3.49 \(\int x^m \cosh ^3(a+b x^n) \, dx\)

Optimal. Leaf size=200 \[ -\frac{e^{3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )}{8 n}-\frac{e^{-3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 b x^n\right )}{8 n} \]

[Out]

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

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Rubi [A]  time = 0.205599, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5363, 5361, 2218} \[ -\frac{e^{3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )}{8 n}-\frac{e^{-3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 b x^n\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Cosh[a + b*x^n]^3,x]

[Out]

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

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 ^3\left (a+b x^n\right ) \, dx &=\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\\ &=\frac{1}{4} \int x^m \cosh \left (3 a+3 b x^n\right ) \, dx+\frac{3}{4} \int x^m \cosh \left (a+b x^n\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 a-3 b x^n} x^m \, dx+\frac{1}{8} \int e^{3 a+3 b x^n} x^m \, dx+\frac{3}{8} \int e^{-a-b x^n} x^m \, dx+\frac{3}{8} \int e^{a+b x^n} x^m \, 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}\\ \end{align*}

Mathematica [A]  time = 0.955639, size = 182, normalized size = 0.91 \[ -\frac{e^{-3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \left (\left (-b x^n\right )^{\frac{m+1}{n}} \left (e^{2 a} 3^{\frac{m+n+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )+\text{Gamma}\left (\frac{m+1}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 b x^n\right )+e^{4 a} 3^{\frac{m+n+1}{n}} \left (b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Cosh[a + b*x^n]^3,x]

[Out]

-(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])))/(8*3^((1 + m)/n)*E^(3*a)*n*(-(b^2*x^(2*n)))^((1 + m)/n))

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( a+b{x}^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.31915, size = 234, normalized size = 1.17 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \cosh \left (b x^{n} + a\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*cosh(b*x^n + a)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh ^{3}{\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)**3,x)

[Out]

Integral(x**m*cosh(a + b*x**n)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*cosh(b*x^n + a)^3, x)