3.25 \(\int x \cosh ^4(a+b x) \, dx\)

Optimal. Leaf size=80 \[ -\frac{\cosh ^4(a+b x)}{16 b^2}-\frac{3 \cosh ^2(a+b x)}{16 b^2}+\frac{x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac{3 x \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac{3 x^2}{16} \]

[Out]

(3*x^2)/16 - (3*Cosh[a + b*x]^2)/(16*b^2) - Cosh[a + b*x]^4/(16*b^2) + (3*x*Cosh[a + b*x]*Sinh[a + b*x])/(8*b)
+ (x*Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)

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Rubi [A]  time = 0.0436869, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3310, 30} \[ -\frac{\cosh ^4(a+b x)}{16 b^2}-\frac{3 \cosh ^2(a+b x)}{16 b^2}+\frac{x \sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac{3 x \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac{3 x^2}{16} \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[a + b*x]^4,x]

[Out]

(3*x^2)/16 - (3*Cosh[a + b*x]^2)/(16*b^2) - Cosh[a + b*x]^4/(16*b^2) + (3*x*Cosh[a + b*x]*Sinh[a + b*x])/(8*b)
+ (x*Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.175357, size = 53, normalized size = 0.66 \[ -\frac{-4 b x (8 \sinh (2 (a+b x))+\sinh (4 (a+b x))+6 b x)+16 \cosh (2 (a+b x))+\cosh (4 (a+b x))}{128 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[a + b*x]^4,x]

[Out]

-(16*Cosh[2*(a + b*x)] + Cosh[4*(a + b*x)] - 4*b*x*(6*b*x + 8*Sinh[2*(a + b*x)] + Sinh[4*(a + b*x)]))/(128*b^2
)

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Maple [A]  time = 0.007, size = 120, normalized size = 1.5 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{ \left ( 3\,bx+3\,a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}+{\frac{3\, \left ( bx+a \right ) ^{2}}{16}}-{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{16}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4}}-a \left ( \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( bx+a \right ) }{8}} \right ) \sinh \left ( bx+a \right ) +{\frac{3\,bx}{8}}+{\frac{3\,a}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*(1/4*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^3+3/8*(b*x+a)*cosh(b*x+a)*sinh(b*x+a)+3/16*(b*x+a)^2-1/16*sinh(b*x+
a)^2*cosh(b*x+a)^2-1/4*cosh(b*x+a)^2-a*((1/4*cosh(b*x+a)^3+3/8*cosh(b*x+a))*sinh(b*x+a)+3/8*b*x+3/8*a))

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Maxima [A]  time = 1.07999, size = 130, normalized size = 1.62 \begin{align*} \frac{3}{16} \, x^{2} + \frac{{\left (4 \, b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{256 \, b^{2}} + \frac{{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{16 \, b^{2}} - \frac{{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{2}} - \frac{{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)^4,x, algorithm="maxima")

[Out]

3/16*x^2 + 1/256*(4*b*x*e^(4*a) - e^(4*a))*e^(4*b*x)/b^2 + 1/16*(2*b*x*e^(2*a) - e^(2*a))*e^(2*b*x)/b^2 - 1/16
*(2*b*x + 1)*e^(-2*b*x - 2*a)/b^2 - 1/256*(4*b*x + 1)*e^(-4*b*x - 4*a)/b^2

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Fricas [A]  time = 2.02112, size = 306, normalized size = 3.82 \begin{align*} \frac{16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 24 \, b^{2} x^{2} - \cosh \left (b x + a\right )^{4} - \sinh \left (b x + a\right )^{4} - 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 8\right )} \sinh \left (b x + a\right )^{2} - 16 \, \cosh \left (b x + a\right )^{2} + 16 \,{\left (b x \cosh \left (b x + a\right )^{3} + 4 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{128 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)^4,x, algorithm="fricas")

[Out]

1/128*(16*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + 24*b^2*x^2 - cosh(b*x + a)^4 - sinh(b*x + a)^4 - 2*(3*cosh(b*x +
a)^2 + 8)*sinh(b*x + a)^2 - 16*cosh(b*x + a)^2 + 16*(b*x*cosh(b*x + a)^3 + 4*b*x*cosh(b*x + a))*sinh(b*x + a)
)/b^2

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Sympy [A]  time = 2.37883, size = 144, normalized size = 1.8 \begin{align*} \begin{cases} \frac{3 x^{2} \sinh ^{4}{\left (a + b x \right )}}{16} - \frac{3 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac{3 x^{2} \cosh ^{4}{\left (a + b x \right )}}{16} - \frac{3 x \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{5 x \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} + \frac{\sinh ^{4}{\left (a + b x \right )}}{4 b^{2}} - \frac{5 \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \cosh ^{4}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)**4,x)

[Out]

Piecewise((3*x**2*sinh(a + b*x)**4/16 - 3*x**2*sinh(a + b*x)**2*cosh(a + b*x)**2/8 + 3*x**2*cosh(a + b*x)**4/1
6 - 3*x*sinh(a + b*x)**3*cosh(a + b*x)/(8*b) + 5*x*sinh(a + b*x)*cosh(a + b*x)**3/(8*b) + sinh(a + b*x)**4/(4*
b**2) - 5*sinh(a + b*x)**2*cosh(a + b*x)**2/(16*b**2), Ne(b, 0)), (x**2*cosh(a)**4/2, True))

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Giac [A]  time = 1.27422, size = 116, normalized size = 1.45 \begin{align*} \frac{3}{16} \, x^{2} + \frac{{\left (4 \, b x - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b^{2}} + \frac{{\left (2 \, b x - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{2}} - \frac{{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{2}} - \frac{{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(b*x+a)^4,x, algorithm="giac")

[Out]

3/16*x^2 + 1/256*(4*b*x - 1)*e^(4*b*x + 4*a)/b^2 + 1/16*(2*b*x - 1)*e^(2*b*x + 2*a)/b^2 - 1/16*(2*b*x + 1)*e^(
-2*b*x - 2*a)/b^2 - 1/256*(4*b*x + 1)*e^(-4*b*x - 4*a)/b^2