exp(3*a)/((b*x)^m)-1/32*5^(-1-m)*x^m*GAMMA(1+m,5*b*x)/b /exp(5*a)/((b*x)^m)
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.84 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {e^{-5 a} x^m \left (-30 e^{6 a} (-b x)^{-m} \Gamma (1+m,-b x)+30 e^{4 a} (b x)^{-m} \Gamma (1+m,b x)+5\ 3^{-m} e^{2 a} \left (-b^2 x^2\right )^{-m} \left (e^{6 a} (b x)^m \Gamma (1+m,-3 b x)-(-b x)^m \Gamma (1+m,3 b x)\right )+3\ 5^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{10 a} (b x)^m \Gamma (1+m,-5 b x)-(-b x)^m \Gamma (1+m,5 b x)\right )\right )}{480 b} \]
(x^m*((-30*E^(6*a)*Gamma[1 + m, -(b*x)])/(-(b*x))^m + (30*E^(4*a)*Gamma[1 + m, b*x])/(b*x)^m + (5*E^(2*a)*(E^(6*a)*(b*x)^m*Gamma[1 + m, -3*b*x] - (- (b*x))^m*Gamma[1 + m, 3*b*x]))/(3^m*(-(b^2*x^2))^m) + (3*(E^(10*a)*(b*x)^m *Gamma[1 + m, -5*b*x] - (-(b*x))^m*Gamma[1 + m, 5*b*x]))/(5^m*(-(b^2*x^2)) ^m)))/(480*b*E^(5*a))
Time = 0.53 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5971, 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.
\(\displaystyle \int x^m \sinh ^2(a+b x) \cosh ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \int \left (-\frac {1}{8} x^m \cosh (a+b x)+\frac {1}{16} x^m \cosh (3 a+3 b x)+\frac {1}{16} x^m \cosh (5 a+5 b x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{5 a} 5^{-m-1} x^m (-b x)^{-m} \Gamma (m+1,-5 b x)}{32 b}+\frac {e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \Gamma (m+1,-3 b x)}{32 b}-\frac {e^a x^m (-b x)^{-m} \Gamma (m+1,-b x)}{16 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+1,b x)}{16 b}-\frac {e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \Gamma (m+1,3 b x)}{32 b}-\frac {e^{-5 a} 5^{-m-1} x^m (b x)^{-m} \Gamma (m+1,5 b x)}{32 b}\) |
(5^(-1 - m)*E^(5*a)*x^m*Gamma[1 + m, -5*b*x])/(32*b*(-(b*x))^m) + (3^(-1 - m)*E^(3*a)*x^m*Gamma[1 + m, -3*b*x])/(32*b*(-(b*x))^m) - (E^a*x^m*Gamma[1 + m, -(b*x)])/(16*b*(-(b*x))^m) + (x^m*Gamma[1 + m, b*x])/(16*b*E^a*(b*x) ^m) - (3^(-1 - m)*x^m*Gamma[1 + m, 3*b*x])/(32*b*E^(3*a)*(b*x)^m) - (5^(-1 - m)*x^m*Gamma[1 + m, 5*b*x])/(32*b*E^(5*a)*(b*x)^m)
3.3.98.3.1 Defintions of rubi rules used
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
\[\int x^{m} \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2}d x\]
Time = 0.10 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.19 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=-\frac {3 \, \cosh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) \Gamma \left (m + 1, 5 \, b x\right ) + 5 \, \cosh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) \Gamma \left (m + 1, 3 \, b x\right ) - 30 \, \cosh \left (m \log \left (b\right ) + a\right ) \Gamma \left (m + 1, b x\right ) + 30 \, \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) - 5 \, \cosh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) \Gamma \left (m + 1, -3 \, b x\right ) - 3 \, \cosh \left (m \log \left (-5 \, b\right ) - 5 \, a\right ) \Gamma \left (m + 1, -5 \, b x\right ) - 3 \, \Gamma \left (m + 1, 5 \, b x\right ) \sinh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) - 5 \, \Gamma \left (m + 1, 3 \, b x\right ) \sinh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) - 30 \, \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) + 5 \, \Gamma \left (m + 1, -3 \, b x\right ) \sinh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) + 3 \, \Gamma \left (m + 1, -5 \, b x\right ) \sinh \left (m \log \left (-5 \, b\right ) - 5 \, a\right ) + 30 \, \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \left (b\right ) + a\right )}{480 \, b} \]
-1/480*(3*cosh(m*log(5*b) + 5*a)*gamma(m + 1, 5*b*x) + 5*cosh(m*log(3*b) + 3*a)*gamma(m + 1, 3*b*x) - 30*cosh(m*log(b) + a)*gamma(m + 1, b*x) + 30*c osh(m*log(-b) - a)*gamma(m + 1, -b*x) - 5*cosh(m*log(-3*b) - 3*a)*gamma(m + 1, -3*b*x) - 3*cosh(m*log(-5*b) - 5*a)*gamma(m + 1, -5*b*x) - 3*gamma(m + 1, 5*b*x)*sinh(m*log(5*b) + 5*a) - 5*gamma(m + 1, 3*b*x)*sinh(m*log(3*b) + 3*a) - 30*gamma(m + 1, -b*x)*sinh(m*log(-b) - a) + 5*gamma(m + 1, -3*b* x)*sinh(m*log(-3*b) - 3*a) + 3*gamma(m + 1, -5*b*x)*sinh(m*log(-5*b) - 5*a ) + 30*gamma(m + 1, b*x)*sinh(m*log(b) + a))/b
\[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\int x^{m} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}\, dx \]
Time = 0.15 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.82 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=-\frac {1}{32} \, \left (5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-5 \, a\right )} \Gamma \left (m + 1, 5 \, b x\right ) - \frac {1}{32} \, \left (3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (m + 1, 3 \, b x\right ) + \frac {1}{16} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) + \frac {1}{16} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) - \frac {1}{32} \, \left (-3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (m + 1, -3 \, b x\right ) - \frac {1}{32} \, \left (-5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (5 \, a\right )} \Gamma \left (m + 1, -5 \, b x\right ) \]
-1/32*(5*b*x)^(-m - 1)*x^(m + 1)*e^(-5*a)*gamma(m + 1, 5*b*x) - 1/32*(3*b* x)^(-m - 1)*x^(m + 1)*e^(-3*a)*gamma(m + 1, 3*b*x) + 1/16*(b*x)^(-m - 1)*x ^(m + 1)*e^(-a)*gamma(m + 1, b*x) + 1/16*(-b*x)^(-m - 1)*x^(m + 1)*e^a*gam ma(m + 1, -b*x) - 1/32*(-3*b*x)^(-m - 1)*x^(m + 1)*e^(3*a)*gamma(m + 1, -3 *b*x) - 1/32*(-5*b*x)^(-m - 1)*x^(m + 1)*e^(5*a)*gamma(m + 1, -5*b*x)
\[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\int x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2 \,d x \]