3.19 \(\int (c+d x) \cosh ^3(a+b x) \, dx\)

Optimal. Leaf size=75 \[ -\frac{d \cosh ^3(a+b x)}{9 b^2}-\frac{2 d \cosh (a+b x)}{3 b^2}+\frac{2 (c+d x) \sinh (a+b x)}{3 b}+\frac{(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]

[Out]

(-2*d*Cosh[a + b*x])/(3*b^2) - (d*Cosh[a + b*x]^3)/(9*b^2) + (2*(c + d*x)*Sinh[a + b*x])/(3*b) + ((c + d*x)*Co
sh[a + b*x]^2*Sinh[a + b*x])/(3*b)

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Rubi [A]  time = 0.0442555, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3310, 3296, 2638} \[ -\frac{d \cosh ^3(a+b x)}{9 b^2}-\frac{2 d \cosh (a+b x)}{3 b^2}+\frac{2 (c+d x) \sinh (a+b x)}{3 b}+\frac{(c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Cosh[a + b*x]^3,x]

[Out]

(-2*d*Cosh[a + b*x])/(3*b^2) - (d*Cosh[a + b*x]^3)/(9*b^2) + (2*(c + d*x)*Sinh[a + b*x])/(3*b) + ((c + d*x)*Co
sh[a + b*x]^2*Sinh[a + b*x])/(3*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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (c+d x) \cosh ^3(a+b x) \, dx &=-\frac{d \cosh ^3(a+b x)}{9 b^2}+\frac{(c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac{2}{3} \int (c+d x) \cosh (a+b x) \, dx\\ &=-\frac{d \cosh ^3(a+b x)}{9 b^2}+\frac{2 (c+d x) \sinh (a+b x)}{3 b}+\frac{(c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac{(2 d) \int \sinh (a+b x) \, dx}{3 b}\\ &=-\frac{2 d \cosh (a+b x)}{3 b^2}-\frac{d \cosh ^3(a+b x)}{9 b^2}+\frac{2 (c+d x) \sinh (a+b x)}{3 b}+\frac{(c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.227353, size = 52, normalized size = 0.69 \[ -\frac{-3 b (c+d x) (9 \sinh (a+b x)+\sinh (3 (a+b x)))+27 d \cosh (a+b x)+d \cosh (3 (a+b x))}{36 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)*Cosh[a + b*x]^3,x]

[Out]

-(27*d*Cosh[a + b*x] + d*Cosh[3*(a + b*x)] - 3*b*(c + d*x)*(9*Sinh[a + b*x] + Sinh[3*(a + b*x)]))/(36*b^2)

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Maple [A]  time = 0.009, size = 115, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ({\frac{d}{b} \left ({\frac{ \left ( 2\,bx+2\,a \right ) \sinh \left ( bx+a \right ) }{3}}+{\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}}-{\frac{7\,\cosh \left ( bx+a \right ) }{9}}-{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{9}} \right ) }-{\frac{da\sinh \left ( bx+a \right ) }{b} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) }+c \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)*cosh(b*x+a)^3,x)

[Out]

1/b*(1/b*d*(2/3*(b*x+a)*sinh(b*x+a)+1/3*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^2-7/9*cosh(b*x+a)-1/9*sinh(b*x+a)^2*co
sh(b*x+a))-1/b*d*a*(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a)+c*(2/3+1/3*cosh(b*x+a)^2)*sinh(b*x+a))

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Maxima [B]  time = 1.07191, size = 193, normalized size = 2.57 \begin{align*} \frac{1}{72} \, d{\left (\frac{{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} + \frac{27 \,{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac{27 \,{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} - \frac{{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac{1}{24} \, c{\left (\frac{e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac{9 \, e^{\left (b x + a\right )}}{b} - \frac{9 \, e^{\left (-b x - a\right )}}{b} - \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cosh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/72*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 + 27*(b*x*e^a - e^a)*e^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x - a)/b
^2 - (3*b*x + 1)*e^(-3*b*x - 3*a)/b^2) + 1/24*c*(e^(3*b*x + 3*a)/b + 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b - e^(-
3*b*x - 3*a)/b)

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Fricas [A]  time = 1.96059, size = 257, normalized size = 3.43 \begin{align*} -\frac{d \cosh \left (b x + a\right )^{3} + 3 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \,{\left (b d x + b c\right )} \sinh \left (b x + a\right )^{3} + 27 \, d \cosh \left (b x + a\right ) - 9 \,{\left (3 \, b d x +{\left (b d x + b c\right )} \cosh \left (b x + a\right )^{2} + 3 \, b c\right )} \sinh \left (b x + a\right )}{36 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cosh(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/36*(d*cosh(b*x + a)^3 + 3*d*cosh(b*x + a)*sinh(b*x + a)^2 - 3*(b*d*x + b*c)*sinh(b*x + a)^3 + 27*d*cosh(b*x
+ a) - 9*(3*b*d*x + (b*d*x + b*c)*cosh(b*x + a)^2 + 3*b*c)*sinh(b*x + a))/b^2

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Sympy [A]  time = 1.33362, size = 126, normalized size = 1.68 \begin{align*} \begin{cases} - \frac{2 c \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac{c \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac{2 d x \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac{d x \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} + \frac{2 d \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{3 b^{2}} - \frac{7 d \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \cosh ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cosh(b*x+a)**3,x)

[Out]

Piecewise((-2*c*sinh(a + b*x)**3/(3*b) + c*sinh(a + b*x)*cosh(a + b*x)**2/b - 2*d*x*sinh(a + b*x)**3/(3*b) + d
*x*sinh(a + b*x)*cosh(a + b*x)**2/b + 2*d*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**2) - 7*d*cosh(a + b*x)**3/(9*b*
*2), Ne(b, 0)), ((c*x + d*x**2/2)*cosh(a)**3, True))

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Giac [A]  time = 1.30403, size = 132, normalized size = 1.76 \begin{align*} \frac{{\left (3 \, b d x + 3 \, b c - d\right )} e^{\left (3 \, b x + 3 \, a\right )}}{72 \, b^{2}} + \frac{3 \,{\left (b d x + b c - d\right )} e^{\left (b x + a\right )}}{8 \, b^{2}} - \frac{3 \,{\left (b d x + b c + d\right )} e^{\left (-b x - a\right )}}{8 \, b^{2}} - \frac{{\left (3 \, b d x + 3 \, b c + d\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{72 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)*cosh(b*x+a)^3,x, algorithm="giac")

[Out]

1/72*(3*b*d*x + 3*b*c - d)*e^(3*b*x + 3*a)/b^2 + 3/8*(b*d*x + b*c - d)*e^(b*x + a)/b^2 - 3/8*(b*d*x + b*c + d)
*e^(-b*x - a)/b^2 - 1/72*(3*b*d*x + 3*b*c + d)*e^(-3*b*x - 3*a)/b^2