3.51 \(\int x^{-1-n} \cosh ^2(a+b x^n) \, dx\)

Optimal. Leaf size=67 \[ \frac {b \sinh (2 a) \text {Chi}\left (2 b x^n\right )}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]

[Out]

-1/2/n/(x^n)-1/2*cosh(2*a+2*b*x^n)/n/(x^n)+b*cosh(2*a)*Shi(2*b*x^n)/n+b*Chi(2*b*x^n)*sinh(2*a)/n

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {b \sinh (2 a) \text {Chi}\left (2 b x^n\right )}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*Cosh[a + b*x^n]^2,x]

[Out]

-1/(2*n*x^n) - Cosh[2*(a + b*x^n)]/(2*n*x^n) + (b*CoshIntegral[2*b*x^n]*Sinh[2*a])/n + (b*Cosh[2*a]*SinhIntegr
al[2*b*x^n])/n

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 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]

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 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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 54, normalized size = 0.81 \[ \frac {x^{-n} \left (b \sinh (2 a) x^n \text {Chi}\left (2 b x^n\right )+b \cosh (2 a) x^n \text {Shi}\left (2 b x^n\right )-\cosh ^2\left (a+b x^n\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)*Cosh[a + b*x^n]^2,x]

[Out]

(-Cosh[a + b*x^n]^2 + b*x^n*CoshIntegral[2*b*x^n]*Sinh[2*a] + b*x^n*Cosh[2*a]*SinhIntegral[2*b*x^n])/(n*x^n)

________________________________________________________________________________________

fricas [B]  time = 0.69, size = 182, normalized size = 2.72 \[ \frac {{\left ({\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (2 \, b \cosh \left (n \log \relax (x)\right ) + 2 \, b \sinh \left (n \log \relax (x)\right )\right ) - {\left ({\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (-2 \, b \cosh \left (n \log \relax (x)\right ) - 2 \, b \sinh \left (n \log \relax (x)\right )\right ) - \cosh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )^{2} - \sinh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )^{2} - 1}{2 \, {\left (n \cosh \left (n \log \relax (x)\right ) + n \sinh \left (n \log \relax (x)\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-n)*cosh(a+b*x^n)^2,x, algorithm="fricas")

[Out]

1/2*(((b*cosh(2*a) + b*sinh(2*a))*cosh(n*log(x)) + (b*cosh(2*a) + b*sinh(2*a))*sinh(n*log(x)))*Ei(2*b*cosh(n*l
og(x)) + 2*b*sinh(n*log(x))) - ((b*cosh(2*a) - b*sinh(2*a))*cosh(n*log(x)) + (b*cosh(2*a) - b*sinh(2*a))*sinh(
n*log(x)))*Ei(-2*b*cosh(n*log(x)) - 2*b*sinh(n*log(x))) - cosh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^2 - si
nh(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a)^2 - 1)/(n*cosh(n*log(x)) + n*sinh(n*log(x)))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-n - 1} \cosh \left (b x^{n} + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-n)*cosh(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate(x^(-n - 1)*cosh(b*x^n + a)^2, x)

________________________________________________________________________________________

maple [A]  time = 0.25, size = 90, normalized size = 1.34 \[ -\frac {x^{-n}}{2 n}-\frac {{\mathrm e}^{-2 a -2 b \,x^{n}} x^{-n}}{4 n}+\frac {b \,{\mathrm e}^{-2 a} \Ei \left (1, 2 b \,x^{n}\right )}{2 n}-\frac {x^{-n} {\mathrm e}^{2 a +2 b \,x^{n}}}{4 n}-\frac {b \,{\mathrm e}^{2 a} \Ei \left (1, -2 b \,x^{n}\right )}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-n)*cosh(a+b*x^n)^2,x)

[Out]

-1/2/n/(x^n)-1/4/n*exp(-2*a-2*b*x^n)/(x^n)+1/2/n*b*exp(-2*a)*Ei(1,2*b*x^n)-1/4/(x^n)*exp(2*a+2*b*x^n)/n-1/2/n*
b*exp(2*a)*Ei(1,-2*b*x^n)

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 47, normalized size = 0.70 \[ -\frac {b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x^{n}\right )}{2 \, n} + \frac {b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x^{n}\right )}{2 \, n} - \frac {1}{2 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-n)*cosh(a+b*x^n)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^2}{x^{n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x^n)^2/x^(n + 1),x)

[Out]

int(cosh(a + b*x^n)^2/x^(n + 1), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-n)*cosh(a+b*x**n)**2,x)

[Out]

Timed out

________________________________________________________________________________________