\(\int x^m \cosh ^2(a+b x^n) \, dx\) [48]

Optimal result

Integrand size = 14, antiderivative size = 128 \[ \int x^m \cosh ^2\left (a+b x^n\right ) \, dx=\frac {x^{1+m}}{2 (1+m)}-\frac {2^{-\frac {1+m+2 n}{n}} e^{2 a} x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 b x^n\right )}{n}-\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 b x^n\right )}{n} \]

[Out]

1/2*x^(1+m)/(1+m)-exp(2*a)*x^(1+m)*GAMMA((1+m)/n,-2*b*x^n)/(2^((1+m+2*n)/n))/n/((-b*x^n)^((1+m)/n))-x^(1+m)*GA
MMA((1+m)/n,2*b*x^n)/(2^((1+m+2*n)/n))/exp(2*a)/n/((b*x^n)^((1+m)/n))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5471, 5469, 2250} \[ \int x^m \cosh ^2\left (a+b x^n\right ) \, dx=-\frac {e^{2 a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-2 b x^n\right )}{n}-\frac {e^{-2 a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},2 b x^n\right )}{n}+\frac {x^{m+1}}{2 (m+1)} \]

[In]

Int[x^m*Cosh[a + b*x^n]^2,x]

[Out]

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

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 5469

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x]
+ Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]

Rule 5471

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^m}{2}+\frac {1}{2} x^m \cosh \left (2 a+2 b x^n\right )\right ) \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}+\frac {1}{2} \int x^m \cosh \left (2 a+2 b x^n\right ) \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}+\frac {1}{4} \int e^{-2 a-2 b x^n} x^m \, dx+\frac {1}{4} \int e^{2 a+2 b x^n} x^m \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}-\frac {2^{-\frac {1+m+2 n}{n}} e^{2 a} x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 b x^n\right )}{n}-\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 b x^n\right )}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int x^m \cosh ^2\left (a+b x^n\right ) \, dx=-\frac {x^{1+m} \left (-2 n+2^{-\frac {1+m}{n}} e^{2 a} (1+m) \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 b x^n\right )+2^{-\frac {1+m}{n}} e^{-2 a} (1+m) \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 b x^n\right )\right )}{4 (1+m) n} \]

[In]

Integrate[x^m*Cosh[a + b*x^n]^2,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^m*cosh(b*x^n + a)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**m*cosh(a + b*x**n)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.79 \[ \int x^m \cosh ^2\left (a+b x^n\right ) \, dx=-\frac {x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (\frac {m + 1}{n}, 2 \, b x^{n}\right )}{4 \, \left (2 \, b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {x^{m + 1} e^{\left (2 \, a\right )} \Gamma \left (\frac {m + 1}{n}, -2 \, b x^{n}\right )}{4 \, \left (-2 \, b x^{n}\right )^{\frac {m + 1}{n}} n} + \frac {x^{m + 1}}{2 \, {\left (m + 1\right )}} \]

[In]

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

[Out]

-1/4*x^(m + 1)*e^(-2*a)*gamma((m + 1)/n, 2*b*x^n)/((2*b*x^n)^((m + 1)/n)*n) - 1/4*x^(m + 1)*e^(2*a)*gamma((m +
1)/n, -2*b*x^n)/((-2*b*x^n)^((m + 1)/n)*n) + 1/2*x^(m + 1)/(m + 1)

Giac [F]

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

[In]

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

[Out]

integrate(x^m*cosh(b*x^n + a)^2, x)

Mupad [F(-1)]

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

[In]

int(x^m*cosh(a + b*x^n)^2,x)

[Out]

int(x^m*cosh(a + b*x^n)^2, x)