3.16 \(\int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx\)

Optimal. Leaf size=69 \[ \frac{\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}-\frac{\sinh (a+b x) \cosh ^3(a+b x)}{8 b}+\frac{\sinh (a+b x) \cosh (a+b x)}{16 b}+\frac{x}{16} \]

[Out]

x/16 + (Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a
+ b*x]^3)/(6*b)

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Rubi [A]  time = 0.0744825, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac{\sinh ^3(a+b x) \cosh ^3(a+b x)}{6 b}-\frac{\sinh (a+b x) \cosh ^3(a+b x)}{8 b}+\frac{\sinh (a+b x) \cosh (a+b x)}{16 b}+\frac{x}{16} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^2*Sinh[a + b*x]^4,x]

[Out]

x/16 + (Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a
+ b*x]^3)/(6*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 ^2(a+b x) \sinh ^4(a+b x) \, dx &=\frac{\cosh ^3(a+b x) \sinh ^3(a+b x)}{6 b}-\frac{1}{2} \int \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx\\ &=-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh ^3(a+b x) \sinh ^3(a+b x)}{6 b}+\frac{1}{8} \int \cosh ^2(a+b x) \, dx\\ &=\frac{\cosh (a+b x) \sinh (a+b x)}{16 b}-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh ^3(a+b x) \sinh ^3(a+b x)}{6 b}+\frac{\int 1 \, dx}{16}\\ &=\frac{x}{16}+\frac{\cosh (a+b x) \sinh (a+b x)}{16 b}-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{8 b}+\frac{\cosh ^3(a+b x) \sinh ^3(a+b x)}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0783301, size = 40, normalized size = 0.58 \[ \frac{-3 \sinh (2 (a+b x))-3 \sinh (4 (a+b x))+\sinh (6 (a+b x))+12 b x}{192 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^2*Sinh[a + b*x]^4,x]

[Out]

(12*b*x - 3*Sinh[2*(a + b*x)] - 3*Sinh[4*(a + b*x)] + Sinh[6*(a + b*x)])/(192*b)

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Maple [A]  time = 0.01, size = 61, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{6}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{8}}+{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{16}}+{\frac{bx}{16}}+{\frac{a}{16}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^2*sinh(b*x+a)^4,x)

[Out]

1/b*(1/6*cosh(b*x+a)^3*sinh(b*x+a)^3-1/8*cosh(b*x+a)^3*sinh(b*x+a)+1/16*cosh(b*x+a)*sinh(b*x+a)+1/16*b*x+1/16*
a)

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Maxima [A]  time = 0.979105, size = 119, normalized size = 1.72 \begin{align*} -\frac{{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac{b x + a}{16 \, b} + \frac{3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/384*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) - 1)*e^(6*b*x + 6*a)/b + 1/16*(b*x + a)/b + 1/384*(3*e^(-2*b*x
- 2*a) + 3*e^(-4*b*x - 4*a) - e^(-6*b*x - 6*a))/b

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Fricas [A]  time = 2.03687, size = 243, normalized size = 3.52 \begin{align*} \frac{3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \,{\left (5 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 6 \, b x + 3 \,{\left (\cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^4,x, algorithm="fricas")

[Out]

1/96*(3*cosh(b*x + a)*sinh(b*x + a)^5 + 2*(5*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^3 + 6*b*x + 3*(c
osh(b*x + a)^5 - 2*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a))/b

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Sympy [A]  time = 3.87096, size = 136, normalized size = 1.97 \begin{align*} \begin{cases} - \frac{x \sinh ^{6}{\left (a + b x \right )}}{16} + \frac{3 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} - \frac{3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} + \frac{x \cosh ^{6}{\left (a + b x \right )}}{16} + \frac{\sinh ^{5}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{16 b} + \frac{\sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} - \frac{\sinh{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text{for}\: b \neq 0 \\x \sinh ^{4}{\left (a \right )} \cosh ^{2}{\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)**2*sinh(b*x+a)**4,x)

[Out]

Piecewise((-x*sinh(a + b*x)**6/16 + 3*x*sinh(a + b*x)**4*cosh(a + b*x)**2/16 - 3*x*sinh(a + b*x)**2*cosh(a + b
*x)**4/16 + x*cosh(a + b*x)**6/16 + sinh(a + b*x)**5*cosh(a + b*x)/(16*b) + sinh(a + b*x)**3*cosh(a + b*x)**3/
(6*b) - sinh(a + b*x)*cosh(a + b*x)**5/(16*b), Ne(b, 0)), (x*sinh(a)**4*cosh(a)**2, True))

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Giac [A]  time = 1.22006, size = 122, normalized size = 1.77 \begin{align*} \frac{24 \, b x -{\left (22 \, e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )} + 24 \, a + e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} - 3 \, e^{\left (2 \, b x + 2 \, a\right )}}{384 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^4,x, algorithm="giac")

[Out]

1/384*(24*b*x - (22*e^(6*b*x + 6*a) - 3*e^(4*b*x + 4*a) - 3*e^(2*b*x + 2*a) + 1)*e^(-6*b*x - 6*a) + 24*a + e^(
6*b*x + 6*a) - 3*e^(4*b*x + 4*a) - 3*e^(2*b*x + 2*a))/b