\(\int \cosh ^3(a+b x^n) \, dx\) [39]

Optimal result

Integrand size = 10, antiderivative size = 150 \[ \int \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3^{-1/n} e^{3 a} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{8 n}-\frac {3^{-1/n} e^{-3 a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 b x^n\right )}{8 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5417, 5415, 2239} \[ \int \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {e^{3 a} 3^{-1/n} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{8 n}-\frac {e^{-3 a} 3^{-1/n} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 b x^n\right )}{8 n} \]

[In]

Int[Cosh[a + b*x^n]^3,x]

[Out]

-1/8*(E^(3*a)*x*Gamma[n^(-1), -3*b*x^n])/(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 2239

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]

Rule 5415

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 5417

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.92 \[ \int \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3^{-1/n} e^{-3 a} x \left (-b^2 x^{2 n}\right )^{-1/n} \left (e^{6 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-3 b x^n\right )+3^{1+\frac {1}{n}} e^{4 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-b x^n\right )+\left (-b x^n\right )^{\frac {1}{n}} \left (3^{1+\frac {1}{n}} e^{2 a} \Gamma \left (\frac {1}{n},b x^n\right )+\Gamma \left (\frac {1}{n},3 b x^n\right )\right )\right )}{8 n} \]

[In]

Integrate[Cosh[a + b*x^n]^3,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(cosh(b*x^n + a)^3, x)

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(a + b*x**n)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.15 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {x e^{\left (-3 \, a\right )} \Gamma \left (\frac {1}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {3 \, x e^{\left (-a\right )} \Gamma \left (\frac {1}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {3 \, x e^{a} \Gamma \left (\frac {1}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {x e^{\left (3 \, a\right )} \Gamma \left (\frac {1}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} \]

[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)

Giac [F]

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

[In]

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

[Out]

integrate(cosh(b*x^n + a)^3, x)

Mupad [F(-1)]

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

[In]

int(cosh(a + b*x^n)^3,x)

[Out]

int(cosh(a + b*x^n)^3, x)