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]

3/128*x+3/128*cosh(b*x+a)*sinh(b*x+a)/b+1/64*cosh(b*x+a)^3*sinh(b*x+a)/b-1/16*cosh(b*x+a)^5*sinh(b*x+a)/b+1/8*
cosh(b*x+a)^5*sinh(b*x+a)^3/b

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Rubi [A]  time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.05, 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)

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fricas [A]  time = 0.39, size = 97, normalized size = 1.08 \[ \frac {7 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + {\left (7 \, \cosh \left (b x + a\right )^{5} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b x + {\left (\cosh \left (b x + a\right )^{7} - 4 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )}{128 \, b} \]

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

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giac [A]  time = 0.23, size = 60, normalized size = 0.67 \[ \frac {3}{128} \, x + \frac {e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} - \frac {e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b} + \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b} - \frac {e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \]

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]

3/128*x + 1/2048*e^(8*b*x + 8*a)/b - 1/256*e^(4*b*x + 4*a)/b + 1/256*e^(-4*b*x - 4*a)/b - 1/2048*e^(-8*b*x - 8
*a)/b

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maple [A]  time = 0.33, size = 74, normalized size = 0.82 \[ \frac {\frac {\left (\sinh ^{3}\left (b x +a \right )\right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{8}-\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{16}+\frac {\left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )}{16}+\frac {3 b x}{128}+\frac {3 a}{128}}{b} \]

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)

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maxima [A]  time = 0.31, size = 66, normalized size = 0.73 \[ -\frac {{\left (8 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} + \frac {3 \, {\left (b x + a\right )}}{128 \, b} + \frac {8 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \]

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

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mupad [B]  time = 0.20, size = 32, normalized size = 0.36 \[ \frac {3\,x}{128}-\frac {\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{128}-\frac {\mathrm {sinh}\left (8\,a+8\,b\,x\right )}{1024}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x)/128 - (sinh(4*a + 4*b*x)/128 - sinh(8*a + 8*b*x)/1024)/b

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sympy [A]  time = 8.29, size = 189, normalized size = 2.10 \[ \begin {cases} \frac {3 x \sinh ^{8}{\left (a + b x \right )}}{128} - \frac {3 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} + \frac {9 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} + \frac {3 x \cosh ^{8}{\left (a + b x \right )}}{128} - \frac {3 \sinh ^{7}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{128 b} + \frac {11 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac {11 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{128 b} - \frac {3 \sinh {\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sinh ^{4}{\relax (a )} \cosh ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]

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))

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