3.20 \(\int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx\)

Optimal. Leaf size=113 \[ \frac{\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}-\frac{\sinh ^3(a+b x) \cosh ^5(a+b x)}{16 b}+\frac{\sinh (a+b x) \cosh ^5(a+b x)}{32 b}-\frac{\sinh (a+b x) \cosh ^3(a+b x)}{128 b}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{256 b}-\frac{3 x}{256} \]

[Out]

(-3*x)/256 - (3*Cosh[a + b*x]*Sinh[a + b*x])/(256*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x
]^5*Sinh[a + b*x])/(32*b) - (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^5)/(10*b
)

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Rubi [A]  time = 0.118413, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac{\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}-\frac{\sinh ^3(a+b x) \cosh ^5(a+b x)}{16 b}+\frac{\sinh (a+b x) \cosh ^5(a+b x)}{32 b}-\frac{\sinh (a+b x) \cosh ^3(a+b x)}{128 b}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{256 b}-\frac{3 x}{256} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^4*Sinh[a + b*x]^6,x]

[Out]

(-3*x)/256 - (3*Cosh[a + b*x]*Sinh[a + b*x])/(256*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x
]^5*Sinh[a + b*x])/(32*b) - (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^5)/(10*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 ^6(a+b x) \, dx &=\frac{\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac{1}{2} \int \cosh ^4(a+b x) \sinh ^4(a+b x) \, dx\\ &=-\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}+\frac{3}{16} \int \cosh ^4(a+b x) \sinh ^2(a+b x) \, dx\\ &=\frac{\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac{1}{32} \int \cosh ^4(a+b x) \, dx\\ &=-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac{3}{128} \int \cosh ^2(a+b x) \, dx\\ &=-\frac{3 \cosh (a+b x) \sinh (a+b x)}{256 b}-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}-\frac{3 \int 1 \, dx}{256}\\ &=-\frac{3 x}{256}-\frac{3 \cosh (a+b x) \sinh (a+b x)}{256 b}-\frac{\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac{\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac{\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac{\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b}\\ \end{align*}

Mathematica [A]  time = 0.172735, size = 62, normalized size = 0.55 \[ \frac{20 \sinh (2 (a+b x))+40 \sinh (4 (a+b x))-10 \sinh (6 (a+b x))-5 \sinh (8 (a+b x))+2 \sinh (10 (a+b x))-120 b x}{10240 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^4*Sinh[a + b*x]^6,x]

[Out]

(-120*b*x + 20*Sinh[2*(a + b*x)] + 40*Sinh[4*(a + b*x)] - 10*Sinh[6*(a + b*x)] - 5*Sinh[8*(a + b*x)] + 2*Sinh[
10*(a + b*x)])/(10240*b)

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Maple [A]  time = 0.013, size = 92, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{5} \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{10}}-{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3} \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{16}}+{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{5}}{32}}-{\frac{\sinh \left ( bx+a \right ) }{32} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( bx+a \right ) }{8}} \right ) }-{\frac{3\,bx}{256}}-{\frac{3\,a}{256}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^4*sinh(b*x+a)^6,x)

[Out]

1/b*(1/10*sinh(b*x+a)^5*cosh(b*x+a)^5-1/16*sinh(b*x+a)^3*cosh(b*x+a)^5+1/32*sinh(b*x+a)*cosh(b*x+a)^5-1/32*(1/
4*cosh(b*x+a)^3+3/8*cosh(b*x+a))*sinh(b*x+a)-3/256*b*x-3/256*a)

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Maxima [A]  time = 1.13314, size = 178, normalized size = 1.58 \begin{align*} -\frac{{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} - 40 \, e^{\left (-6 \, b x - 6 \, a\right )} - 20 \, e^{\left (-8 \, b x - 8 \, a\right )} - 2\right )} e^{\left (10 \, b x + 10 \, a\right )}}{20480 \, b} - \frac{3 \,{\left (b x + a\right )}}{256 \, b} - \frac{20 \, e^{\left (-2 \, b x - 2 \, a\right )} + 40 \, e^{\left (-4 \, b x - 4 \, a\right )} - 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + 2 \, e^{\left (-10 \, b x - 10 \, a\right )}}{20480 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^4*sinh(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/20480*(5*e^(-2*b*x - 2*a) + 10*e^(-4*b*x - 4*a) - 40*e^(-6*b*x - 6*a) - 20*e^(-8*b*x - 8*a) - 2)*e^(10*b*x
+ 10*a)/b - 3/256*(b*x + a)/b - 1/20480*(20*e^(-2*b*x - 2*a) + 40*e^(-4*b*x - 4*a) - 10*e^(-6*b*x - 6*a) - 5*e
^(-8*b*x - 8*a) + 2*e^(-10*b*x - 10*a))/b

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Fricas [A]  time = 2.07082, size = 544, normalized size = 4.81 \begin{align*} \frac{5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + 10 \,{\left (6 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} +{\left (126 \, \cosh \left (b x + a\right )^{5} - 70 \, \cosh \left (b x + a\right )^{3} - 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 10 \,{\left (6 \, \cosh \left (b x + a\right )^{7} - 7 \, \cosh \left (b x + a\right )^{5} - 5 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 5 \,{\left (\cosh \left (b x + a\right )^{9} - 2 \, \cosh \left (b x + a\right )^{7} - 3 \, \cosh \left (b x + a\right )^{5} + 8 \, \cosh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{2560 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^4*sinh(b*x+a)^6,x, algorithm="fricas")

[Out]

1/2560*(5*cosh(b*x + a)*sinh(b*x + a)^9 + 10*(6*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^7 + (126*cosh(b
*x + a)^5 - 70*cosh(b*x + a)^3 - 15*cosh(b*x + a))*sinh(b*x + a)^5 + 10*(6*cosh(b*x + a)^7 - 7*cosh(b*x + a)^5
- 5*cosh(b*x + a)^3 + 4*cosh(b*x + a))*sinh(b*x + a)^3 - 30*b*x + 5*(cosh(b*x + a)^9 - 2*cosh(b*x + a)^7 - 3*
cosh(b*x + a)^5 + 8*cosh(b*x + a)^3 + 2*cosh(b*x + a))*sinh(b*x + a))/b

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Sympy [A]  time = 31.0621, size = 231, normalized size = 2.04 \begin{align*} \begin{cases} \frac{3 x \sinh ^{10}{\left (a + b x \right )}}{256} - \frac{15 x \sinh ^{8}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{256} + \frac{15 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{128} - \frac{15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{128} + \frac{15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{8}{\left (a + b x \right )}}{256} - \frac{3 x \cosh ^{10}{\left (a + b x \right )}}{256} - \frac{3 \sinh ^{9}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{256 b} + \frac{7 \sinh ^{7}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac{\sinh ^{5}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{10 b} - \frac{7 \sinh ^{3}{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} + \frac{3 \sinh{\left (a + b x \right )} \cosh ^{9}{\left (a + b x \right )}}{256 b} & \text{for}\: b \neq 0 \\x \sinh ^{6}{\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)**6,x)

[Out]

Piecewise((3*x*sinh(a + b*x)**10/256 - 15*x*sinh(a + b*x)**8*cosh(a + b*x)**2/256 + 15*x*sinh(a + b*x)**6*cosh
(a + b*x)**4/128 - 15*x*sinh(a + b*x)**4*cosh(a + b*x)**6/128 + 15*x*sinh(a + b*x)**2*cosh(a + b*x)**8/256 - 3
*x*cosh(a + b*x)**10/256 - 3*sinh(a + b*x)**9*cosh(a + b*x)/(256*b) + 7*sinh(a + b*x)**7*cosh(a + b*x)**3/(128
*b) + sinh(a + b*x)**5*cosh(a + b*x)**5/(10*b) - 7*sinh(a + b*x)**3*cosh(a + b*x)**7/(128*b) + 3*sinh(a + b*x)
*cosh(a + b*x)**9/(256*b), Ne(b, 0)), (x*sinh(a)**6*cosh(a)**4, True))

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Giac [A]  time = 1.22946, size = 184, normalized size = 1.63 \begin{align*} -\frac{240 \, b x -{\left (274 \, e^{\left (10 \, b x + 10 \, a\right )} - 20 \, e^{\left (8 \, b x + 8 \, a\right )} - 40 \, e^{\left (6 \, b x + 6 \, a\right )} + 10 \, e^{\left (4 \, b x + 4 \, a\right )} + 5 \, e^{\left (2 \, b x + 2 \, a\right )} - 2\right )} e^{\left (-10 \, b x - 10 \, a\right )} + 240 \, a - 2 \, e^{\left (10 \, b x + 10 \, a\right )} + 5 \, e^{\left (8 \, b x + 8 \, a\right )} + 10 \, e^{\left (6 \, b x + 6 \, a\right )} - 40 \, e^{\left (4 \, b x + 4 \, a\right )} - 20 \, e^{\left (2 \, b x + 2 \, a\right )}}{20480 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^4*sinh(b*x+a)^6,x, algorithm="giac")

[Out]

-1/20480*(240*b*x - (274*e^(10*b*x + 10*a) - 20*e^(8*b*x + 8*a) - 40*e^(6*b*x + 6*a) + 10*e^(4*b*x + 4*a) + 5*
e^(2*b*x + 2*a) - 2)*e^(-10*b*x - 10*a) + 240*a - 2*e^(10*b*x + 10*a) + 5*e^(8*b*x + 8*a) + 10*e^(6*b*x + 6*a)
- 40*e^(4*b*x + 4*a) - 20*e^(2*b*x + 2*a))/b