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

Optimal. Leaf size=134 \[ \frac{\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac{\sinh ^3(a+b x) \cosh ^7(a+b x)}{24 b}+\frac{\sinh (a+b x) \cosh ^7(a+b x)}{64 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{384 b}-\frac{5 \sinh (a+b x) \cosh ^3(a+b x)}{1536 b}-\frac{5 \sinh (a+b x) \cosh (a+b x)}{1024 b}-\frac{5 x}{1024} \]

[Out]

(-5*x)/1024 - (5*Cosh[a + b*x]*Sinh[a + b*x])/(1024*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(1536*b) - (Cosh[a
+ b*x]^5*Sinh[a + b*x])/(384*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^7*Sinh[a + b*x]^3)/(
24*b) + (Cosh[a + b*x]^7*Sinh[a + b*x]^5)/(12*b)

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Rubi [A]  time = 0.132118, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac{\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac{\sinh ^3(a+b x) \cosh ^7(a+b x)}{24 b}+\frac{\sinh (a+b x) \cosh ^7(a+b x)}{64 b}-\frac{\sinh (a+b x) \cosh ^5(a+b x)}{384 b}-\frac{5 \sinh (a+b x) \cosh ^3(a+b x)}{1536 b}-\frac{5 \sinh (a+b x) \cosh (a+b x)}{1024 b}-\frac{5 x}{1024} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*x)/1024 - (5*Cosh[a + b*x]*Sinh[a + b*x])/(1024*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(1536*b) - (Cosh[a
+ b*x]^5*Sinh[a + b*x])/(384*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^7*Sinh[a + b*x]^3)/(
24*b) + (Cosh[a + b*x]^7*Sinh[a + b*x]^5)/(12*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 ^6(a+b x) \, dx &=\frac{\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac{5}{12} \int \cosh ^6(a+b x) \sinh ^4(a+b x) \, dx\\ &=-\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}+\frac{1}{8} \int \cosh ^6(a+b x) \sinh ^2(a+b x) \, dx\\ &=\frac{\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac{1}{64} \int \cosh ^6(a+b x) \, dx\\ &=-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac{5}{384} \int \cosh ^4(a+b x) \, dx\\ &=-\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{1536 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac{5}{512} \int \cosh ^2(a+b x) \, dx\\ &=-\frac{5 \cosh (a+b x) \sinh (a+b x)}{1024 b}-\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{1536 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}-\frac{5 \int 1 \, dx}{1024}\\ &=-\frac{5 x}{1024}-\frac{5 \cosh (a+b x) \sinh (a+b x)}{1024 b}-\frac{5 \cosh ^3(a+b x) \sinh (a+b x)}{1536 b}-\frac{\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac{\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac{\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac{\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b}\\ \end{align*}

Mathematica [A]  time = 0.0821472, size = 43, normalized size = 0.32 \[ \frac{45 \sinh (4 (a+b x))-9 \sinh (8 (a+b x))+\sinh (12 (a+b x))-120 a-120 b x}{24576 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-120*a - 120*b*x + 45*Sinh[4*(a + b*x)] - 9*Sinh[8*(a + b*x)] + Sinh[12*(a + b*x)])/(24576*b)

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Maple [A]  time = 0.013, size = 102, 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 ) ^{7}}{12}}-{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3} \left ( \cosh \left ( bx+a \right ) \right ) ^{7}}{24}}+{\frac{\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{7}}{64}}-{\frac{\sinh \left ( bx+a \right ) }{64} \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}{1024}}-{\frac{5\,a}{1024}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/12*sinh(b*x+a)^5*cosh(b*x+a)^7-1/24*sinh(b*x+a)^3*cosh(b*x+a)^7+1/64*sinh(b*x+a)*cosh(b*x+a)^7-1/64*(1/
6*cosh(b*x+a)^5+5/24*cosh(b*x+a)^3+5/16*cosh(b*x+a))*sinh(b*x+a)-5/1024*b*x-5/1024*a)

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Maxima [A]  time = 1.0582, size = 116, normalized size = 0.87 \begin{align*} -\frac{{\left (9 \, e^{\left (-4 \, b x - 4 \, a\right )} - 45 \, e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )} e^{\left (12 \, b x + 12 \, a\right )}}{49152 \, b} - \frac{5 \,{\left (b x + a\right )}}{1024 \, b} - \frac{45 \, e^{\left (-4 \, b x - 4 \, a\right )} - 9 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )}}{49152 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49152*(9*e^(-4*b*x - 4*a) - 45*e^(-8*b*x - 8*a) - 1)*e^(12*b*x + 12*a)/b - 5/1024*(b*x + a)/b - 1/49152*(45
*e^(-4*b*x - 4*a) - 9*e^(-8*b*x - 8*a) + e^(-12*b*x - 12*a))/b

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Fricas [A]  time = 2.0948, size = 498, normalized size = 3.72 \begin{align*} \frac{55 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{9} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{11} + 18 \,{\left (11 \, \cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} + 18 \,{\left (11 \, \cosh \left (b x + a\right )^{7} - 7 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{5} +{\left (55 \, \cosh \left (b x + a\right )^{9} - 126 \, \cosh \left (b x + a\right )^{5} + 45 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 3 \,{\left (\cosh \left (b x + a\right )^{11} - 6 \, \cosh \left (b x + a\right )^{7} + 15 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + 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)^6,x, algorithm="fricas")

[Out]

1/6144*(55*cosh(b*x + a)^3*sinh(b*x + a)^9 + 3*cosh(b*x + a)*sinh(b*x + a)^11 + 18*(11*cosh(b*x + a)^5 - cosh(
b*x + a))*sinh(b*x + a)^7 + 18*(11*cosh(b*x + a)^7 - 7*cosh(b*x + a)^3)*sinh(b*x + a)^5 + (55*cosh(b*x + a)^9
- 126*cosh(b*x + a)^5 + 45*cosh(b*x + a))*sinh(b*x + a)^3 - 30*b*x + 3*(cosh(b*x + a)^11 - 6*cosh(b*x + a)^7 +
15*cosh(b*x + a)^3)*sinh(b*x + a))/b

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Sympy [A]  time = 79.0808, size = 277, normalized size = 2.07 \begin{align*} \begin{cases} - \frac{5 x \sinh ^{12}{\left (a + b x \right )}}{1024} + \frac{15 x \sinh ^{10}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{512} - \frac{75 x \sinh ^{8}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{1024} + \frac{25 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{256} - \frac{75 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{8}{\left (a + b x \right )}}{1024} + \frac{15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{10}{\left (a + b x \right )}}{512} - \frac{5 x \cosh ^{12}{\left (a + b x \right )}}{1024} + \frac{5 \sinh ^{11}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{1024 b} - \frac{85 \sinh ^{9}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3072 b} + \frac{33 \sinh ^{7}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{512 b} + \frac{33 \sinh ^{5}{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{512 b} - \frac{85 \sinh ^{3}{\left (a + b x \right )} \cosh ^{9}{\left (a + b x \right )}}{3072 b} + \frac{5 \sinh{\left (a + b x \right )} \cosh ^{11}{\left (a + b x \right )}}{1024 b} & \text{for}\: b \neq 0 \\x \sinh ^{6}{\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)**6,x)

[Out]

Piecewise((-5*x*sinh(a + b*x)**12/1024 + 15*x*sinh(a + b*x)**10*cosh(a + b*x)**2/512 - 75*x*sinh(a + b*x)**8*c
osh(a + b*x)**4/1024 + 25*x*sinh(a + b*x)**6*cosh(a + b*x)**6/256 - 75*x*sinh(a + b*x)**4*cosh(a + b*x)**8/102
4 + 15*x*sinh(a + b*x)**2*cosh(a + b*x)**10/512 - 5*x*cosh(a + b*x)**12/1024 + 5*sinh(a + b*x)**11*cosh(a + b*
x)/(1024*b) - 85*sinh(a + b*x)**9*cosh(a + b*x)**3/(3072*b) + 33*sinh(a + b*x)**7*cosh(a + b*x)**5/(512*b) + 3
3*sinh(a + b*x)**5*cosh(a + b*x)**7/(512*b) - 85*sinh(a + b*x)**3*cosh(a + b*x)**9/(3072*b) + 5*sinh(a + b*x)*
cosh(a + b*x)**11/(1024*b), Ne(b, 0)), (x*sinh(a)**6*cosh(a)**6, True))

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Giac [A]  time = 1.21426, size = 124, normalized size = 0.93 \begin{align*} -\frac{240 \, b x -{\left (110 \, e^{\left (12 \, b x + 12 \, a\right )} - 45 \, e^{\left (8 \, b x + 8 \, a\right )} + 9 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-12 \, b x - 12 \, a\right )} + 240 \, a - e^{\left (12 \, b x + 12 \, a\right )} + 9 \, e^{\left (8 \, b x + 8 \, a\right )} - 45 \, e^{\left (4 \, b x + 4 \, a\right )}}{49152 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49152*(240*b*x - (110*e^(12*b*x + 12*a) - 45*e^(8*b*x + 8*a) + 9*e^(4*b*x + 4*a) - 1)*e^(-12*b*x - 12*a) +
240*a - e^(12*b*x + 12*a) + 9*e^(8*b*x + 8*a) - 45*e^(4*b*x + 4*a))/b