### 3.327 $$\int x^2 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx$$

Optimal. Leaf size=105 $\frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}-\frac{3 \cosh (2 a+2 b x)}{128 b^3}+\frac{\cosh (6 a+6 b x)}{3456 b^3}-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}$

[Out]

(-3*Cosh[2*a + 2*b*x])/(128*b^3) - (3*x^2*Cosh[2*a + 2*b*x])/(64*b) + Cosh[6*a + 6*b*x]/(3456*b^3) + (x^2*Cosh
[6*a + 6*b*x])/(192*b) + (3*x*Sinh[2*a + 2*b*x])/(64*b^2) - (x*Sinh[6*a + 6*b*x])/(576*b^2)

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Rubi [A]  time = 0.140026, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.15, Rules used = {5448, 3296, 2638} $\frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}-\frac{3 \cosh (2 a+2 b x)}{128 b^3}+\frac{\cosh (6 a+6 b x)}{3456 b^3}-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*Cosh[2*a + 2*b*x])/(128*b^3) - (3*x^2*Cosh[2*a + 2*b*x])/(64*b) + Cosh[6*a + 6*b*x]/(3456*b^3) + (x^2*Cosh
[6*a + 6*b*x])/(192*b) + (3*x*Sinh[2*a + 2*b*x])/(64*b^2) - (x*Sinh[6*a + 6*b*x])/(576*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 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 ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} x^2 \sinh (2 a+2 b x)+\frac{1}{32} x^2 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int x^2 \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int x^2 \sinh (2 a+2 b x) \, dx\\ &=-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}-\frac{\int x \cosh (6 a+6 b x) \, dx}{96 b}+\frac{3 \int x \cosh (2 a+2 b x) \, dx}{32 b}\\ &=-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}+\frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}+\frac{\int \sinh (6 a+6 b x) \, dx}{576 b^2}-\frac{3 \int \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac{3 \cosh (2 a+2 b x)}{128 b^3}-\frac{3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac{\cosh (6 a+6 b x)}{3456 b^3}+\frac{x^2 \cosh (6 a+6 b x)}{192 b}+\frac{3 x \sinh (2 a+2 b x)}{64 b^2}-\frac{x \sinh (6 a+6 b x)}{576 b^2}\\ \end{align*}

Mathematica [A]  time = 0.219113, size = 72, normalized size = 0.69 $\frac{-81 \left (2 b^2 x^2+1\right ) \cosh (2 (a+b x))+\left (18 b^2 x^2+1\right ) \cosh (6 (a+b x))+6 b x (27 \sinh (2 (a+b x))-\sinh (6 (a+b x)))}{3456 b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-81*(1 + 2*b^2*x^2)*Cosh[2*(a + b*x)] + (1 + 18*b^2*x^2)*Cosh[6*(a + b*x)] + 6*b*x*(27*Sinh[2*(a + b*x)] - Si
nh[6*(a + b*x)]))/(3456*b^3)

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Maple [B]  time = 0.009, size = 358, normalized size = 3.4 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{6}}-{\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( bx+a \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{12}}-{\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{18}}+{\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{18}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{108}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{216}}-{\frac{5\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{108}}+{\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{12}}+{\frac{ \left ( bx+a \right ) ^{2}}{24}}-2\,a \left ( 1/6\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{4}-1/12\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/12\, \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/36\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}+1/36\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) +1/24\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) +1/24\,bx+a/24 \right ) +{a}^{2} \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^2*cosh(b*x+a)^3*sinh(b*x+a)^3,x)

[Out]

1/b^3*(1/6*(b*x+a)^2*sinh(b*x+a)^2*cosh(b*x+a)^4-1/12*(b*x+a)^2*sinh(b*x+a)^2*cosh(b*x+a)^2-1/12*(b*x+a)^2*cos
h(b*x+a)^2-1/18*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^5+1/18*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^3+1/108*cosh(b*x+a)^4*s
inh(b*x+a)^2-1/216*cosh(b*x+a)^2*sinh(b*x+a)^2-5/108*cosh(b*x+a)^2+1/12*(b*x+a)*cosh(b*x+a)*sinh(b*x+a)+1/24*(
b*x+a)^2-2*a*(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)*co
sh(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^2*(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 = 1.06654, size = 171, normalized size = 1.63 \begin{align*} \frac{{\left (18 \, b^{2} x^{2} e^{\left (6 \, a\right )} - 6 \, b x e^{\left (6 \, a\right )} + e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{6912 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{256 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac{{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6912*(18*b^2*x^2*e^(6*a) - 6*b*x*e^(6*a) + e^(6*a))*e^(6*b*x)/b^3 - 3/256*(2*b^2*x^2*e^(2*a) - 2*b*x*e^(2*a)
+ e^(2*a))*e^(2*b*x)/b^3 - 3/256*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b^3 + 1/6912*(18*b^2*x^2 + 6*b*x +
1)*e^(-6*b*x - 6*a)/b^3

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Fricas [B]  time = 1.75665, size = 531, normalized size = 5.06 \begin{align*} -\frac{120 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 36 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} -{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{6} - 15 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} -{\left (18 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 81 \,{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 3 \,{\left (5 \,{\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 54 \, b^{2} x^{2} - 27\right )} \sinh \left (b x + a\right )^{2} + 36 \,{\left (b x \cosh \left (b x + a\right )^{5} - 9 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3456*(120*b*x*cosh(b*x + a)^3*sinh(b*x + a)^3 + 36*b*x*cosh(b*x + a)*sinh(b*x + a)^5 - (18*b^2*x^2 + 1)*cos
h(b*x + a)^6 - 15*(18*b^2*x^2 + 1)*cosh(b*x + a)^2*sinh(b*x + a)^4 - (18*b^2*x^2 + 1)*sinh(b*x + a)^6 + 81*(2*
b^2*x^2 + 1)*cosh(b*x + a)^2 - 3*(5*(18*b^2*x^2 + 1)*cosh(b*x + a)^4 - 54*b^2*x^2 - 27)*sinh(b*x + a)^2 + 36*(
b*x*cosh(b*x + a)^5 - 9*b*x*cosh(b*x + a))*sinh(b*x + a))/b^3

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Sympy [A]  time = 13.4604, size = 223, normalized size = 2.12 \begin{align*} \begin{cases} - \frac{x^{2} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x^{2} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac{x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac{x^{2} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac{x \sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{12 b^{2}} - \frac{2 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{x \sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{12 b^{2}} - \frac{5 \sinh ^{6}{\left (a + b x \right )}}{108 b^{3}} + \frac{7 \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{72 b^{3}} - \frac{\sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{24 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\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)**3*sinh(b*x+a)**3,x)

[Out]

Piecewise((-x**2*sinh(a + b*x)**6/(24*b) + x**2*sinh(a + b*x)**4*cosh(a + b*x)**2/(8*b) + x**2*sinh(a + b*x)**
2*cosh(a + b*x)**4/(8*b) - x**2*cosh(a + b*x)**6/(24*b) + x*sinh(a + b*x)**5*cosh(a + b*x)/(12*b**2) - 2*x*sin
h(a + b*x)**3*cosh(a + b*x)**3/(9*b**2) + x*sinh(a + b*x)*cosh(a + b*x)**5/(12*b**2) - 5*sinh(a + b*x)**6/(108
*b**3) + 7*sinh(a + b*x)**4*cosh(a + b*x)**2/(72*b**3) - sinh(a + b*x)**2*cosh(a + b*x)**4/(24*b**3), Ne(b, 0)
), (x**3*sinh(a)**3*cosh(a)**3/3, True))

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Giac [A]  time = 1.22813, size = 153, normalized size = 1.46 \begin{align*} \frac{{\left (18 \, b^{2} x^{2} - 6 \, b x + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{6912 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{256 \, b^{3}} - \frac{3 \,{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac{{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6912*(18*b^2*x^2 - 6*b*x + 1)*e^(6*b*x + 6*a)/b^3 - 3/256*(2*b^2*x^2 - 2*b*x + 1)*e^(2*b*x + 2*a)/b^3 - 3/25
6*(2*b^2*x^2 + 2*b*x + 1)*e^(-2*b*x - 2*a)/b^3 + 1/6912*(18*b^2*x^2 + 6*b*x + 1)*e^(-6*b*x - 6*a)/b^3