\(\int x \sinh (a+b x^n) \, dx\) [59]

Optimal result

Integrand size = 10, antiderivative size = 75 \[ \int x \sinh \left (a+b x^n\right ) \, dx=-\frac {e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )}{2 n}+\frac {e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )}{2 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5468, 2250} \[ \int x \sinh \left (a+b x^n\right ) \, dx=\frac {e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )}{2 n}-\frac {e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )}{2 n} \]

[In]

Int[x*Sinh[a + b*x^n],x]

[Out]

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

Rule 2250

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

Rule 5468

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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int x \sinh \left (a+b x^n\right ) \, dx=\frac {e^{-a} x^2 \left (-e^{2 a} \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )+\left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )\right )}{2 n} \]

[In]

Integrate[x*Sinh[a + b*x^n],x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.55 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92

method	result	size
meijerg	\(\frac {x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{n}\right ], \left [\frac {1}{2}, 1+\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{2}+\frac {x^{2+n} b \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {1}{n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {1}{n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )}{2+n}\)	\(69\)

[In]

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

[Out]

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

Fricas [F]

\[ \int x \sinh \left (a+b x^n\right ) \, dx=\int { x \sinh \left (b x^{n} + a\right ) \,d x } \]

[In]

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

[Out]

integral(x*sinh(b*x^n + a), x)

Sympy [F]

\[ \int x \sinh \left (a+b x^n\right ) \, dx=\int x \sinh {\left (a + b x^{n} \right )}\, dx \]

[In]

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

[Out]

Integral(x*sinh(a + b*x**n), x)

Maxima [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int x \sinh \left (a+b x^n\right ) \, dx=\frac {x^{2} e^{\left (-a\right )} \Gamma \left (\frac {2}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\frac {2}{n}} n} - \frac {x^{2} e^{a} \Gamma \left (\frac {2}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\frac {2}{n}} n} \]

[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)

Giac [F]

\[ \int x \sinh \left (a+b x^n\right ) \, dx=\int { x \sinh \left (b x^{n} + a\right ) \,d x } \]

[In]

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

[Out]

integrate(x*sinh(b*x^n + a), x)

Mupad [F(-1)]

Timed out. \[ \int x \sinh \left (a+b x^n\right ) \, dx=\int x\,\mathrm {sinh}\left (a+b\,x^n\right ) \,d x \]

[In]

int(x*sinh(a + b*x^n),x)

[Out]

int(x*sinh(a + b*x^n), x)