\(\int \cosh (a+b x^n) \, dx\) [35]

Optimal result

Integrand size = 8, antiderivative size = 67 \[ \int \cosh \left (a+b x^n\right ) \, dx=-\frac {e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{2 n}-\frac {e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{2 n} \]

[Out]

-1/2*exp(a)*x*GAMMA(1/n,-b*x^n)/n/((-b*x^n)^(1/n))-1/2*x*GAMMA(1/n,b*x^n)/exp(a)/n/((b*x^n)^(1/n))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5415, 2239} \[ \int \cosh \left (a+b x^n\right ) \, dx=-\frac {e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{2 n}-\frac {e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{2 n} \]

[In]

Int[Cosh[a + b*x^n],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93 \[ \int \cosh \left (a+b x^n\right ) \, dx=-\frac {e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )+e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{2 n} \]

[In]

Integrate[Cosh[a + b*x^n],x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.10

method	result	size
meijerg	\(x \operatorname {hypergeom}\left (\left [\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )+\frac {x^{n +1} b \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{n +1}\)	\(74\)

[In]

int(cosh(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

x*hypergeom([1/2/n],[1/2,1+1/2/n],1/4*x^(2*n)*b^2)*cosh(a)+1/(n+1)*x^(n+1)*b*hypergeom([1/2+1/2/n],[3/2,3/2+1/
2/n],1/4*x^(2*n)*b^2)*sinh(a)

Fricas [F]

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

[In]

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

[Out]

integral(cosh(b*x^n + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral(cosh(a + b*x**n), x)

Maxima [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \cosh \left (a+b x^n\right ) \, dx=-\frac {x e^{\left (-a\right )} \Gamma \left (\frac {1}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {x e^{a} \Gamma \left (\frac {1}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\left (\frac {1}{n}\right )} n} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(cosh(b*x^n + a), x)

Mupad [F(-1)]

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

[In]

int(cosh(a + b*x^n),x)

[Out]

int(cosh(a + b*x^n), x)