3.48 \(\int x^m \cosh ^2(a+b x^n) \, dx\)

Optimal. Leaf size=128 \[ -\frac{e^{2 a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-2 b x^n\right )}{n}-\frac{e^{-2 a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},2 b x^n\right )}{n}+\frac{x^{m+1}}{2 (m+1)} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.155311, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5363, 5361, 2218} \[ -\frac{e^{2 a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-2 b x^n\right )}{n}-\frac{e^{-2 a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},2 b x^n\right )}{n}+\frac{x^{m+1}}{2 (m+1)} \]

Antiderivative was successfully verified.

[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 5363

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]

Rule 5361

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 2218

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

Rubi steps

\begin{align*} \int x^m \cosh ^2\left (a+b x^n\right ) \, dx &=\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]  time = 0.24899, size = 116, normalized size = 0.91 \[ -\frac{x^{m+1} \left (e^{2 a} (m+1) 2^{-\frac{m+1}{n}} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-2 b x^n\right )+e^{-2 a} (m+1) 2^{-\frac{m+1}{n}} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},2 b x^n\right )-2 n\right )}{4 (m+1) n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Cosh[a + b*x^n]^2,x]

[Out]

-(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))))/(4*(1 + m)*n)

________________________________________________________________________________________

Maple [F]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( a+b{x}^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \cosh \left (b x^{n} + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh ^{2}{\left (a + b x^{n} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*cosh(a + b*x**n)**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh \left (b x^{n} + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*cosh(b*x^n + a)^2, x)