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

Optimal. Leaf size=113 \[ \frac{3 b \sinh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{3 b \sinh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 b \cosh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{3 b \cosh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n}-\frac{3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac{x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.225329, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5363, 5321, 3297, 3303, 3298, 3301} \[ \frac{3 b \sinh (a) \text{Chi}\left (b x^n\right )}{4 n}+\frac{3 b \sinh (3 a) \text{Chi}\left (3 b x^n\right )}{4 n}+\frac{3 b \cosh (a) \text{Shi}\left (b x^n\right )}{4 n}+\frac{3 b \cosh (3 a) \text{Shi}\left (3 b x^n\right )}{4 n}-\frac{3 x^{-n} \cosh \left (a+b x^n\right )}{4 n}-\frac{x^{-n} \cosh \left (3 \left (a+b x^n\right )\right )}{4 n} \]

Antiderivative was successfully verified.

[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 5363

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]

Rule 5321

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 3297

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 3303

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 3298

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 3301

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]

Rubi steps

\begin{align*} \int x^{-1-n} \cosh ^3\left (a+b x^n\right ) \, dx &=\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{\operatorname{Subst}\left (\int \frac{\cosh (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{4 n}+\frac{3 \operatorname{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) \operatorname{Subst}\left (\int \frac{\sinh (a+b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac{(3 b) \operatorname{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)) \operatorname{Subst}\left (\int \frac{\sinh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac{(3 b \cosh (3 a)) \operatorname{Subst}\left (\int \frac{\sinh (3 b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac{(3 b \sinh (a)) \operatorname{Subst}\left (\int \frac{\cosh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac{(3 b \sinh (3 a)) \operatorname{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]  time = 0.15244, size = 97, normalized size = 0.86 \[ \frac{x^{-n} \left (3 b \sinh (a) x^n \text{Chi}\left (b x^n\right )+3 b \sinh (3 a) x^n \text{Chi}\left (3 b x^n\right )+3 b \cosh (a) x^n \text{Shi}\left (b x^n\right )+3 b \cosh (3 a) x^n \text{Shi}\left (3 b x^n\right )-3 \cosh \left (a+b x^n\right )-\cosh \left (3 \left (a+b x^n\right )\right )\right )}{4 n} \]

Antiderivative was successfully verified.

[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]  time = 0.102, size = 152, normalized size = 1.4 \begin{align*} -{\frac{{{\rm e}^{-3\,a-3\,b{x}^{n}}}}{8\,n{x}^{n}}}+{\frac{3\,b{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,b{x}^{n} \right ) }{8\,n}}-{\frac{3\,{{\rm e}^{-a-b{x}^{n}}}}{8\,n{x}^{n}}}+{\frac{3\,b{{\rm e}^{-a}}{\it Ei} \left ( 1,b{x}^{n} \right ) }{8\,n}}-{\frac{{{\rm e}^{3\,a+3\,b{x}^{n}}}}{8\,n{x}^{n}}}-{\frac{3\,b{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,b{x}^{n} \right ) }{8\,n}}-{\frac{3\,{{\rm e}^{a+b{x}^{n}}}}{8\,n{x}^{n}}}-{\frac{3\,{{\rm e}^{a}}b{\it Ei} \left ( 1,-b{x}^{n} \right ) }{8\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.33236, size = 95, normalized size = 0.84 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [B]  time = 1.85079, size = 1045, normalized size = 9.25 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{-n - 1} \cosh \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)