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

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

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Rubi [A]  time = 0.122997, antiderivative size = 67, normalized size of antiderivative = 1., 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 = {5362, 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)*Sinh[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]*SinhIntegra
l[2*b*x^n])/n

Rule 5362

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(
e*x)^m, (a + b*Sinh[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} \sinh ^2\left (a+b x^n\right ) \, dx &=\int \left (-\frac{1}{2} x^{-1-n}+\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 = 0.130863, size = 54, normalized size = 0.81 \[ \frac{x^{-n} \left (b \sinh (2 a) x^n \text{Chi}\left (2 b x^n\right )+b \cosh (2 a) x^n \text{Shi}\left (2 b x^n\right )-\sinh ^2\left (a+b x^n\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.073, size = 90, normalized size = 1.3 \begin{align*}{\frac{1}{2\,n{x}^{n}}}-{\frac{{{\rm e}^{-2\,a-2\,b{x}^{n}}}}{4\,n{x}^{n}}}+{\frac{b{{\rm e}^{-2\,a}}{\it Ei} \left ( 1,2\,b{x}^{n} \right ) }{2\,n}}-{\frac{{{\rm e}^{2\,a+2\,b{x}^{n}}}}{4\,n{x}^{n}}}-{\frac{b{{\rm e}^{2\,a}}{\it Ei} \left ( 1,-2\,b{x}^{n} \right ) }{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-n)*sinh(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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.89354, size = 567, normalized size = 8.46 \begin{align*} \frac{{\left ({\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (2 \, b \cosh \left (n \log \left (x\right )\right ) + 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) -{\left ({\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \left (x\right )\right ) +{\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )}{\rm Ei}\left (-2 \, b \cosh \left (n \log \left (x\right )\right ) - 2 \, b \sinh \left (n \log \left (x\right )\right )\right ) - \cosh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} - \sinh \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a\right )^{2} + 1}{2 \,{\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)*sinh(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)))

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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)*sinh(a+b*x**n)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{-n - 1} \sinh \left (b x^{n} + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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