3.1.58 \(\int x^2 \sinh (a+b x^n) \, dx\) [58]

Optimal. Leaf size=75 \[ -\frac {e^a x^3 \left (-b x^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-b x^n\right )}{2 n}+\frac {e^{-a} x^3 \left (b x^n\right )^{-3/n} \Gamma \left (\frac {3}{n},b x^n\right )}{2 n} \]

[Out]

-1/2*exp(a)*x^3*GAMMA(3/n,-b*x^n)/n/((-b*x^n)^(3/n))+1/2*x^3*GAMMA(3/n,b*x^n)/exp(a)/n/((b*x^n)^(3/n))

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Rubi [A]
time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {5468, 2250} \begin {gather*} \frac {e^{-a} x^3 \left (b x^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},b x^n\right )}{2 n}-\frac {e^a x^3 \left (-b x^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-b x^n\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sinh[a + b*x^n],x]

[Out]

-1/2*(E^a*x^3*Gamma[3/n, -(b*x^n)])/(n*(-(b*x^n))^(3/n)) + (x^3*Gamma[3/n, b*x^n])/(2*E^a*n*(b*x^n)^(3/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*} \int x^2 \sinh \left (a+b x^n\right ) \, dx &=-\left (\frac {1}{2} \int e^{-a-b x^n} x^2 \, dx\right )+\frac {1}{2} \int e^{a+b x^n} x^2 \, dx\\ &=-\frac {e^a x^3 \left (-b x^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-b x^n\right )}{2 n}+\frac {e^{-a} x^3 \left (b x^n\right )^{-3/n} \Gamma \left (\frac {3}{n},b x^n\right )}{2 n}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 88, normalized size = 1.17 \begin {gather*} -\frac {x^3 \left (-b^2 x^{2 n}\right )^{-3/n} \left (-\left (-b x^n\right )^{3/n} \Gamma \left (\frac {3}{n},b x^n\right ) (\cosh (a)-\sinh (a))+\left (b x^n\right )^{3/n} \Gamma \left (\frac {3}{n},-b x^n\right ) (\cosh (a)+\sinh (a))\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sinh[a + b*x^n],x]

[Out]

-1/2*(x^3*(-((-(b*x^n))^(3/n)*Gamma[3/n, b*x^n]*(Cosh[a] - Sinh[a])) + (b*x^n)^(3/n)*Gamma[3/n, -(b*x^n)]*(Cos
h[a] + Sinh[a])))/(n*(-(b^2*x^(2*n)))^(3/n))

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.34, size = 77, normalized size = 1.03

method	result	size
meijerg	\(\frac {x^{3} \hypergeom \left (\left [\frac {3}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {3}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{3}+\frac {x^{n +3} b \hypergeom \left (\left [\frac {1}{2}+\frac {3}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {3}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )}{n +3}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sinh(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.08, size = 73, normalized size = 0.97 \begin {gather*} \frac {x^{3} e^{\left (-a\right )} \Gamma \left (\frac {3}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\frac {3}{n}} n} - \frac {x^{3} e^{a} \Gamma \left (\frac {3}{n}, -b x^{n}\right )}{2 \, \left (-b x^{n}\right )^{\frac {3}{n}} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^3*e^(-a)*gamma(3/n, b*x^n)/((b*x^n)^(3/n)*n) - 1/2*x^3*e^a*gamma(3/n, -b*x^n)/((-b*x^n)^(3/n)*n)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*sinh(b*x^n + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sinh {\left (a + b x^{n} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sinh(a + b*x**n), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*sinh(b*x^n + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\mathrm {sinh}\left (a+b\,x^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sinh(a + b*x^n),x)

[Out]

int(x^2*sinh(a + b*x^n), x)

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