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

Optimal. Leaf size=67 \[ \frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5 \sinh (a+b x) \cosh ^3(a+b x)}{24 b}+\frac {5 \sinh (a+b x) \cosh (a+b x)}{16 b}+\frac {5 x}{16} \]

[Out]

5/16*x+5/16*cosh(b*x+a)*sinh(b*x+a)/b+5/24*cosh(b*x+a)^3*sinh(b*x+a)/b+1/6*cosh(b*x+a)^5*sinh(b*x+a)/b

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Rubi [A]  time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2635, 8} \[ \frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5 \sinh (a+b x) \cosh ^3(a+b x)}{24 b}+\frac {5 \sinh (a+b x) \cosh (a+b x)}{16 b}+\frac {5 x}{16} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^6,x]

[Out]

(5*x)/16 + (5*Cosh[a + b*x]*Sinh[a + b*x])/(16*b) + (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(24*b) + (Cosh[a + b*x]^
5*Sinh[a + b*x])/(6*b)

Rule 8

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

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 ^6(a+b x) \, dx &=\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}+\frac {5}{6} \int \cosh ^4(a+b x) \, dx\\ &=\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{24 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}+\frac {5}{8} \int \cosh ^2(a+b x) \, dx\\ &=\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}+\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{24 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}+\frac {5 \int 1 \, dx}{16}\\ &=\frac {5 x}{16}+\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}+\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{24 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{6 b}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 43, normalized size = 0.64 \[ \frac {45 \sinh (2 (a+b x))+9 \sinh (4 (a+b x))+\sinh (6 (a+b x))+60 a+60 b x}{192 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^6,x]

[Out]

(60*a + 60*b*x + 45*Sinh[2*(a + b*x)] + 9*Sinh[4*(a + b*x)] + Sinh[6*(a + b*x)])/(192*b)

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fricas [A]  time = 0.61, size = 90, normalized size = 1.34 \[ \frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 30 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{5} + 6 \, \cosh \left (b x + a\right )^{3} + 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(3*cosh(b*x + a)*sinh(b*x + a)^5 + 2*(5*cosh(b*x + a)^3 + 9*cosh(b*x + a))*sinh(b*x + a)^3 + 30*b*x + 3*(
cosh(b*x + a)^5 + 6*cosh(b*x + a)^3 + 15*cosh(b*x + a))*sinh(b*x + a))/b

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giac [A]  time = 0.14, size = 88, normalized size = 1.31 \[ \frac {5}{16} \, x + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {3 \, e^{\left (4 \, b x + 4 \, a\right )}}{128 \, b} + \frac {15 \, e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} - \frac {15 \, e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} - \frac {3 \, e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} - \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*x + 1/384*e^(6*b*x + 6*a)/b + 3/128*e^(4*b*x + 4*a)/b + 15/128*e^(2*b*x + 2*a)/b - 15/128*e^(-2*b*x - 2*a
)/b - 3/128*e^(-4*b*x - 4*a)/b - 1/384*e^(-6*b*x - 6*a)/b

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maple [A]  time = 0.22, size = 49, normalized size = 0.73 \[ \frac {\left (\frac {\left (\cosh ^{5}\left (b x +a \right )\right )}{6}+\frac {5 \left (\cosh ^{3}\left (b x +a \right )\right )}{24}+\frac {5 \cosh \left (b x +a \right )}{16}\right ) \sinh \left (b x +a \right )+\frac {5 b x}{16}+\frac {5 a}{16}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*((1/6*cosh(b*x+a)^5+5/24*cosh(b*x+a)^3+5/16*cosh(b*x+a))*sinh(b*x+a)+5/16*b*x+5/16*a)

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maxima [A]  time = 0.31, size = 86, normalized size = 1.28 \[ \frac {{\left (9 \, e^{\left (-2 \, b x - 2 \, a\right )} + 45 \, e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {5 \, {\left (b x + a\right )}}{16 \, b} - \frac {45 \, e^{\left (-2 \, b x - 2 \, a\right )} + 9 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(9*e^(-2*b*x - 2*a) + 45*e^(-4*b*x - 4*a) + 1)*e^(6*b*x + 6*a)/b + 5/16*(b*x + a)/b - 1/384*(45*e^(-2*b*
x - 2*a) + 9*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a))/b

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mupad [B]  time = 0.97, size = 42, normalized size = 0.63 \[ \frac {5\,x}{16}+\frac {\frac {15\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}+\frac {3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^6,x)

[Out]

(5*x)/16 + ((15*sinh(2*a + 2*b*x))/64 + (3*sinh(4*a + 4*b*x))/64 + sinh(6*a + 6*b*x)/192)/b

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sympy [A]  time = 2.99, size = 139, normalized size = 2.07 \[ \begin {cases} - \frac {5 x \sinh ^{6}{\left (a + b x \right )}}{16} + \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} - \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} + \frac {5 x \cosh ^{6}{\left (a + b x \right )}}{16} + \frac {5 \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b} - \frac {5 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac {11 \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \cosh ^{6}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-5*x*sinh(a + b*x)**6/16 + 15*x*sinh(a + b*x)**4*cosh(a + b*x)**2/16 - 15*x*sinh(a + b*x)**2*cosh(a
+ b*x)**4/16 + 5*x*cosh(a + b*x)**6/16 + 5*sinh(a + b*x)**5*cosh(a + b*x)/(16*b) - 5*sinh(a + b*x)**3*cosh(a
+ b*x)**3/(6*b) + 11*sinh(a + b*x)*cosh(a + b*x)**5/(16*b), Ne(b, 0)), (x*cosh(a)**6, True))

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