3.17 \(\int \cosh ^2(a+b x) \sinh ^6(a+b x) \, dx\)

Optimal. Leaf size=92 \[ \frac {\sinh ^5(a+b x) \cosh ^3(a+b x)}{8 b}-\frac {5 \sinh ^3(a+b x) \cosh ^3(a+b x)}{48 b}+\frac {5 \sinh (a+b x) \cosh ^3(a+b x)}{64 b}-\frac {5 \sinh (a+b x) \cosh (a+b x)}{128 b}-\frac {5 x}{128} \]

[Out]

-5/128*x-5/128*cosh(b*x+a)*sinh(b*x+a)/b+5/64*cosh(b*x+a)^3*sinh(b*x+a)/b-5/48*cosh(b*x+a)^3*sinh(b*x+a)^3/b+1
/8*cosh(b*x+a)^3*sinh(b*x+a)^5/b

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Rubi [A]  time = 0.10, antiderivative size = 92, 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} \[ \frac {\sinh ^5(a+b x) \cosh ^3(a+b x)}{8 b}-\frac {5 \sinh ^3(a+b x) \cosh ^3(a+b x)}{48 b}+\frac {5 \sinh (a+b x) \cosh ^3(a+b x)}{64 b}-\frac {5 \sinh (a+b x) \cosh (a+b x)}{128 b}-\frac {5 x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^2*Sinh[a + b*x]^6,x]

[Out]

(-5*x)/128 - (5*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) + (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(64*b) - (5*Cosh[a +
b*x]^3*Sinh[a + b*x]^3)/(48*b) + (Cosh[a + b*x]^3*Sinh[a + b*x]^5)/(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 ^2(a+b x) \sinh ^6(a+b x) \, dx &=\frac {\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}-\frac {5}{8} \int \cosh ^2(a+b x) \sinh ^4(a+b x) \, dx\\ &=-\frac {5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac {\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}+\frac {5}{16} \int \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx\\ &=\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac {5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac {\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}-\frac {5}{64} \int \cosh ^2(a+b x) \, dx\\ &=-\frac {5 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac {5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac {\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}-\frac {5 \int 1 \, dx}{128}\\ &=-\frac {5 x}{128}-\frac {5 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac {5 \cosh ^3(a+b x) \sinh ^3(a+b x)}{48 b}+\frac {\cosh ^3(a+b x) \sinh ^5(a+b x)}{8 b}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 52, normalized size = 0.57 \[ \frac {48 \sinh (2 (a+b x))+24 \sinh (4 (a+b x))-16 \sinh (6 (a+b x))+3 \sinh (8 (a+b x))-120 b x}{3072 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^2*Sinh[a + b*x]^6,x]

[Out]

(-120*b*x + 48*Sinh[2*(a + b*x)] + 24*Sinh[4*(a + b*x)] - 16*Sinh[6*(a + b*x)] + 3*Sinh[8*(a + b*x)])/(3072*b)

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fricas [A]  time = 0.43, size = 138, normalized size = 1.50 \[ \frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + 3 \, {\left (7 \, \cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + {\left (21 \, \cosh \left (b x + a\right )^{5} - 40 \, \cosh \left (b x + a\right )^{3} + 12 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 15 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{7} - 4 \, \cosh \left (b x + a\right )^{5} + 4 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{384 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^6,x, algorithm="fricas")

[Out]

1/384*(3*cosh(b*x + a)*sinh(b*x + a)^7 + 3*(7*cosh(b*x + a)^3 - 4*cosh(b*x + a))*sinh(b*x + a)^5 + (21*cosh(b*
x + a)^5 - 40*cosh(b*x + a)^3 + 12*cosh(b*x + a))*sinh(b*x + a)^3 - 15*b*x + 3*(cosh(b*x + a)^7 - 4*cosh(b*x +
a)^5 + 4*cosh(b*x + a)^3 + 4*cosh(b*x + a))*sinh(b*x + a))/b

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giac [A]  time = 0.21, size = 116, normalized size = 1.26 \[ -\frac {5}{128} \, x + \frac {e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} - \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b} + \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} - \frac {e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^6,x, algorithm="giac")

[Out]

-5/128*x + 1/2048*e^(8*b*x + 8*a)/b - 1/384*e^(6*b*x + 6*a)/b + 1/256*e^(4*b*x + 4*a)/b + 1/128*e^(2*b*x + 2*a
)/b - 1/128*e^(-2*b*x - 2*a)/b - 1/256*e^(-4*b*x - 4*a)/b + 1/384*e^(-6*b*x - 6*a)/b - 1/2048*e^(-8*b*x - 8*a)
/b

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maple [A]  time = 0.09, size = 79, normalized size = 0.86 \[ \frac {\frac {\left (\sinh ^{5}\left (b x +a \right )\right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{8}-\frac {5 \left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{48}+\frac {5 \left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{64}-\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{128}-\frac {5 b x}{128}-\frac {5 a}{128}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x+a)^2*sinh(b*x+a)^6,x)

[Out]

1/b*(1/8*sinh(b*x+a)^5*cosh(b*x+a)^3-5/48*cosh(b*x+a)^3*sinh(b*x+a)^3+5/64*cosh(b*x+a)^3*sinh(b*x+a)-5/128*cos
h(b*x+a)*sinh(b*x+a)-5/128*b*x-5/128*a)

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maxima [A]  time = 0.32, size = 110, normalized size = 1.20 \[ -\frac {{\left (16 \, e^{\left (-2 \, b x - 2 \, a\right )} - 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 48 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3\right )} e^{\left (8 \, b x + 8 \, a\right )}}{6144 \, b} - \frac {5 \, {\left (b x + a\right )}}{128 \, b} - \frac {48 \, e^{\left (-2 \, b x - 2 \, a\right )} + 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 16 \, e^{\left (-6 \, b x - 6 \, a\right )} + 3 \, e^{\left (-8 \, b x - 8 \, a\right )}}{6144 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)^2*sinh(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/6144*(16*e^(-2*b*x - 2*a) - 24*e^(-4*b*x - 4*a) - 48*e^(-6*b*x - 6*a) - 3)*e^(8*b*x + 8*a)/b - 5/128*(b*x +
a)/b - 1/6144*(48*e^(-2*b*x - 2*a) + 24*e^(-4*b*x - 4*a) - 16*e^(-6*b*x - 6*a) + 3*e^(-8*b*x - 8*a))/b

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mupad [B]  time = 1.73, size = 53, normalized size = 0.58 \[ \frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{128}-\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}+\frac {\mathrm {sinh}\left (8\,a+8\,b\,x\right )}{1024}}{b}-\frac {5\,x}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^2*sinh(a + b*x)^6,x)

[Out]

(sinh(2*a + 2*b*x)/64 + sinh(4*a + 4*b*x)/128 - sinh(6*a + 6*b*x)/192 + sinh(8*a + 8*b*x)/1024)/b - (5*x)/128

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sympy [A]  time = 8.32, size = 189, normalized size = 2.05 \[ \begin {cases} - \frac {5 x \sinh ^{8}{\left (a + b x \right )}}{128} + \frac {5 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} - \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} + \frac {5 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} - \frac {5 x \cosh ^{8}{\left (a + b x \right )}}{128} + \frac {5 \sinh ^{7}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{128 b} + \frac {73 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{384 b} - \frac {55 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{384 b} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sinh ^{6}{\relax (a )} \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)**2*sinh(b*x+a)**6,x)

[Out]

Piecewise((-5*x*sinh(a + b*x)**8/128 + 5*x*sinh(a + b*x)**6*cosh(a + b*x)**2/32 - 15*x*sinh(a + b*x)**4*cosh(a
+ b*x)**4/64 + 5*x*sinh(a + b*x)**2*cosh(a + b*x)**6/32 - 5*x*cosh(a + b*x)**8/128 + 5*sinh(a + b*x)**7*cosh(
a + b*x)/(128*b) + 73*sinh(a + b*x)**5*cosh(a + b*x)**3/(384*b) - 55*sinh(a + b*x)**3*cosh(a + b*x)**5/(384*b)
+ 5*sinh(a + b*x)*cosh(a + b*x)**7/(128*b), Ne(b, 0)), (x*sinh(a)**6*cosh(a)**2, True))

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