3.62 \(\int \frac{\sinh (a+b x^n)}{x^2} \, dx\)

Optimal. Leaf size=71 \[ \frac{e^{-a} \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},b x^n\right )}{2 n x}-\frac{e^a \left (-b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b x^n\right )}{2 n x} \]

[Out]

-(E^a*(-(b*x^n))^n^(-1)*Gamma[-n^(-1), -(b*x^n)])/(2*n*x) + ((b*x^n)^n^(-1)*Gamma[-n^(-1), b*x^n])/(2*E^a*n*x)

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Rubi [A]  time = 0.0634738, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5360, 2218} \[ \frac{e^{-a} \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},b x^n\right )}{2 n x}-\frac{e^a \left (-b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b x^n\right )}{2 n x} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^n]/x^2,x]

[Out]

-(E^a*(-(b*x^n))^n^(-1)*Gamma[-n^(-1), -(b*x^n)])/(2*n*x) + ((b*x^n)^n^(-1)*Gamma[-n^(-1), b*x^n])/(2*E^a*n*x)

Rule 5360

Int[((e_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x]
- Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\sinh \left (a+b x^n\right )}{x^2} \, dx &=-\left (\frac{1}{2} \int \frac{e^{-a-b x^n}}{x^2} \, dx\right )+\frac{1}{2} \int \frac{e^{a+b x^n}}{x^2} \, dx\\ &=-\frac{e^a \left (-b x^n\right )^{\frac{1}{n}} \Gamma \left (-\frac{1}{n},-b x^n\right )}{2 n x}+\frac{e^{-a} \left (b x^n\right )^{\frac{1}{n}} \Gamma \left (-\frac{1}{n},b x^n\right )}{2 n x}\\ \end{align*}

Mathematica [A]  time = 0.0672384, size = 68, normalized size = 0.96 \[ \frac{(\cosh (a)-\sinh (a)) \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},b x^n\right )-(\sinh (a)+\cosh (a)) \left (-b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b x^n\right )}{2 n x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^n]/x^2,x]

[Out]

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

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Maple [C]  time = 0.077, size = 77, normalized size = 1.1 \begin{align*} -{\frac{\sinh \left ( a \right ) }{x}{\mbox{$_1$F$_2$}(-{\frac{1}{2\,n}};\,{\frac{1}{2}},1-{\frac{1}{2\,n}};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}}+{\frac{{x}^{-1+n}b\cosh \left ( a \right ) }{-1+n}{\mbox{$_1$F$_2$}({\frac{1}{2}}-{\frac{1}{2\,n}};\,{\frac{3}{2}},{\frac{3}{2}}-{\frac{1}{2\,n}};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b*x^n)/x^2,x)

[Out]

-1/x*hypergeom([-1/2/n],[1/2,1-1/2/n],1/4*x^(2*n)*b^2)*sinh(a)+1/(-1+n)*x^(-1+n)*b*hypergeom([1/2-1/2/n],[3/2,
3/2-1/2/n],1/4*x^(2*n)*b^2)*cosh(a)

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Maxima [A]  time = 1.19774, size = 88, normalized size = 1.24 \begin{align*} \frac{\left (b x^{n}\right )^{\left (\frac{1}{n}\right )} e^{\left (-a\right )} \Gamma \left (-\frac{1}{n}, b x^{n}\right )}{2 \, n x} - \frac{\left (-b x^{n}\right )^{\left (\frac{1}{n}\right )} e^{a} \Gamma \left (-\frac{1}{n}, -b x^{n}\right )}{2 \, n x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sinh \left (b x^{n} + a\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*x^n)/x^2,x, algorithm="fricas")

[Out]

integral(sinh(b*x^n + a)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**n)/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(b*x^n + a)/x^2, x)