\(\int x^{-1-n} \cosh ^3(a+b x^n) \, dx\) [52]

Optimal result

Integrand size = 18, antiderivative size = 113 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n}+\frac {3 b \text {Chi}\left (b x^n\right ) \sinh (a)}{4 n}+\frac {3 b \text {Chi}\left (3 b x^n\right ) \sinh (3 a)}{4 n}+\frac {3 b \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]

[Out]

-3/4*cosh(a+b*x^n)/n/(x^n)-1/4*cosh(3*a+3*b*x^n)/n/(x^n)+3/4*b*cosh(a)*Shi(b*x^n)/n+3/4*b*cosh(3*a)*Shi(3*b*x^
n)/n+3/4*b*Chi(b*x^n)*sinh(a)/n+3/4*b*Chi(3*b*x^n)*sinh(3*a)/n

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5471, 5429, 3378, 3384, 3379, 3382} \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\frac {3 b \sinh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {3 b \sinh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}+\frac {3 b \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n} \]

[In]

Int[x^(-1 - n)*Cosh[a + b*x^n]^3,x]

[Out]

(-3*Cosh[a + b*x^n])/(4*n*x^n) - Cosh[3*(a + b*x^n)]/(4*n*x^n) + (3*b*CoshIntegral[b*x^n]*Sinh[a])/(4*n) + (3*
b*CoshIntegral[3*b*x^n]*Sinh[3*a])/(4*n) + (3*b*Cosh[a]*SinhIntegral[b*x^n])/(4*n) + (3*b*Cosh[3*a]*SinhIntegr
al[3*b*x^n])/(4*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]))

Rule 5471

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{4} x^{-1-n} \cosh \left (a+b x^n\right )+\frac {1}{4} x^{-1-n} \cosh \left (3 a+3 b x^n\right )\right ) \, dx \\ & = \frac {1}{4} \int x^{-1-n} \cosh \left (3 a+3 b x^n\right ) \, dx+\frac {3}{4} \int x^{-1-n} \cosh \left (a+b x^n\right ) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{4 n}+\frac {3 \text {Subst}\left (\int \frac {\cosh (a+b x)}{x^2} \, dx,x,x^n\right )}{4 n} \\ & = -\frac {3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n}+\frac {(3 b) \text {Subst}\left (\int \frac {\sinh (a+b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b) \text {Subst}\left (\int \frac {\sinh (3 a+3 b x)}{x} \, dx,x,x^n\right )}{4 n} \\ & = -\frac {3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n}+\frac {(3 b \cosh (a)) \text {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b \cosh (3 a)) \text {Subst}\left (\int \frac {\sinh (3 b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b \sinh (a)) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b \sinh (3 a)) \text {Subst}\left (\int \frac {\cosh (3 b x)}{x} \, dx,x,x^n\right )}{4 n} \\ & = -\frac {3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n}+\frac {3 b \text {Chi}\left (b x^n\right ) \sinh (a)}{4 n}+\frac {3 b \text {Chi}\left (3 b x^n\right ) \sinh (3 a)}{4 n}+\frac {3 b \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\frac {x^{-n} \left (-3 \cosh \left (a+b x^n\right )-\cosh \left (3 \left (a+b x^n\right )\right )+3 b x^n \text {Chi}\left (b x^n\right ) \sinh (a)+3 b x^n \text {Chi}\left (3 b x^n\right ) \sinh (3 a)+3 b x^n \cosh (a) \text {Shi}\left (b x^n\right )+3 b x^n \cosh (3 a) \text {Shi}\left (3 b x^n\right )\right )}{4 n} \]

[In]

Integrate[x^(-1 - n)*Cosh[a + b*x^n]^3,x]

[Out]

(-3*Cosh[a + b*x^n] - Cosh[3*(a + b*x^n)] + 3*b*x^n*CoshIntegral[b*x^n]*Sinh[a] + 3*b*x^n*CoshIntegral[3*b*x^n
]*Sinh[3*a] + 3*b*x^n*Cosh[a]*SinhIntegral[b*x^n] + 3*b*x^n*Cosh[3*a]*SinhIntegral[3*b*x^n])/(4*n*x^n)

Maple [A] (verified)

Time = 7.61 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12

method	result	size
risch	\(-\frac {\left (3 b \,{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{-3 a} \operatorname {Ei}_{1}\left (3 b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b \,x^{n}\right ) x^{n}+3 b \,{\mathrm e}^{3 a} \operatorname {Ei}_{1}\left (-3 b \,x^{n}\right ) x^{n}+{\mathrm e}^{-3 a -3 b \,x^{n}}+3 \,{\mathrm e}^{-a -b \,x^{n}}+{\mathrm e}^{3 a +3 b \,x^{n}}+3 \,{\mathrm e}^{a +b \,x^{n}}\right ) x^{-n}}{8 n}\)	\(126\)

[In]

int(x^(-1-n)*cosh(a+b*x^n)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (102) = 204\).

Time = 0.26 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.83 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {2 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{3} + 6 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right ) \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - 3 \, {\left ({\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (3 \, b \cosh \left (n \log \left (x\right )\right ) + 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) - 3 \, {\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 ) + 3 \, {\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 ) + 3 \, {\left ({\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} {\rm Ei}\left (-3 \, b \cosh \left (n \log \left (x\right )\right ) - 3 \, b \sinh \left (n \log \left (x\right )\right )\right ) + 6 \, \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )}{8 \, {\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)^3,x, algorithm="fricas")

[Out]

-1/8*(2*cosh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^3 + 6*cosh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)*sinh
(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^2 - 3*((b*cosh(3*a) + b*sinh(3*a))*cosh(n*log(x)) + (b*cosh(3*a) + b
*sinh(3*a))*sinh(n*log(x)))*Ei(3*b*cosh(n*log(x)) + 3*b*sinh(n*log(x))) - 3*((b*cosh(a) + b*sinh(a))*cosh(n*lo
g(x)) + (b*cosh(a) + b*sinh(a))*sinh(n*log(x)))*Ei(b*cosh(n*log(x)) + b*sinh(n*log(x))) + 3*((b*cosh(a) - b*si
nh(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))) + 3*(
(b*cosh(3*a) - b*sinh(3*a))*cosh(n*log(x)) + (b*cosh(3*a) - b*sinh(3*a))*sinh(n*log(x)))*Ei(-3*b*cosh(n*log(x)
) - 3*b*sinh(n*log(x))) + 6*cosh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a))/(n*cosh(n*log(x)) + n*sinh(n*log(x)
))

Sympy [F]

\[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\int x^{- n - 1} \cosh ^{3}{\left (a + b x^{n} \right )}\, dx \]

[In]

integrate(x**(-1-n)*cosh(a+b*x**n)**3,x)

[Out]

Integral(x**(-n - 1)*cosh(a + b*x**n)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.62 \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=-\frac {3 \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{n}\right )}{8 \, n} - \frac {3 \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{8 \, n} + \frac {3 \, b e^{a} \Gamma \left (-1, -b x^{n}\right )}{8 \, n} + \frac {3 \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{n}\right )}{8 \, n} \]

[In]

integrate(x^(-1-n)*cosh(a+b*x^n)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cosh \left (b x^{n} + a\right )^{3} \,d x } \]

[In]

integrate(x^(-1-n)*cosh(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate(x^(-n - 1)*cosh(b*x^n + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^3}{x^{n+1}} \,d x \]

[In]

int(cosh(a + b*x^n)^3/x^(n + 1),x)

[Out]

int(cosh(a + b*x^n)^3/x^(n + 1), x)