### 3.19 $$\int \cosh ^4(a+b x) \sinh ^4(a+b x) \, dx$$

Optimal. Leaf size=90 $\frac{\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{16 b}+\frac{\sinh (a+b x) \cosh ^3(a+b x)}{64 b}+\frac{3 \sinh (a+b x) \cosh (a+b x)}{128 b}+\frac{3 x}{128}$

[Out]

(3*x)/128 + (3*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^
5*Sinh[a + b*x])/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(8*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0839527, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {2568, 2635, 8} $\frac{\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{16 b}+\frac{\sinh (a+b x) \cosh ^3(a+b x)}{64 b}+\frac{3 \sinh (a+b x) \cosh (a+b x)}{128 b}+\frac{3 x}{128}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]^4*Sinh[a + b*x]^4,x]

[Out]

(3*x)/128 + (3*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^
5*Sinh[a + b*x])/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(8*b)

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cosh ^4(a+b x) \sinh ^4(a+b x) \, dx &=\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}-\frac{3}{8} \int \cosh ^4(a+b x) \sinh ^2(a+b x) \, dx\\ &=-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac{1}{16} \int \cosh ^4(a+b x) \, dx\\ &=\frac{\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac{3}{64} \int \cosh ^2(a+b x) \, dx\\ &=\frac{3 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac{3 \int 1 \, dx}{128}\\ &=\frac{3 x}{128}+\frac{3 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0428134, size = 33, normalized size = 0.37 $\frac{24 (a+b x)-8 \sinh (4 (a+b x))+\sinh (8 (a+b x))}{1024 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[a + b*x]^4*Sinh[a + b*x]^4,x]

[Out]

(24*(a + b*x) - 8*Sinh[4*(a + b*x)] + Sinh[8*(a + b*x)])/(1024*b)

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 74, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3} \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{8}}-{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{16}}+{\frac{\sinh \left ( bx+a \right ) }{16} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( bx+a \right ) }{8}} \right ) }+{\frac{3\,bx}{128}}+{\frac{3\,a}{128}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^4*sinh(b*x+a)^4,x)

[Out]

1/b*(1/8*sinh(b*x+a)^3*cosh(b*x+a)^5-1/16*sinh(b*x+a)*cosh(b*x+a)^5+1/16*(1/4*cosh(b*x+a)^3+3/8*cosh(b*x+a))*s
inh(b*x+a)+3/128*b*x+3/128*a)

________________________________________________________________________________________

Maxima [A]  time = 1.21238, size = 89, normalized size = 0.99 \begin{align*} -\frac{{\left (8 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} + \frac{3 \,{\left (b x + a\right )}}{128 \, b} + \frac{8 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^4*sinh(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/2048*(8*e^(-4*b*x - 4*a) - 1)*e^(8*b*x + 8*a)/b + 3/128*(b*x + a)/b + 1/2048*(8*e^(-4*b*x - 4*a) - e^(-8*b*
x - 8*a))/b

________________________________________________________________________________________

Fricas [A]  time = 2.06411, size = 263, normalized size = 2.92 \begin{align*} \frac{7 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} +{\left (7 \, \cosh \left (b x + a\right )^{5} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b x +{\left (\cosh \left (b x + a\right )^{7} - 4 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )}{128 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^4*sinh(b*x+a)^4,x, algorithm="fricas")

[Out]

1/128*(7*cosh(b*x + a)^3*sinh(b*x + a)^5 + cosh(b*x + a)*sinh(b*x + a)^7 + (7*cosh(b*x + a)^5 - 4*cosh(b*x + a
))*sinh(b*x + a)^3 + 3*b*x + (cosh(b*x + a)^7 - 4*cosh(b*x + a)^3)*sinh(b*x + a))/b

________________________________________________________________________________________

Sympy [A]  time = 12.0885, size = 189, normalized size = 2.1 \begin{align*} \begin{cases} \frac{3 x \sinh ^{8}{\left (a + b x \right )}}{128} - \frac{3 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} + \frac{9 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} - \frac{3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} + \frac{3 x \cosh ^{8}{\left (a + b x \right )}}{128} - \frac{3 \sinh ^{7}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{128 b} + \frac{11 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac{11 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{128 b} - \frac{3 \sinh{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sinh ^{4}{\left (a \right )} \cosh ^{4}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**4*sinh(b*x+a)**4,x)

[Out]

Piecewise((3*x*sinh(a + b*x)**8/128 - 3*x*sinh(a + b*x)**6*cosh(a + b*x)**2/32 + 9*x*sinh(a + b*x)**4*cosh(a +
b*x)**4/64 - 3*x*sinh(a + b*x)**2*cosh(a + b*x)**6/32 + 3*x*cosh(a + b*x)**8/128 - 3*sinh(a + b*x)**7*cosh(a
+ b*x)/(128*b) + 11*sinh(a + b*x)**5*cosh(a + b*x)**3/(128*b) + 11*sinh(a + b*x)**3*cosh(a + b*x)**5/(128*b) -
3*sinh(a + b*x)*cosh(a + b*x)**7/(128*b), Ne(b, 0)), (x*sinh(a)**4*cosh(a)**4, True))

________________________________________________________________________________________

Giac [A]  time = 1.2121, size = 92, normalized size = 1.02 \begin{align*} \frac{48 \, b x -{\left (18 \, e^{\left (8 \, b x + 8 \, a\right )} - 8 \, e^{\left (4 \, b x + 4 \, a\right )} + 1\right )} e^{\left (-8 \, b x - 8 \, a\right )} + 48 \, a + e^{\left (8 \, b x + 8 \, a\right )} - 8 \, e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^4*sinh(b*x+a)^4,x, algorithm="giac")

[Out]

1/2048*(48*b*x - (18*e^(8*b*x + 8*a) - 8*e^(4*b*x + 4*a) + 1)*e^(-8*b*x - 8*a) + 48*a + e^(8*b*x + 8*a) - 8*e^
(4*b*x + 4*a))/b