### 3.281 $$\int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx$$

Optimal. Leaf size=117 $\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{2 \cosh ^3(a+b x)}{27 b^4}+\frac{14 \cosh (a+b x)}{9 b^4}+\frac{x^3 \sinh ^3(a+b x)}{3 b}$

[Out]

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

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Rubi [A]  time = 0.14488, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.278, Rules used = {5372, 3311, 3296, 2638, 2633} $\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{x^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{2 \cosh ^3(a+b x)}{27 b^4}+\frac{14 \cosh (a+b x)}{9 b^4}+\frac{x^3 \sinh ^3(a+b x)}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3*Cosh[a + b*x]*Sinh[a + b*x]^2,x]

[Out]

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

Rule 5372

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

Rule 3311

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int x^3 \cosh (a+b x) \sinh ^2(a+b x) \, dx &=\frac{x^3 \sinh ^3(a+b x)}{3 b}-\frac{\int x^2 \sinh ^3(a+b x) \, dx}{b}\\ &=-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}-\frac{2 \int \sinh ^3(a+b x) \, dx}{9 b^3}+\frac{2 \int x^2 \sinh (a+b x) \, dx}{3 b}\\ &=\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}+\frac{2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{9 b^4}-\frac{4 \int x \cosh (a+b x) \, dx}{3 b^2}\\ &=\frac{2 \cosh (a+b x)}{9 b^4}+\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{2 \cosh ^3(a+b x)}{27 b^4}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}+\frac{4 \int \sinh (a+b x) \, dx}{3 b^3}\\ &=\frac{14 \cosh (a+b x)}{9 b^4}+\frac{2 x^2 \cosh (a+b x)}{3 b^2}-\frac{2 \cosh ^3(a+b x)}{27 b^4}-\frac{4 x \sinh (a+b x)}{3 b^3}-\frac{x^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b^2}+\frac{2 x \sinh ^3(a+b x)}{9 b^3}+\frac{x^3 \sinh ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.398752, size = 84, normalized size = 0.72 $\frac{81 \left (b^2 x^2+2\right ) \cosh (a+b x)-\left (9 b^2 x^2+2\right ) \cosh (3 (a+b x))+6 b x \sinh (a+b x) \left (\left (3 b^2 x^2+2\right ) \cosh (2 (a+b x))-3 b^2 x^2-26\right )}{108 b^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3*Cosh[a + b*x]*Sinh[a + b*x]^2,x]

[Out]

(81*(2 + b^2*x^2)*Cosh[a + b*x] - (2 + 9*b^2*x^2)*Cosh[3*(a + b*x)] + 6*b*x*(-26 - 3*b^2*x^2 + (2 + 3*b^2*x^2)
*Cosh[2*(a + b*x)])*Sinh[a + b*x])/(108*b^4)

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Maple [B]  time = 0.005, size = 334, normalized size = 2.9 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{ \left ( bx+a \right ) ^{3}\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}}-{\frac{ \left ( bx+a \right ) ^{3}\sinh \left ( bx+a \right ) }{3}}-{\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{2\, \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 ( 14\,bx+14\,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{40\,\cosh \left ( bx+a \right ) }{27}}-3\,a \left ( 1/3\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/3\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) -2/9\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) +4/9\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) +{\frac{2\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{27}}-{\frac{14\,\sinh \left ( bx+a \right ) }{27}} \right ) +3\,{a}^{2} \left ( 1/3\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/3\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) -1/9\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}+2/9\,\cosh \left ( bx+a \right ) \right ) -{a}^{3} \left ({\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}}-{\frac{\sinh \left ( bx+a \right ) }{3}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^4*(1/3*(b*x+a)^3*sinh(b*x+a)*cosh(b*x+a)^2-1/3*(b*x+a)^3*sinh(b*x+a)-1/3*(b*x+a)^2*sinh(b*x+a)^2*cosh(b*x+
a)+2/3*(b*x+a)^2*cosh(b*x+a)+2/9*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-14/9*(b*x+a)*sinh(b*x+a)-2/27*cosh(b*x+a)*s
inh(b*x+a)^2+40/27*cosh(b*x+a)-3*a*(1/3*(b*x+a)^2*sinh(b*x+a)*cosh(b*x+a)^2-1/3*(b*x+a)^2*sinh(b*x+a)-2/9*(b*x
+a)*sinh(b*x+a)^2*cosh(b*x+a)+4/9*(b*x+a)*cosh(b*x+a)+2/27*sinh(b*x+a)*cosh(b*x+a)^2-14/27*sinh(b*x+a))+3*a^2*
(1/3*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-1/3*(b*x+a)*sinh(b*x+a)-1/9*cosh(b*x+a)*sinh(b*x+a)^2+2/9*cosh(b*x+a))-
a^3*(1/3*sinh(b*x+a)*cosh(b*x+a)^2-1/3*sinh(b*x+a)))

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Maxima [A]  time = 1.13066, size = 216, normalized size = 1.85 \begin{align*} \frac{{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 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^{4}} - \frac{{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{4}} + \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x)/b^4 - 1/8*(b^3*x^3*e^a - 3
*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 + 1/8*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x - a)/b^4 - 1/216
*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4

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Fricas [A]  time = 1.76391, size = 332, normalized size = 2.84 \begin{align*} -\frac{{\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} - 3 \,{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \sinh \left (b x + a\right )^{3} - 81 \,{\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 9 \,{\left (3 \, b^{3} x^{3} -{\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} + 18 \, b x\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108*((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 - 3*(3*b^3*x^3 + 2*b
*x)*sinh(b*x + a)^3 - 81*(b^2*x^2 + 2)*cosh(b*x + a) + 9*(3*b^3*x^3 - (3*b^3*x^3 + 2*b*x)*cosh(b*x + a)^2 + 18
*b*x)*sinh(b*x + a))/b^4

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Sympy [A]  time = 4.19061, size = 146, normalized size = 1.25 \begin{align*} \begin{cases} \frac{x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} - \frac{x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b^{2}} + \frac{2 x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac{14 x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac{4 x \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac{14 \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{9 b^{4}} + \frac{40 \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh ^{2}{\left (a \right )} \cosh{\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+a)*sinh(b*x+a)**2,x)

[Out]

Piecewise((x**3*sinh(a + b*x)**3/(3*b) - x**2*sinh(a + b*x)**2*cosh(a + b*x)/b**2 + 2*x**2*cosh(a + b*x)**3/(3
*b**2) + 14*x*sinh(a + b*x)**3/(9*b**3) - 4*x*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**3) - 14*sinh(a + b*x)**2*co
sh(a + b*x)/(9*b**4) + 40*cosh(a + b*x)**3/(27*b**4), Ne(b, 0)), (x**4*sinh(a)**2*cosh(a)/4, True))

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Giac [A]  time = 1.162, size = 189, normalized size = 1.62 \begin{align*} \frac{{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} - \frac{{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} + \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} - \frac{{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(9*b^3*x^3 - 9*b^2*x^2 + 6*b*x - 2)*e^(3*b*x + 3*a)/b^4 - 1/8*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*e^(b*x +
a)/b^4 + 1/8*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x - a)/b^4 - 1/216*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e
^(-3*b*x - 3*a)/b^4