### 3.21 $$\int \cosh ^6(a+b x) \sinh ^2(a+b x) \, dx$$

Optimal. Leaf size=88 $\frac{\sinh (a+b x) \cosh ^7(a+b x)}{8 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{48 b}-\frac{5 \sinh (a+b x) \cosh ^3(a+b x)}{192 b}-\frac{5 \sinh (a+b x) \cosh (a+b x)}{128 b}-\frac{5 x}{128}$

[Out]

(-5*x)/128 - (5*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(192*b) - (Cosh[a + b
*x]^5*Sinh[a + b*x])/(48*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(8*b)

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Rubi [A]  time = 0.0643681, antiderivative size = 88, 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} $\frac{\sinh (a+b x) \cosh ^7(a+b x)}{8 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{48 b}-\frac{5 \sinh (a+b x) \cosh ^3(a+b x)}{192 b}-\frac{5 \sinh (a+b x) \cosh (a+b x)}{128 b}-\frac{5 x}{128}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]^6*Sinh[a + b*x]^2,x]

[Out]

(-5*x)/128 - (5*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(192*b) - (Cosh[a + b
*x]^5*Sinh[a + b*x])/(48*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(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 ^6(a+b x) \sinh ^2(a+b x) \, dx &=\frac{\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac{1}{8} \int \cosh ^6(a+b x) \, dx\\ &=-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac{5}{48} \int \cosh ^4(a+b x) \, dx\\ &=-\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{192 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac{5}{64} \int \cosh ^2(a+b x) \, dx\\ &=-\frac{5 \cosh (a+b x) \sinh (a+b x)}{128 b}-\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{192 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac{5 \int 1 \, dx}{128}\\ &=-\frac{5 x}{128}-\frac{5 \cosh (a+b x) \sinh (a+b x)}{128 b}-\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{192 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0942673, size = 52, normalized size = 0.59 $\frac{-48 \sinh (2 (a+b x))+24 \sinh (4 (a+b x))+16 \sinh (6 (a+b x))+3 \sinh (8 (a+b x))-120 b x}{3072 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[a + b*x]^6*Sinh[a + b*x]^2,x]

[Out]

(-120*b*x - 48*Sinh[2*(a + b*x)] + 24*Sinh[4*(a + b*x)] + 16*Sinh[6*(a + b*x)] + 3*Sinh[8*(a + b*x)])/(3072*b)

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Maple [A]  time = 0.012, size = 66, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{7}}{8}}-{\frac{\sinh \left ( bx+a \right ) }{8} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{6}}+{\frac{5\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{24}}+{\frac{5\,\cosh \left ( bx+a \right ) }{16}} \right ) }-{\frac{5\,bx}{128}}-{\frac{5\,a}{128}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^6*sinh(b*x+a)^2,x)

[Out]

1/b*(1/8*sinh(b*x+a)*cosh(b*x+a)^7-1/8*(1/6*cosh(b*x+a)^5+5/24*cosh(b*x+a)^3+5/16*cosh(b*x+a))*sinh(b*x+a)-5/1
28*b*x-5/128*a)

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Maxima [A]  time = 1.00818, size = 149, normalized size = 1.69 \begin{align*} \frac{{\left (16 \, e^{\left (-2 \, b x - 2 \, a\right )} + 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 48 \, e^{\left (-6 \, b x - 6 \, a\right )} + 3\right )} e^{\left (8 \, b x + 8 \, a\right )}}{6144 \, b} - \frac{5 \,{\left (b x + a\right )}}{128 \, b} + \frac{48 \, e^{\left (-2 \, b x - 2 \, a\right )} - 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 16 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )}}{6144 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^6*sinh(b*x+a)^2,x, algorithm="maxima")

[Out]

1/6144*(16*e^(-2*b*x - 2*a) + 24*e^(-4*b*x - 4*a) - 48*e^(-6*b*x - 6*a) + 3)*e^(8*b*x + 8*a)/b - 5/128*(b*x +
a)/b + 1/6144*(48*e^(-2*b*x - 2*a) - 24*e^(-4*b*x - 4*a) - 16*e^(-6*b*x - 6*a) - 3*e^(-8*b*x - 8*a))/b

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Fricas [A]  time = 2.05741, size = 382, normalized size = 4.34 \begin{align*} \frac{3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + 3 \,{\left (7 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} +{\left (21 \, \cosh \left (b x + a\right )^{5} + 40 \, \cosh \left (b x + a\right )^{3} + 12 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 15 \, b x + 3 \,{\left (\cosh \left (b x + a\right )^{7} + 4 \, \cosh \left (b x + a\right )^{5} + 4 \, \cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{384 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^6*sinh(b*x+a)^2,x, algorithm="fricas")

[Out]

1/384*(3*cosh(b*x + a)*sinh(b*x + a)^7 + 3*(7*cosh(b*x + a)^3 + 4*cosh(b*x + a))*sinh(b*x + a)^5 + (21*cosh(b*
x + a)^5 + 40*cosh(b*x + a)^3 + 12*cosh(b*x + a))*sinh(b*x + a)^3 - 15*b*x + 3*(cosh(b*x + a)^7 + 4*cosh(b*x +
a)^5 + 4*cosh(b*x + a)^3 - 4*cosh(b*x + a))*sinh(b*x + a))/b

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Sympy [A]  time = 12.1828, size = 189, normalized size = 2.15 \begin{align*} \begin{cases} - \frac{5 x \sinh ^{8}{\left (a + b x \right )}}{128} + \frac{5 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} - \frac{15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} + \frac{5 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} - \frac{5 x \cosh ^{8}{\left (a + b x \right )}}{128} + \frac{5 \sinh ^{7}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{128 b} - \frac{55 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{384 b} + \frac{73 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{384 b} + \frac{5 \sinh{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sinh ^{2}{\left (a \right )} \cosh ^{6}{\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)**6*sinh(b*x+a)**2,x)

[Out]

Piecewise((-5*x*sinh(a + b*x)**8/128 + 5*x*sinh(a + b*x)**6*cosh(a + b*x)**2/32 - 15*x*sinh(a + b*x)**4*cosh(a
+ b*x)**4/64 + 5*x*sinh(a + b*x)**2*cosh(a + b*x)**6/32 - 5*x*cosh(a + b*x)**8/128 + 5*sinh(a + b*x)**7*cosh(
a + b*x)/(128*b) - 55*sinh(a + b*x)**5*cosh(a + b*x)**3/(384*b) + 73*sinh(a + b*x)**3*cosh(a + b*x)**5/(384*b)
+ 5*sinh(a + b*x)*cosh(a + b*x)**7/(128*b), Ne(b, 0)), (x*sinh(a)**2*cosh(a)**6, True))

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Giac [A]  time = 1.22557, size = 154, normalized size = 1.75 \begin{align*} -\frac{240 \, b x -{\left (250 \, e^{\left (8 \, b x + 8 \, a\right )} + 48 \, e^{\left (6 \, b x + 6 \, a\right )} - 24 \, e^{\left (4 \, b x + 4 \, a\right )} - 16 \, e^{\left (2 \, b x + 2 \, a\right )} - 3\right )} e^{\left (-8 \, b x - 8 \, a\right )} + 240 \, a - 3 \, e^{\left (8 \, b x + 8 \, a\right )} - 16 \, e^{\left (6 \, b x + 6 \, a\right )} - 24 \, e^{\left (4 \, b x + 4 \, a\right )} + 48 \, e^{\left (2 \, b x + 2 \, a\right )}}{6144 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^6*sinh(b*x+a)^2,x, algorithm="giac")

[Out]

-1/6144*(240*b*x - (250*e^(8*b*x + 8*a) + 48*e^(6*b*x + 6*a) - 24*e^(4*b*x + 4*a) - 16*e^(2*b*x + 2*a) - 3)*e^
(-8*b*x - 8*a) + 240*a - 3*e^(8*b*x + 8*a) - 16*e^(6*b*x + 6*a) - 24*e^(4*b*x + 4*a) + 48*e^(2*b*x + 2*a))/b