\(\int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx\) [256]

Optimal result

Integrand size = 20, antiderivative size = 209 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {5^{-1-m} e^{5 a} x^m (-b x)^{-m} \Gamma (1+m,-5 b x)}{32 b}+\frac {3^{-1-m} e^{3 a} x^m (-b x)^{-m} \Gamma (1+m,-3 b x)}{32 b}-\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{16 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{16 b}-\frac {3^{-1-m} e^{-3 a} x^m (b x)^{-m} \Gamma (1+m,3 b x)}{32 b}-\frac {5^{-1-m} e^{-5 a} x^m (b x)^{-m} \Gamma (1+m,5 b x)}{32 b} \] Output:

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)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.84 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {e^{-5 a} x^m \left (-30 e^{6 a} (-b x)^{-m} \Gamma (1+m,-b x)+30 e^{4 a} (b x)^{-m} \Gamma (1+m,b x)+5\ 3^{-m} e^{2 a} \left (-b^2 x^2\right )^{-m} \left (e^{6 a} (b x)^m \Gamma (1+m,-3 b x)-(-b x)^m \Gamma (1+m,3 b x)\right )+3\ 5^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{10 a} (b x)^m \Gamma (1+m,-5 b x)-(-b x)^m \Gamma (1+m,5 b x)\right )\right )}{480 b} \] Input:

Integrate[x^m*Cosh[a + b*x]^3*Sinh[a + b*x]^2,x]
 

Output:

(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))
 

Rubi [A] (verified)

Time = 0.55 (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

\(\displaystyle \frac {e^{5 a} 5^{-m-1} x^m (-b x)^{-m} \Gamma (m+1,-5 b x)}{32 b}+\frac {e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \Gamma (m+1,-3 b x)}{32 b}-\frac {e^a x^m (-b x)^{-m} \Gamma (m+1,-b x)}{16 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+1,b x)}{16 b}-\frac {e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \Gamma (m+1,3 b x)}{32 b}-\frac {e^{-5 a} 5^{-m-1} x^m (b x)^{-m} \Gamma (m+1,5 b x)}{32 b}\)

Input:

Int[x^m*Cosh[a + b*x]^3*Sinh[a + b*x]^2,x]
 

Output:

(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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

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]
 

Maple [F]

\[\int x^{m} \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2}d x\]

Input:

int(x^m*cosh(b*x+a)^3*sinh(b*x+a)^2,x)
 

Output:

int(x^m*cosh(b*x+a)^3*sinh(b*x+a)^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.19 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=-\frac {3 \, \cosh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) \Gamma \left (m + 1, 5 \, b x\right ) + 5 \, \cosh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) \Gamma \left (m + 1, 3 \, b x\right ) - 30 \, \cosh \left (m \log \left (b\right ) + a\right ) \Gamma \left (m + 1, b x\right ) + 30 \, \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) - 5 \, \cosh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) \Gamma \left (m + 1, -3 \, b x\right ) - 3 \, \cosh \left (m \log \left (-5 \, b\right ) - 5 \, a\right ) \Gamma \left (m + 1, -5 \, b x\right ) - 3 \, \Gamma \left (m + 1, 5 \, b x\right ) \sinh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) - 5 \, \Gamma \left (m + 1, 3 \, b x\right ) \sinh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) - 30 \, \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) + 5 \, \Gamma \left (m + 1, -3 \, b x\right ) \sinh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) + 3 \, \Gamma \left (m + 1, -5 \, b x\right ) \sinh \left (m \log \left (-5 \, b\right ) - 5 \, a\right ) + 30 \, \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \left (b\right ) + a\right )}{480 \, b} \] Input:

integrate(x^m*cosh(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="fricas")
 

Output:

-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
 

Sympy [F]

\[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\int x^{m} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}\, dx \] Input:

integrate(x**m*cosh(b*x+a)**3*sinh(b*x+a)**2,x)
 

Output:

Integral(x**m*sinh(a + b*x)**2*cosh(a + b*x)**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.82 \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=-\frac {1}{32} \, \left (5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-5 \, a\right )} \Gamma \left (m + 1, 5 \, b x\right ) - \frac {1}{32} \, \left (3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (m + 1, 3 \, b x\right ) + \frac {1}{16} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) + \frac {1}{16} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) - \frac {1}{32} \, \left (-3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (m + 1, -3 \, b x\right ) - \frac {1}{32} \, \left (-5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (5 \, a\right )} \Gamma \left (m + 1, -5 \, b x\right ) \] Input:

integrate(x^m*cosh(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="maxima")
 

Output:

-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)
 

Giac [F]

\[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{2} \,d x } \] Input:

integrate(x^m*cosh(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="giac")
 

Output:

integrate(x^m*cosh(b*x + a)^3*sinh(b*x + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\int x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2 \,d x \] Input:

int(x^m*cosh(a + b*x)^3*sinh(a + b*x)^2,x)
 

Output:

int(x^m*cosh(a + b*x)^3*sinh(a + b*x)^2, x)
 

Reduce [F]

\[ \int x^m \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {3 x^{m} e^{10 b x +10 a}+5 x^{m} e^{8 b x +8 a}-30 x^{m} e^{6 b x +6 a}-3 e^{5 b x +10 a} \left (\int \frac {x^{m} e^{5 b x}}{x}d x \right ) m -5 e^{5 b x +8 a} \left (\int \frac {x^{m} e^{3 b x}}{x}d x \right ) m +30 e^{5 b x +6 a} \left (\int \frac {x^{m} e^{b x}}{x}d x \right ) m -30 e^{5 b x +4 a} \left (\int \frac {x^{m}}{e^{b x} x}d x \right ) m +5 e^{5 b x +2 a} \left (\int \frac {x^{m}}{e^{3 b x} x}d x \right ) m +3 e^{5 b x} \left (\int \frac {x^{m}}{e^{5 b x} x}d x \right ) m +30 x^{m} e^{4 b x +4 a}-5 x^{m} e^{2 b x +2 a}-3 x^{m}}{480 e^{5 b x +5 a} b} \] Input:

int(x^m*cosh(b*x+a)^3*sinh(b*x+a)^2,x)
 

Output:

(3*x**m*e**(10*a + 10*b*x) + 5*x**m*e**(8*a + 8*b*x) - 30*x**m*e**(6*a + 6 
*b*x) - 3*e**(10*a + 5*b*x)*int((x**m*e**(5*b*x))/x,x)*m - 5*e**(8*a + 5*b 
*x)*int((x**m*e**(3*b*x))/x,x)*m + 30*e**(6*a + 5*b*x)*int((x**m*e**(b*x)) 
/x,x)*m - 30*e**(4*a + 5*b*x)*int(x**m/(e**(b*x)*x),x)*m + 5*e**(2*a + 5*b 
*x)*int(x**m/(e**(3*b*x)*x),x)*m + 3*e**(5*b*x)*int(x**m/(e**(5*b*x)*x),x) 
*m + 30*x**m*e**(4*a + 4*b*x) - 5*x**m*e**(2*a + 2*b*x) - 3*x**m)/(480*e** 
(5*a + 5*b*x)*b)