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

Optimal. Leaf size=75 \[ \frac{e^{-a} x^2 \left (b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},b x^n\right )}{2 n}-\frac{e^a x^2 \left (-b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b x^n\right )}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.0406622, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5360, 2218} \[ \frac{e^{-a} x^2 \left (b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},b x^n\right )}{2 n}-\frac{e^a x^2 \left (-b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x*Sinh[a + b*x^n],x]

[Out]

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

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

Mathematica [A]  time = 0.0811381, size = 88, normalized size = 1.17 \[ -\frac{x^2 \left (-b^2 x^{2 n}\right )^{-2/n} \left ((\sinh (a)+\cosh (a)) \left (b x^n\right )^{2/n} \text{Gamma}\left (\frac{2}{n},-b x^n\right )-(\cosh (a)-\sinh (a)) \left (-b x^n\right )^{2/n} \text{Gamma}\left (\frac{2}{n},b x^n\right )\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sinh[a + b*x^n],x]

[Out]

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

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Maple [C]  time = 0.065, size = 69, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}\sinh \left ( a \right ) }{2}{\mbox{$_1$F$_2$}({n}^{-1};\,{\frac{1}{2}},1+{n}^{-1};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}}+{\frac{{x}^{n+2}b\cosh \left ( a \right ) }{n+2}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{n}^{-1};\,{\frac{3}{2}},{\frac{3}{2}}+{n}^{-1};\,{\frac{{x}^{2\,n}{b}^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.21475, size = 99, normalized size = 1.32 \begin{align*} \frac{x^{2} e^{\left (-a\right )} \Gamma \left (\frac{2}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\frac{2}{n}} n} - \frac{x^{2} e^{a} \Gamma \left (\frac{2}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\frac{2}{n}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*sinh(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sinh(a + b*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*sinh(b*x^n + a), x)