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

Optimal. Leaf size=150 \[ -\frac{e^{3 a} 3^{-1/n} x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},b x^n\right )}{8 n}-\frac{e^{-3 a} 3^{-1/n} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},3 b x^n\right )}{8 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0795542, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5309, 5307, 2208} \[ -\frac{e^{3 a} 3^{-1/n} x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},b x^n\right )}{8 n}-\frac{e^{-3 a} 3^{-1/n} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},3 b x^n\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x^n]^3,x]

[Out]

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

Rule 5309

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

Rule 5307

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

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

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

Mathematica [A]  time = 0.600603, size = 138, normalized size = 0.92 \[ -\frac{e^{-3 a} 3^{-1/n} x \left (-b^2 x^{2 n}\right )^{-1/n} \left (\left (-b x^n\right )^{\frac{1}{n}} \left (e^{2 a} 3^{\frac{1}{n}+1} \text{Gamma}\left (\frac{1}{n},b x^n\right )+\text{Gamma}\left (\frac{1}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-3 b x^n\right )+e^{4 a} 3^{\frac{1}{n}+1} \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-b x^n\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x^n]^3,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.101, size = 0, normalized size = 0. \begin{align*} \int \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(cosh(a+b*x^n)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.20309, size = 169, normalized size = 1.13 \begin{align*} -\frac{x e^{\left (-3 \, a\right )} \Gamma \left (\frac{1}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\left (\frac{1}{n}\right )} n} - \frac{3 \, x e^{\left (-a\right )} \Gamma \left (\frac{1}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\left (\frac{1}{n}\right )} n} - \frac{3 \, x e^{a} \Gamma \left (\frac{1}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\left (\frac{1}{n}\right )} n} - \frac{x e^{\left (3 \, a\right )} \Gamma \left (\frac{1}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\left (\frac{1}{n}\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh(b*x^n + a)^3, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(a + b*x**n)**3, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cosh(b*x^n + a)^3, x)