### 3.328 $$\int x \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx$$

Optimal. Leaf size=67 $\frac{3 \sinh (2 a+2 b x)}{128 b^2}-\frac{\sinh (6 a+6 b x)}{1152 b^2}-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b}$

[Out]

(-3*x*Cosh[2*a + 2*b*x])/(64*b) + (x*Cosh[6*a + 6*b*x])/(192*b) + (3*Sinh[2*a + 2*b*x])/(128*b^2) - Sinh[6*a +
6*b*x]/(1152*b^2)

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Rubi [A]  time = 0.0726938, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {5448, 3296, 2637} $\frac{3 \sinh (2 a+2 b x)}{128 b^2}-\frac{\sinh (6 a+6 b x)}{1152 b^2}-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*x*Cosh[2*a + 2*b*x])/(64*b) + (x*Cosh[6*a + 6*b*x])/(192*b) + (3*Sinh[2*a + 2*b*x])/(128*b^2) - Sinh[6*a +
6*b*x]/(1152*b^2)

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} x \sinh (2 a+2 b x)+\frac{1}{32} x \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int x \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int x \sinh (2 a+2 b x) \, dx\\ &=-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b}-\frac{\int \cosh (6 a+6 b x) \, dx}{192 b}+\frac{3 \int \cosh (2 a+2 b x) \, dx}{64 b}\\ &=-\frac{3 x \cosh (2 a+2 b x)}{64 b}+\frac{x \cosh (6 a+6 b x)}{192 b}+\frac{3 \sinh (2 a+2 b x)}{128 b^2}-\frac{\sinh (6 a+6 b x)}{1152 b^2}\\ \end{align*}

Mathematica [A]  time = 0.143971, size = 50, normalized size = 0.75 $-\frac{-27 \sinh (2 (a+b x))+\sinh (6 (a+b x))+54 b x \cosh (2 (a+b x))-6 b x \cosh (6 (a+b x))}{1152 b^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(54*b*x*Cosh[2*(a + b*x)] - 6*b*x*Cosh[6*(a + b*x)] - 27*Sinh[2*(a + b*x)] + Sinh[6*(a + b*x)])/(1152*b^2)

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Maple [B]  time = 0.009, size = 170, normalized size = 2.5 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{6}}-{\frac{ \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{36}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{36}}+{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{24}}+{\frac{bx}{24}}+{\frac{a}{24}}-a \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{6}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*(1/6*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)^4-1/12*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)^2-1/12*(b*x+a)*cosh(b*x+
a)^2-1/36*sinh(b*x+a)*cosh(b*x+a)^5+1/36*cosh(b*x+a)^3*sinh(b*x+a)+1/24*cosh(b*x+a)*sinh(b*x+a)+1/24*b*x+1/24*
a-a*(1/6*cosh(b*x+a)^4*sinh(b*x+a)^2-1/12*cosh(b*x+a)^2*sinh(b*x+a)^2-1/12*cosh(b*x+a)^2))

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Maxima [A]  time = 1.0983, size = 123, normalized size = 1.84 \begin{align*} \frac{{\left (6 \, b x e^{\left (6 \, a\right )} - e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{2304 \, b^{2}} - \frac{3 \,{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{256 \, b^{2}} - \frac{3 \,{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{2}} + \frac{{\left (6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{2304 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2304*(6*b*x*e^(6*a) - e^(6*a))*e^(6*b*x)/b^2 - 3/256*(2*b*x*e^(2*a) - e^(2*a))*e^(2*b*x)/b^2 - 3/256*(2*b*x
+ 1)*e^(-2*b*x - 2*a)/b^2 + 1/2304*(6*b*x + 1)*e^(-6*b*x - 6*a)/b^2

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Fricas [B]  time = 1.80394, size = 408, normalized size = 6.09 \begin{align*} \frac{3 \, b x \cosh \left (b x + a\right )^{6} + 45 \, b x \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + 3 \, b x \sinh \left (b x + a\right )^{6} - 10 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - 27 \, b x \cosh \left (b x + a\right )^{2} + 9 \,{\left (5 \, b x \cosh \left (b x + a\right )^{4} - 3 \, b x\right )} \sinh \left (b x + a\right )^{2} - 3 \,{\left (\cosh \left (b x + a\right )^{5} - 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{576 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(3*b*x*cosh(b*x + a)^6 + 45*b*x*cosh(b*x + a)^2*sinh(b*x + a)^4 + 3*b*x*sinh(b*x + a)^6 - 10*cosh(b*x +
a)^3*sinh(b*x + a)^3 - 3*cosh(b*x + a)*sinh(b*x + a)^5 - 27*b*x*cosh(b*x + a)^2 + 9*(5*b*x*cosh(b*x + a)^4 - 3
*b*x)*sinh(b*x + a)^2 - 3*(cosh(b*x + a)^5 - 9*cosh(b*x + a))*sinh(b*x + a))/b^2

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Sympy [A]  time = 7.62976, size = 148, normalized size = 2.21 \begin{align*} \begin{cases} - \frac{x \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac{x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac{x \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac{\sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{24 b^{2}} - \frac{\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{\sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{24 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\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)**3*sinh(b*x+a)**3,x)

[Out]

Piecewise((-x*sinh(a + b*x)**6/(24*b) + x*sinh(a + b*x)**4*cosh(a + b*x)**2/(8*b) + x*sinh(a + b*x)**2*cosh(a
+ b*x)**4/(8*b) - x*cosh(a + b*x)**6/(24*b) + sinh(a + b*x)**5*cosh(a + b*x)/(24*b**2) - sinh(a + b*x)**3*cosh
(a + b*x)**3/(9*b**2) + sinh(a + b*x)*cosh(a + b*x)**5/(24*b**2), Ne(b, 0)), (x**2*sinh(a)**3*cosh(a)**3/2, Tr
ue))

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Giac [A]  time = 1.15879, size = 109, normalized size = 1.63 \begin{align*} \frac{{\left (6 \, b x - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{2304 \, b^{2}} - \frac{3 \,{\left (2 \, b x - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{256 \, b^{2}} - \frac{3 \,{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{2}} + \frac{{\left (6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{2304 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2304*(6*b*x - 1)*e^(6*b*x + 6*a)/b^2 - 3/256*(2*b*x - 1)*e^(2*b*x + 2*a)/b^2 - 3/256*(2*b*x + 1)*e^(-2*b*x -
2*a)/b^2 + 1/2304*(6*b*x + 1)*e^(-6*b*x - 6*a)/b^2