### 3.262 $$\int x \cosh ^2(a+b x) \sinh (a+b x) \, dx$$

Optimal. Leaf size=45 $-\frac{\sinh ^3(a+b x)}{9 b^2}-\frac{\sinh (a+b x)}{3 b^2}+\frac{x \cosh ^3(a+b x)}{3 b}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0313528, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {5373, 2633} $-\frac{\sinh ^3(a+b x)}{9 b^2}-\frac{\sinh (a+b x)}{3 b^2}+\frac{x \cosh ^3(a+b x)}{3 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*Cosh[a + b*x]^3)/(3*b) - Sinh[a + b*x]/(3*b^2) - Sinh[a + b*x]^3/(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 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 \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac{x \cosh ^3(a+b x)}{3 b}-\frac{\int \cosh ^3(a+b x) \, dx}{3 b}\\ &=\frac{x \cosh ^3(a+b x)}{3 b}-\frac{i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{3 b^2}\\ &=\frac{x \cosh ^3(a+b x)}{3 b}-\frac{\sinh (a+b x)}{3 b^2}-\frac{\sinh ^3(a+b x)}{9 b^2}\\ \end{align*}

Mathematica [A]  time = 0.136738, size = 46, normalized size = 1.02 $-\frac{9 \sinh (a+b x)+\sinh (3 (a+b x))-9 b x \cosh (a+b x)-3 b x \cosh (3 (a+b x))}{36 b^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(-9*b*x*Cosh[a + b*x] - 3*b*x*Cosh[3*(a + b*x)] + 9*Sinh[a + b*x] + Sinh[3*(a + b*x)])/(36*b^2)

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 92, normalized size = 2. \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}+{\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) }{3}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{9}}-{\frac{2\,\sinh \left ( bx+a \right ) }{9}}-a \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*cosh(b*x+a)^2*sinh(b*x+a),x)

[Out]

1/b^2*(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*(1/3*cosh(b*x+a)*sinh(b*x+a)^2+1/3*cosh(b*x+a)))

________________________________________________________________________________________

Maxima [B]  time = 1.19734, size = 113, normalized size = 2.51 \begin{align*} \frac{{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{72 \, b^{2}} + \frac{{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{8 \, b^{2}} + \frac{{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} + \frac{{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 + 1/8*(b*x*e^a - e^a)*e^(b*x)/b^2 + 1/8*(b*x + 1)*e^(-b*x - a)/b^
2 + 1/72*(3*b*x + 1)*e^(-3*b*x - 3*a)/b^2

________________________________________________________________________________________

Fricas [A]  time = 2.30944, size = 205, normalized size = 4.56 \begin{align*} \frac{3 \, b x \cosh \left (b x + a\right )^{3} + 9 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 9 \, b x \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )^{3} - 3 \,{\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )}{36 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(3*b*x*cosh(b*x + a)^3 + 9*b*x*cosh(b*x + a)*sinh(b*x + a)^2 + 9*b*x*cosh(b*x + a) - sinh(b*x + a)^3 - 3*
(cosh(b*x + a)^2 + 3)*sinh(b*x + a))/b^2

________________________________________________________________________________________

Sympy [A]  time = 1.13427, size = 61, normalized size = 1.36 \begin{align*} \begin{cases} \frac{x \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{2 \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac{\sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh{\left (a \right )} \cosh ^{2}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*cosh(a + b*x)**3/(3*b) + 2*sinh(a + b*x)**3/(9*b**2) - sinh(a + b*x)*cosh(a + b*x)**2/(3*b**2), N
e(b, 0)), (x**2*sinh(a)*cosh(a)**2/2, True))

________________________________________________________________________________________

Giac [A]  time = 1.16303, size = 103, normalized size = 2.29 \begin{align*} \frac{{\left (3 \, b x - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{72 \, b^{2}} + \frac{{\left (b x - 1\right )} e^{\left (b x + a\right )}}{8 \, b^{2}} + \frac{{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} + \frac{{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(3*b*x - 1)*e^(3*b*x + 3*a)/b^2 + 1/8*(b*x - 1)*e^(b*x + a)/b^2 + 1/8*(b*x + 1)*e^(-b*x - a)/b^2 + 1/72*(
3*b*x + 1)*e^(-3*b*x - 3*a)/b^2