\(\int x^{-1-n} \cosh (a+b x^n) \, dx\) [50]

Optimal result

Integrand size = 16, antiderivative size = 45 \[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=-\frac {x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac {b \text {Chi}\left (b x^n\right ) \sinh (a)}{n}+\frac {b \cosh (a) \text {Shi}\left (b x^n\right )}{n} \]

[Out]

-cosh(a+b*x^n)/n/(x^n)+b*cosh(a)*Shi(b*x^n)/n+b*Chi(b*x^n)*sinh(a)/n

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5429, 3378, 3384, 3379, 3382} \[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=\frac {b \sinh (a) \text {Chi}\left (b x^n\right )}{n}+\frac {b \cosh (a) \text {Shi}\left (b x^n\right )}{n}-\frac {x^{-n} \cosh \left (a+b x^n\right )}{n} \]

[In]

Int[x^(-1 - n)*Cosh[a + b*x^n],x]

[Out]

-(Cosh[a + b*x^n]/(n*x^n)) + (b*CoshIntegral[b*x^n]*Sinh[a])/n + (b*Cosh[a]*SinhIntegral[b*x^n])/n

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5429

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (a+b x)}{x^2} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac {b \text {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac {(b \cosh (a)) \text {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^n\right )}{n}+\frac {(b \sinh (a)) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-n} \cosh \left (a+b x^n\right )}{n}+\frac {b \text {Chi}\left (b x^n\right ) \sinh (a)}{n}+\frac {b \cosh (a) \text {Shi}\left (b x^n\right )}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=\frac {x^{-n} \left (-\cosh \left (a+b x^n\right )+b x^n \text {Chi}\left (b x^n\right ) \sinh (a)+b x^n \cosh (a) \text {Shi}\left (b x^n\right )\right )}{n} \]

[In]

Integrate[x^(-1 - n)*Cosh[a + b*x^n],x]

[Out]

(-Cosh[a + b*x^n] + b*x^n*CoshIntegral[b*x^n]*Sinh[a] + b*x^n*Cosh[a]*SinhIntegral[b*x^n])/(n*x^n)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.40

method	result	size
risch	\(-\frac {\left (-b \,{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{n}\right ) x^{n}+b \,{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b \,x^{n}\right ) x^{n}+{\mathrm e}^{-a -b \,x^{n}}+{\mathrm e}^{a +b \,x^{n}}\right ) x^{-n}}{2 n}\)	\(63\)

[In]

int(x^(-1-n)*cosh(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (45) = 90\).

Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.11 \[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=\frac {{\left ({\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (a\right ) + b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right ) - {\left ({\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (a\right ) - b \sinh \left (a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \left (x\right )\right ) - b \sinh \left (n \log \left (x\right )\right )\right ) - 2 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )}{2 \, {\left (n \cosh \left (n \log \left (x\right )\right ) + n \sinh \left (n \log \left (x\right )\right )\right )}} \]

[In]

integrate(x^(-1-n)*cosh(a+b*x^n),x, algorithm="fricas")

[Out]

1/2*(((b*cosh(a) + b*sinh(a))*cosh(n*log(x)) + (b*cosh(a) + b*sinh(a))*sinh(n*log(x)))*Ei(b*cosh(n*log(x)) + b
*sinh(n*log(x))) - ((b*cosh(a) - b*sinh(a))*cosh(n*log(x)) + (b*cosh(a) - b*sinh(a))*sinh(n*log(x)))*Ei(-b*cos
h(n*log(x)) - b*sinh(n*log(x))) - 2*cosh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a))/(n*cosh(n*log(x)) + n*sinh(
n*log(x)))

Sympy [F]

\[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=\int x^{- n - 1} \cosh {\left (a + b x^{n} \right )}\, dx \]

[In]

integrate(x**(-1-n)*cosh(a+b*x**n),x)

[Out]

Integral(x**(-n - 1)*cosh(a + b*x**n), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=-\frac {b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{2 \, n} + \frac {b e^{a} \Gamma \left (-1, -b x^{n}\right )}{2 \, n} \]

[In]

integrate(x^(-1-n)*cosh(a+b*x^n),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cosh \left (b x^{n} + a\right ) \,d x } \]

[In]

integrate(x^(-1-n)*cosh(a+b*x^n),x, algorithm="giac")

[Out]

integrate(x^(-n - 1)*cosh(b*x^n + a), x)

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x^n\right )}{x^{n+1}} \,d x \]

[In]

int(cosh(a + b*x^n)/x^(n + 1),x)

[Out]

int(cosh(a + b*x^n)/x^(n + 1), x)