3.8 \(\int x^3 \cosh ^2(a+b x^2) \, dx\)

Optimal. Leaf size=51 \[ -\frac{\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \sinh \left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{4 b}+\frac{x^4}{8} \]

[Out]

x^4/8 - Cosh[a + b*x^2]^2/(8*b^2) + (x^2*Cosh[a + b*x^2]*Sinh[a + b*x^2])/(4*b)

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Rubi [A]  time = 0.0513967, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5321, 3310, 30} \[ -\frac{\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \sinh \left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{4 b}+\frac{x^4}{8} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Cosh[a + b*x^2]^2,x]

[Out]

x^4/8 - Cosh[a + b*x^2]^2/(8*b^2) + (x^2*Cosh[a + b*x^2]*Sinh[a + b*x^2])/(4*b)

Rule 5321

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^3 \cosh ^2\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \cosh ^2(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}+\frac{1}{4} \operatorname{Subst}\left (\int x \, dx,x,x^2\right )\\ &=\frac{x^4}{8}-\frac{\cosh ^2\left (a+b x^2\right )}{8 b^2}+\frac{x^2 \cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.100742, size = 40, normalized size = 0.78 \[ -\frac{\cosh \left (2 \left (a+b x^2\right )\right )-2 b x^2 \left (\sinh \left (2 \left (a+b x^2\right )\right )+b x^2\right )}{16 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cosh[a + b*x^2]^2,x]

[Out]

-(Cosh[2*(a + b*x^2)] - 2*b*x^2*(b*x^2 + Sinh[2*(a + b*x^2)]))/(16*b^2)

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Maple [A]  time = 0.03, size = 55, normalized size = 1.1 \begin{align*}{\frac{{x}^{4}}{8}}+{\frac{ \left ( 2\,b{x}^{2}-1 \right ){{\rm e}^{2\,b{x}^{2}+2\,a}}}{32\,{b}^{2}}}-{\frac{ \left ( 2\,b{x}^{2}+1 \right ){{\rm e}^{-2\,b{x}^{2}-2\,a}}}{32\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(b*x^2+a)^2,x)

[Out]

1/8*x^4+1/32*(2*b*x^2-1)/b^2*exp(2*b*x^2+2*a)-1/32*(2*b*x^2+1)/b^2*exp(-2*b*x^2-2*a)

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Maxima [A]  time = 1.0953, size = 80, normalized size = 1.57 \begin{align*} \frac{1}{8} \, x^{4} + \frac{{\left (2 \, b x^{2} e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x^{2}\right )}}{32 \, b^{2}} - \frac{{\left (2 \, b x^{2} + 1\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{32 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/8*x^4 + 1/32*(2*b*x^2*e^(2*a) - e^(2*a))*e^(2*b*x^2)/b^2 - 1/32*(2*b*x^2 + 1)*e^(-2*b*x^2 - 2*a)/b^2

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Fricas [A]  time = 1.76402, size = 140, normalized size = 2.75 \begin{align*} \frac{2 \, b^{2} x^{4} + 4 \, b x^{2} \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) - \cosh \left (b x^{2} + a\right )^{2} - \sinh \left (b x^{2} + a\right )^{2}}{16 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/16*(2*b^2*x^4 + 4*b*x^2*cosh(b*x^2 + a)*sinh(b*x^2 + a) - cosh(b*x^2 + a)^2 - sinh(b*x^2 + a)^2)/b^2

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Sympy [A]  time = 2.17481, size = 78, normalized size = 1.53 \begin{align*} \begin{cases} - \frac{x^{4} \sinh ^{2}{\left (a + b x^{2} \right )}}{8} + \frac{x^{4} \cosh ^{2}{\left (a + b x^{2} \right )}}{8} + \frac{x^{2} \sinh{\left (a + b x^{2} \right )} \cosh{\left (a + b x^{2} \right )}}{4 b} - \frac{\cosh ^{2}{\left (a + b x^{2} \right )}}{8 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cosh ^{2}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cosh(b*x**2+a)**2,x)

[Out]

Piecewise((-x**4*sinh(a + b*x**2)**2/8 + x**4*cosh(a + b*x**2)**2/8 + x**2*sinh(a + b*x**2)*cosh(a + b*x**2)/(
4*b) - cosh(a + b*x**2)**2/(8*b**2), Ne(b, 0)), (x**4*cosh(a)**2/4, True))

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Giac [B]  time = 1.23386, size = 163, normalized size = 3.2 \begin{align*} \frac{4 \,{\left (b x^{2} + a\right )}^{2} - 8 \,{\left (b x^{2} + a\right )} a + 2 \,{\left (b x^{2} + a\right )} e^{\left (2 \, b x^{2} + 2 \, a\right )} - 2 \, a e^{\left (2 \, b x^{2} + 2 \, a\right )} - 2 \,{\left (b x^{2} + a\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )} + 2 \, a e^{\left (-2 \, b x^{2} - 2 \, a\right )} - e^{\left (2 \, b x^{2} + 2 \, a\right )} - e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{32 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/32*(4*(b*x^2 + a)^2 - 8*(b*x^2 + a)*a + 2*(b*x^2 + a)*e^(2*b*x^2 + 2*a) - 2*a*e^(2*b*x^2 + 2*a) - 2*(b*x^2 +
a)*e^(-2*b*x^2 - 2*a) + 2*a*e^(-2*b*x^2 - 2*a) - e^(2*b*x^2 + 2*a) - e^(-2*b*x^2 - 2*a))/b^2