3.60 \(\int \sinh (a+b x^n) \, dx\)

Optimal. Leaf size=67 \[ \frac{e^{-a} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},b x^n\right )}{2 n}-\frac{e^a x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b x^n\right )}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.0170829, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5306, 2208} \[ \frac{e^{-a} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},b x^n\right )}{2 n}-\frac{e^a x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^n],x]

[Out]

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

Rule 5306

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

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

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

Mathematica [A]  time = 0.0804836, size = 79, normalized size = 1.18 \[ \frac{\left (-b^2 x^{2 n}\right )^{-1/n} \left (x (\cosh (a)-\sinh (a)) \left (-b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},b x^n\right )-x (\sinh (a)+\cosh (a)) \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-b x^n\right )\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^n],x]

[Out]

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

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Maple [C]  time = 0.052, size = 74, normalized size = 1.1 \begin{align*} x{\mbox{$_1$F$_2$}({\frac{1}{2\,n}};\,{\frac{1}{2}},1+{\frac{1}{2\,n}};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}\sinh \left ( a \right ) +{\frac{{x}^{n+1}b\cosh \left ( a \right ) }{n+1}{\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)

[Out]

x*hypergeom([1/2/n],[1/2,1+1/2/n],1/4*x^(2*n)*b^2)*sinh(a)+1/(n+1)*x^(n+1)*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 = 1.17206, size = 82, normalized size = 1.22 \begin{align*} \frac{x e^{\left (-a\right )} \Gamma \left (\frac{1}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\left (\frac{1}{n}\right )} n} - \frac{x e^{a} \Gamma \left (\frac{1}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\left (\frac{1}{n}\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sinh(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(b*x^n + a), x)