3.261 \(\int x^2 \cosh ^2(a+b x) \sinh (a+b x) \, dx\)

Optimal. Leaf size=83 \[ -\frac{4 x \sinh (a+b x)}{9 b^2}+\frac{2 \cosh ^3(a+b x)}{27 b^3}+\frac{4 \cosh (a+b x)}{9 b^3}-\frac{2 x \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2}+\frac{x^2 \cosh ^3(a+b x)}{3 b} \]

[Out]

(4*Cosh[a + b*x])/(9*b^3) + (2*Cosh[a + b*x]^3)/(27*b^3) + (x^2*Cosh[a + b*x]^3)/(3*b) - (4*x*Sinh[a + b*x])/(
9*b^2) - (2*x*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^2)

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Rubi [A]  time = 0.0705722, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5373, 3310, 3296, 2638} \[ -\frac{4 x \sinh (a+b x)}{9 b^2}+\frac{2 \cosh ^3(a+b x)}{27 b^3}+\frac{4 \cosh (a+b x)}{9 b^3}-\frac{2 x \sinh (a+b x) \cosh ^2(a+b x)}{9 b^2}+\frac{x^2 \cosh ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cosh[a + b*x]^2*Sinh[a + b*x],x]

[Out]

(4*Cosh[a + b*x])/(9*b^3) + (2*Cosh[a + b*x]^3)/(27*b^3) + (x^2*Cosh[a + b*x]^3)/(3*b) - (4*x*Sinh[a + b*x])/(
9*b^2) - (2*x*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^2)

Rule 5373

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(x^(m -
n + 1)*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

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^2 \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac{x^2 \cosh ^3(a+b x)}{3 b}-\frac{2 \int x \cosh ^3(a+b x) \, dx}{3 b}\\ &=\frac{2 \cosh ^3(a+b x)}{27 b^3}+\frac{x^2 \cosh ^3(a+b x)}{3 b}-\frac{2 x \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2}-\frac{4 \int x \cosh (a+b x) \, dx}{9 b}\\ &=\frac{2 \cosh ^3(a+b x)}{27 b^3}+\frac{x^2 \cosh ^3(a+b x)}{3 b}-\frac{4 x \sinh (a+b x)}{9 b^2}-\frac{2 x \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2}+\frac{4 \int \sinh (a+b x) \, dx}{9 b^2}\\ &=\frac{4 \cosh (a+b x)}{9 b^3}+\frac{2 \cosh ^3(a+b x)}{27 b^3}+\frac{x^2 \cosh ^3(a+b x)}{3 b}-\frac{4 x \sinh (a+b x)}{9 b^2}-\frac{2 x \cosh ^2(a+b x) \sinh (a+b x)}{9 b^2}\\ \end{align*}

Mathematica [A]  time = 0.199815, size = 65, normalized size = 0.78 \[ \frac{27 \left (b^2 x^2+2\right ) \cosh (a+b x)+\left (9 b^2 x^2+2\right ) \cosh (3 (a+b x))-6 b x (9 \sinh (a+b x)+\sinh (3 (a+b x)))}{108 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cosh[a + b*x]^2*Sinh[a + b*x],x]

[Out]

(27*(2 + b^2*x^2)*Cosh[a + b*x] + (2 + 9*b^2*x^2)*Cosh[3*(a + b*x)] - 6*b*x*(9*Sinh[a + b*x] + Sinh[3*(a + b*x
)]))/(108*b^3)

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Maple [B]  time = 0.006, size = 193, normalized size = 2.3 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{ \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}-{\frac{ \left ( 2\,bx+2\,a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{9}}-{\frac{ \left ( 4\,bx+4\,a \right ) \sinh \left ( bx+a \right ) }{9}}+{\frac{2\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{27}}+{\frac{14\,\cosh \left ( bx+a \right ) }{27}}-2\,a \left ( 1/3\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) +1/3\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) -1/9\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-2/9\,\sinh \left ( bx+a \right ) \right ) +{a}^{2} \left ({\frac{\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}}+{\frac{\cosh \left ( bx+a \right ) }{3}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^3*(1/3*(b*x+a)^2*sinh(b*x+a)^2*cosh(b*x+a)+1/3*(b*x+a)^2*cosh(b*x+a)-2/9*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2
-4/9*(b*x+a)*sinh(b*x+a)+2/27*cosh(b*x+a)*sinh(b*x+a)^2+14/27*cosh(b*x+a)-2*a*(1/3*(b*x+a)*sinh(b*x+a)^2*cosh(
b*x+a)+1/3*(b*x+a)*cosh(b*x+a)-1/9*sinh(b*x+a)*cosh(b*x+a)^2-2/9*sinh(b*x+a))+a^2*(1/3*cosh(b*x+a)*sinh(b*x+a)
^2+1/3*cosh(b*x+a)))

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Maxima [A]  time = 1.11215, size = 165, normalized size = 1.99 \begin{align*} \frac{{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{216 \, b^{3}} + \frac{{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{3}} + \frac{{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} + \frac{{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2*e^(3*a))*e^(3*b*x)/b^3 + 1/8*(b^2*x^2*e^a - 2*b*x*e^a + 2*e^a)*e^
(b*x)/b^3 + 1/8*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 + 1/216*(9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^3

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Fricas [A]  time = 1.97761, size = 273, normalized size = 3.29 \begin{align*} -\frac{6 \, b x \sinh \left (b x + a\right )^{3} -{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 27 \,{\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 18 \,{\left (b x \cosh \left (b x + a\right )^{2} + 3 \, b x\right )} \sinh \left (b x + a\right )}{108 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108*(6*b*x*sinh(b*x + a)^3 - (9*b^2*x^2 + 2)*cosh(b*x + a)^3 - 3*(9*b^2*x^2 + 2)*cosh(b*x + a)*sinh(b*x + a
)^2 - 27*(b^2*x^2 + 2)*cosh(b*x + a) + 18*(b*x*cosh(b*x + a)^2 + 3*b*x)*sinh(b*x + a))/b^3

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Sympy [A]  time = 2.25007, size = 105, normalized size = 1.27 \begin{align*} \begin{cases} \frac{x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{4 x \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac{2 x \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} - \frac{4 \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{9 b^{3}} + \frac{14 \cosh ^{3}{\left (a + b x \right )}}{27 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \sinh{\left (a \right )} \cosh ^{2}{\left (a \right )}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2*cosh(a + b*x)**3/(3*b) + 4*x*sinh(a + b*x)**3/(9*b**2) - 2*x*sinh(a + b*x)*cosh(a + b*x)**2/(3
*b**2) - 4*sinh(a + b*x)**2*cosh(a + b*x)/(9*b**3) + 14*cosh(a + b*x)**3/(27*b**3), Ne(b, 0)), (x**3*sinh(a)*c
osh(a)**2/3, True))

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Giac [A]  time = 1.14968, size = 146, normalized size = 1.76 \begin{align*} \frac{{\left (9 \, b^{2} x^{2} - 6 \, b x + 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{3}} + \frac{{\left (b^{2} x^{2} - 2 \, b x + 2\right )} e^{\left (b x + a\right )}}{8 \, b^{3}} + \frac{{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} + \frac{{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(9*b^2*x^2 - 6*b*x + 2)*e^(3*b*x + 3*a)/b^3 + 1/8*(b^2*x^2 - 2*b*x + 2)*e^(b*x + a)/b^3 + 1/8*(b^2*x^2 +
2*b*x + 2)*e^(-b*x - a)/b^3 + 1/216*(9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^3