\(\int x^m \cosh ^3(a+b x^n) \, dx\) [49]

Optimal result

Integrand size = 14, antiderivative size = 200 \[ \int x^m \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3^{-\frac {1+m}{n}} e^{3 a} x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{8 n}-\frac {3^{-\frac {1+m}{n}} e^{-3 a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 b x^n\right )}{8 n} \] Output:

-1/8*exp(3*a)*x^(1+m)*GAMMA((1+m)/n,-3*b*x^n)/(3^((1+m)/n))/n/((-b*x^n)^(( 
1+m)/n))-3/8*exp(a)*x^(1+m)*GAMMA((1+m)/n,-b*x^n)/n/((-b*x^n)^((1+m)/n))-3 
/8*x^(1+m)*GAMMA((1+m)/n,b*x^n)/exp(a)/n/((b*x^n)^((1+m)/n))-1/8*x^(1+m)*G 
AMMA((1+m)/n,3*b*x^n)/(3^((1+m)/n))/exp(3*a)/n/((b*x^n)^((1+m)/n))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91 \[ \int x^m \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3^{-\frac {1+m}{n}} e^{-3 a} x^{1+m} \left (-b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \left (e^{6 a} \left (b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )+3^{\frac {1+m+n}{n}} e^{4 a} \left (b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )+\left (-b x^n\right )^{\frac {1+m}{n}} \left (3^{\frac {1+m+n}{n}} e^{2 a} \Gamma \left (\frac {1+m}{n},b x^n\right )+\Gamma \left (\frac {1+m}{n},3 b x^n\right )\right )\right )}{8 n} \] Input:

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

Output:

-1/8*(x^(1 + m)*(E^(6*a)*(b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, -3*b*x^n] + 
3^((1 + m + n)/n)*E^(4*a)*(b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, -(b*x^n)] + 
 (-(b*x^n))^((1 + m)/n)*(3^((1 + m + n)/n)*E^(2*a)*Gamma[(1 + m)/n, b*x^n] 
 + Gamma[(1 + m)/n, 3*b*x^n])))/(3^((1 + m)/n)*E^(3*a)*n*(-(b^2*x^(2*n)))^ 
((1 + m)/n))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5886, 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.

\(\displaystyle \int x^m \cosh ^3\left (a+b x^n\right ) \, dx\)

\(\Big \downarrow \) 5886

\(\displaystyle \int \left (\frac {3}{4} x^m \cosh \left (a+b x^n\right )+\frac {1}{4} x^m \cosh \left (3 a+3 b x^n\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e^{3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},b x^n\right )}{8 n}-\frac {e^{-3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},3 b x^n\right )}{8 n}\)

Input:

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

Output:

-1/8*(E^(3*a)*x^(1 + m)*Gamma[(1 + m)/n, -3*b*x^n])/(3^((1 + m)/n)*n*(-(b* 
x^n))^((1 + m)/n)) - (3*E^a*x^(1 + m)*Gamma[(1 + m)/n, -(b*x^n)])/(8*n*(-( 
b*x^n))^((1 + m)/n)) - (3*x^(1 + m)*Gamma[(1 + m)/n, b*x^n])/(8*E^a*n*(b*x 
^n)^((1 + m)/n)) - (x^(1 + m)*Gamma[(1 + m)/n, 3*b*x^n])/(8*3^((1 + m)/n)* 
E^(3*a)*n*(b*x^n)^((1 + m)/n))
 

Defintions of rubi rules used

rule 2009

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

rule 5886

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), 
x_Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Cosh[c + d*x^n])^p, x], x 
] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x^m*cosh(b*x^n + a)^3, x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.86 \[ \int x^m \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (\frac {m + 1}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {3 \, x^{m + 1} e^{\left (-a\right )} \Gamma \left (\frac {m + 1}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {3 \, x^{m + 1} e^{a} \Gamma \left (\frac {m + 1}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (\frac {m + 1}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\frac {m + 1}{n}} n} \] Input:

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

Output:

-1/8*x^(m + 1)*e^(-3*a)*gamma((m + 1)/n, 3*b*x^n)/((3*b*x^n)^((m + 1)/n)*n 
) - 3/8*x^(m + 1)*e^(-a)*gamma((m + 1)/n, b*x^n)/((b*x^n)^((m + 1)/n)*n) - 
 3/8*x^(m + 1)*e^a*gamma((m + 1)/n, -b*x^n)/((-b*x^n)^((m + 1)/n)*n) - 1/8 
*x^(m + 1)*e^(3*a)*gamma((m + 1)/n, -3*b*x^n)/((-3*b*x^n)^((m + 1)/n)*n)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(x**m*cosh(x**n*b + a)**3,x)