3.70 \(\int x \sinh ^3(a+b x^n) \, dx\)

Optimal. Leaf size=166 \[ -\frac{e^{3 a} 9^{-1/n} x^2 \left (-b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-3 b x^n\right )}{8 n}+\frac{3 e^a x^2 \left (-b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^2 \left (b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},b x^n\right )}{8 n}+\frac{e^{-3 a} 9^{-1/n} x^2 \left (b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},3 b x^n\right )}{8 n} \]

[Out]

-(E^(3*a)*x^2*Gamma[2/n, -3*b*x^n])/(8*9^n^(-1)*n*(-(b*x^n))^(2/n)) + (3*E^a*x^2*Gamma[2/n, -(b*x^n)])/(8*n*(-
(b*x^n))^(2/n)) - (3*x^2*Gamma[2/n, b*x^n])/(8*E^a*n*(b*x^n)^(2/n)) + (x^2*Gamma[2/n, 3*b*x^n])/(8*9^n^(-1)*E^
(3*a)*n*(b*x^n)^(2/n))

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Rubi [A]  time = 0.13314, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5362, 5360, 2218} \[ -\frac{e^{3 a} 9^{-1/n} x^2 \left (-b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-3 b x^n\right )}{8 n}+\frac{3 e^a x^2 \left (-b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^2 \left (b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},b x^n\right )}{8 n}+\frac{e^{-3 a} 9^{-1/n} x^2 \left (b x^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},3 b x^n\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Int[x*Sinh[a + b*x^n]^3,x]

[Out]

-(E^(3*a)*x^2*Gamma[2/n, -3*b*x^n])/(8*9^n^(-1)*n*(-(b*x^n))^(2/n)) + (3*E^a*x^2*Gamma[2/n, -(b*x^n)])/(8*n*(-
(b*x^n))^(2/n)) - (3*x^2*Gamma[2/n, b*x^n])/(8*E^a*n*(b*x^n)^(2/n)) + (x^2*Gamma[2/n, 3*b*x^n])/(8*9^n^(-1)*E^
(3*a)*n*(b*x^n)^(2/n))

Rule 5362

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(
e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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 ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac{3}{4} x \sinh \left (a+b x^n\right )+\frac{1}{4} x \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} \int x \sinh \left (3 a+3 b x^n\right ) \, dx-\frac{3}{4} \int x \sinh \left (a+b x^n\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 a-3 b x^n} x \, dx\right )+\frac{1}{8} \int e^{3 a+3 b x^n} x \, dx+\frac{3}{8} \int e^{-a-b x^n} x \, dx-\frac{3}{8} \int e^{a+b x^n} x \, dx\\ &=-\frac{9^{-1/n} e^{3 a} x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac{2}{n},-3 b x^n\right )}{8 n}+\frac{3 e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac{2}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac{2}{n},b x^n\right )}{8 n}+\frac{9^{-1/n} e^{-3 a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac{2}{n},3 b x^n\right )}{8 n}\\ \end{align*}

Mathematica [A]  time = 1.55575, size = 161, normalized size = 0.97 \[ -\frac{e^{-3 a} 9^{-1/n} x^2 \left (-b^2 x^{2 n}\right )^{-2/n} \left (\left (-b x^n\right )^{2/n} \left (e^{2 a} 3^{\frac{n+2}{n}} \text{Gamma}\left (\frac{2}{n},b x^n\right )-\text{Gamma}\left (\frac{2}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{2/n} \text{Gamma}\left (\frac{2}{n},-3 b x^n\right )-e^{4 a} 3^{\frac{n+2}{n}} \left (b x^n\right )^{2/n} \text{Gamma}\left (\frac{2}{n},-b x^n\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sinh[a + b*x^n]^3,x]

[Out]

-(x^2*(E^(6*a)*(b*x^n)^(2/n)*Gamma[2/n, -3*b*x^n] - 3^((2 + n)/n)*E^(4*a)*(b*x^n)^(2/n)*Gamma[2/n, -(b*x^n)] +
(-(b*x^n))^(2/n)*(3^((2 + n)/n)*E^(2*a)*Gamma[2/n, b*x^n] - Gamma[2/n, 3*b*x^n])))/(8*9^n^(-1)*E^(3*a)*n*(-(b
^2*x^(2*n)))^(2/n))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int x \left ( \sinh \left ( a+b{x}^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(a+b*x^n)^3,x)

[Out]

int(x*sinh(a+b*x^n)^3,x)

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Maxima [A]  time = 1.27054, size = 201, normalized size = 1.21 \begin{align*} \frac{x^{2} e^{\left (-3 \, a\right )} \Gamma \left (\frac{2}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\frac{2}{n}} n} - \frac{3 \, x^{2} e^{\left (-a\right )} \Gamma \left (\frac{2}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\frac{2}{n}} n} + \frac{3 \, x^{2} e^{a} \Gamma \left (\frac{2}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\frac{2}{n}} n} - \frac{x^{2} e^{\left (3 \, a\right )} \Gamma \left (\frac{2}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, 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)^3,x, algorithm="maxima")

[Out]

1/8*x^2*e^(-3*a)*gamma(2/n, 3*b*x^n)/((3*b*x^n)^(2/n)*n) - 3/8*x^2*e^(-a)*gamma(2/n, b*x^n)/((b*x^n)^(2/n)*n)
+ 3/8*x^2*e^a*gamma(2/n, -b*x^n)/((-b*x^n)^(2/n)*n) - 1/8*x^2*e^(3*a)*gamma(2/n, -3*b*x^n)/((-3*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 )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(a+b*x^n)^3,x, algorithm="fricas")

[Out]

integral(x*sinh(b*x^n + a)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sinh ^{3}{\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)**3,x)

[Out]

Integral(x*sinh(a + b*x**n)**3, 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 )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate(x*sinh(b*x^n + a)^3, x)