3.1.19 \(\int \cosh ^4(a+b x) \sinh ^4(a+b x) \, dx\) [19]

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

[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 = 0.06, 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 = {2648, 2715, 8} \begin {gather*} \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} \end {gather*}

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 2648

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 2715

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] && Integ
erQ[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 = 0.03, size = 33, normalized size = 0.37 \begin {gather*} \frac {24 (a+b x)-8 \sinh (4 (a+b x))+\sinh (8 (a+b x))}{1024 b} \end {gather*}

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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Maple [A]
time = 1.74, size = 33, normalized size = 0.37

method	result	size
default	\(\frac {3 x}{128}-\frac {\sinh \left (4 b x +4 a \right )}{128 b}+\frac {\sinh \left (8 b x +8 a \right )}{1024 b}\)	\(33\)
risch	\(\frac {3 x}{128}+\frac {{\mathrm e}^{8 b x +8 a}}{2048 b}-\frac {{\mathrm e}^{4 b x +4 a}}{256 b}+\frac {{\mathrm e}^{-4 b x -4 a}}{256 b}-\frac {{\mathrm e}^{-8 b x -8 a}}{2048 b}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*x-1/128/b*sinh(4*b*x+4*a)+1/1024/b*sinh(8*b*x+8*a)

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Maxima [A]
time = 0.28, size = 66, normalized size = 0.73 \begin {gather*} -\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 {gather*}

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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Fricas [A]
time = 0.35, size = 97, normalized size = 1.08 \begin {gather*} \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 {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (80) = 160\).
time = 0.92, size = 189, normalized size = 2.10 \begin {gather*} \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 {gather*}

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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Giac [A]
time = 0.40, size = 60, normalized size = 0.67 \begin {gather*} \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} \end {gather*}

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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Mupad [B]
time = 0.20, size = 32, normalized size = 0.36 \begin {gather*} \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} \end {gather*}

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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