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

Optimal. Leaf size=79 \[ -\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}-\frac{\cosh \left (a+b x^2\right )}{3 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{6 b} \]

[Out]

-Cosh[a + b*x^2]/(3*b^2) - Cosh[a + b*x^2]^3/(18*b^2) + (x^2*Sinh[a + b*x^2])/(3*b) + (x^2*Cosh[a + b*x^2]^2*S
inh[a + b*x^2])/(6*b)

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Rubi [A]  time = 0.0771761, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5321, 3310, 3296, 2638} \[ -\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}-\frac{\cosh \left (a+b x^2\right )}{3 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Cosh[a + b*x^2]^3,x]

[Out]

-Cosh[a + b*x^2]/(3*b^2) - Cosh[a + b*x^2]^3/(18*b^2) + (x^2*Sinh[a + b*x^2])/(3*b) + (x^2*Cosh[a + b*x^2]^2*S
inh[a + b*x^2])/(6*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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \cosh ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{6 b}+\frac{1}{3} \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,x^2\right )\\ &=-\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{6 b}-\frac{\operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,x^2\right )}{3 b}\\ &=-\frac{\cosh \left (a+b x^2\right )}{3 b^2}-\frac{\cosh ^3\left (a+b x^2\right )}{18 b^2}+\frac{x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac{x^2 \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.163931, size = 55, normalized size = 0.7 \[ -\frac{-3 b x^2 \left (9 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )\right )+27 \cosh \left (a+b x^2\right )+\cosh \left (3 \left (a+b x^2\right )\right )}{72 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cosh[a + b*x^2]^3,x]

[Out]

-(27*Cosh[a + b*x^2] + Cosh[3*(a + b*x^2)] - 3*b*x^2*(9*Sinh[a + b*x^2] + Sinh[3*(a + b*x^2)]))/(72*b^2)

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Maple [A]  time = 0.047, size = 93, normalized size = 1.2 \begin{align*}{\frac{ \left ( 3\,b{x}^{2}-1 \right ){{\rm e}^{3\,b{x}^{2}+3\,a}}}{144\,{b}^{2}}}+{\frac{ \left ( 3\,b{x}^{2}-3 \right ){{\rm e}^{b{x}^{2}+a}}}{16\,{b}^{2}}}-{\frac{ \left ( 3\,b{x}^{2}+3 \right ){{\rm e}^{-b{x}^{2}-a}}}{16\,{b}^{2}}}-{\frac{ \left ( 3\,b{x}^{2}+1 \right ){{\rm e}^{-3\,b{x}^{2}-3\,a}}}{144\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(3*b*x^2-1)/b^2*exp(3*b*x^2+3*a)+3/16*(b*x^2-1)/b^2*exp(b*x^2+a)-3/16*(b*x^2+1)/b^2*exp(-b*x^2-a)-1/144*
(3*b*x^2+1)/b^2*exp(-3*b*x^2-3*a)

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Maxima [A]  time = 1.16948, size = 135, normalized size = 1.71 \begin{align*} \frac{{\left (3 \, b x^{2} e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x^{2}\right )}}{144 \, b^{2}} + \frac{3 \,{\left (b x^{2} e^{a} - e^{a}\right )} e^{\left (b x^{2}\right )}}{16 \, b^{2}} - \frac{3 \,{\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{16 \, b^{2}} - \frac{{\left (3 \, b x^{2} + 1\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(3*b*x^2*e^(3*a) - e^(3*a))*e^(3*b*x^2)/b^2 + 3/16*(b*x^2*e^a - e^a)*e^(b*x^2)/b^2 - 3/16*(b*x^2 + 1)*e^
(-b*x^2 - a)/b^2 - 1/144*(3*b*x^2 + 1)*e^(-3*b*x^2 - 3*a)/b^2

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Fricas [A]  time = 1.83854, size = 234, normalized size = 2.96 \begin{align*} \frac{3 \, b x^{2} \sinh \left (b x^{2} + a\right )^{3} - \cosh \left (b x^{2} + a\right )^{3} - 3 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{2} + 9 \,{\left (b x^{2} \cosh \left (b x^{2} + a\right )^{2} + 3 \, b x^{2}\right )} \sinh \left (b x^{2} + a\right ) - 27 \, \cosh \left (b x^{2} + a\right )}{72 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(3*b*x^2*sinh(b*x^2 + a)^3 - cosh(b*x^2 + a)^3 - 3*cosh(b*x^2 + a)*sinh(b*x^2 + a)^2 + 9*(b*x^2*cosh(b*x^
2 + a)^2 + 3*b*x^2)*sinh(b*x^2 + a) - 27*cosh(b*x^2 + a))/b^2

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Sympy [A]  time = 4.04589, size = 92, normalized size = 1.16 \begin{align*} \begin{cases} - \frac{x^{2} \sinh ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac{x^{2} \sinh{\left (a + b x^{2} \right )} \cosh ^{2}{\left (a + b x^{2} \right )}}{2 b} + \frac{\sinh ^{2}{\left (a + b x^{2} \right )} \cosh{\left (a + b x^{2} \right )}}{3 b^{2}} - \frac{7 \cosh ^{3}{\left (a + b x^{2} \right )}}{18 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cosh ^{3}{\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)**3,x)

[Out]

Piecewise((-x**2*sinh(a + b*x**2)**3/(3*b) + x**2*sinh(a + b*x**2)*cosh(a + b*x**2)**2/(2*b) + sinh(a + b*x**2
)**2*cosh(a + b*x**2)/(3*b**2) - 7*cosh(a + b*x**2)**3/(18*b**2), Ne(b, 0)), (x**4*cosh(a)**3/4, True))

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Giac [B]  time = 1.37563, size = 250, normalized size = 3.16 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )} e^{\left (3 \, b x^{2} + 3 \, a\right )} - 3 \, a e^{\left (3 \, b x^{2} + 3 \, a\right )} + 27 \,{\left (b x^{2} + a\right )} e^{\left (b x^{2} + a\right )} - 27 \, a e^{\left (b x^{2} + a\right )} - 27 \,{\left (b x^{2} + a\right )} e^{\left (-b x^{2} - a\right )} + 27 \, a e^{\left (-b x^{2} - a\right )} - 3 \,{\left (b x^{2} + a\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )} + 3 \, a e^{\left (-3 \, b x^{2} - 3 \, a\right )} - e^{\left (3 \, b x^{2} + 3 \, a\right )} - 27 \, e^{\left (b x^{2} + a\right )} - 27 \, e^{\left (-b x^{2} - a\right )} - e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(3*(b*x^2 + a)*e^(3*b*x^2 + 3*a) - 3*a*e^(3*b*x^2 + 3*a) + 27*(b*x^2 + a)*e^(b*x^2 + a) - 27*a*e^(b*x^2
+ a) - 27*(b*x^2 + a)*e^(-b*x^2 - a) + 27*a*e^(-b*x^2 - a) - 3*(b*x^2 + a)*e^(-3*b*x^2 - 3*a) + 3*a*e^(-3*b*x^
2 - 3*a) - e^(3*b*x^2 + 3*a) - 27*e^(b*x^2 + a) - 27*e^(-b*x^2 - a) - e^(-3*b*x^2 - 3*a))/b^2