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

Optimal. Leaf size=124 \[ -\frac{3 a x^2 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}-\frac{6 a \cosh (c+d x)}{d^4}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{12 b x^2 \sinh (c+d x)}{d^3}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{24 b \sinh (c+d x)}{d^5}-\frac{24 b x \cosh (c+d x)}{d^4}+\frac{b x^4 \sinh (c+d x)}{d} \]

[Out]

(-6*a*Cosh[c + d*x])/d^4 - (24*b*x*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + d*x])/d^2 - (4*b*x^3*Cosh[c + d*x])/
d^2 + (24*b*Sinh[c + d*x])/d^5 + (6*a*x*Sinh[c + d*x])/d^3 + (12*b*x^2*Sinh[c + d*x])/d^3 + (a*x^3*Sinh[c + d*
x])/d + (b*x^4*Sinh[c + d*x])/d

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Rubi [A]  time = 0.320678, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6742, 3296, 2638, 2637} \[ -\frac{3 a x^2 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}-\frac{6 a \cosh (c+d x)}{d^4}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{12 b x^2 \sinh (c+d x)}{d^3}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{24 b \sinh (c+d x)}{d^5}-\frac{24 b x \cosh (c+d x)}{d^4}+\frac{b x^4 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)*Cosh[c + d*x],x]

[Out]

(-6*a*Cosh[c + d*x])/d^4 - (24*b*x*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + d*x])/d^2 - (4*b*x^3*Cosh[c + d*x])/
d^2 + (24*b*Sinh[c + d*x])/d^5 + (6*a*x*Sinh[c + d*x])/d^3 + (12*b*x^2*Sinh[c + d*x])/d^3 + (a*x^3*Sinh[c + d*
x])/d + (b*x^4*Sinh[c + d*x])/d

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^3 (a+b x) \cosh (c+d x) \, dx &=\int \left (a x^3 \cosh (c+d x)+b x^4 \cosh (c+d x)\right ) \, dx\\ &=a \int x^3 \cosh (c+d x) \, dx+b \int x^4 \cosh (c+d x) \, dx\\ &=\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}-\frac{(3 a) \int x^2 \sinh (c+d x) \, dx}{d}-\frac{(4 b) \int x^3 \sinh (c+d x) \, dx}{d}\\ &=-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}+\frac{(6 a) \int x \cosh (c+d x) \, dx}{d^2}+\frac{(12 b) \int x^2 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{12 b x^2 \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}-\frac{(6 a) \int \sinh (c+d x) \, dx}{d^3}-\frac{(24 b) \int x \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{24 b x \cosh (c+d x)}{d^4}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{12 b x^2 \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}+\frac{(24 b) \int \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{24 b x \cosh (c+d x)}{d^4}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{4 b x^3 \cosh (c+d x)}{d^2}+\frac{24 b \sinh (c+d x)}{d^5}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{12 b x^2 \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^4 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.139607, size = 82, normalized size = 0.66 \[ \frac{\left (a d^2 x \left (d^2 x^2+6\right )+b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \sinh (c+d x)-d \left (3 a \left (d^2 x^2+2\right )+4 b x \left (d^2 x^2+6\right )\right ) \cosh (c+d x)}{d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(d*(3*a*(2 + d^2*x^2) + 4*b*x*(6 + d^2*x^2))*Cosh[c + d*x]) + (a*d^2*x*(6 + d^2*x^2) + b*(24 + 12*d^2*x^2 +
d^4*x^4))*Sinh[c + d*x])/d^5

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Maple [B]  time = 0.011, size = 356, normalized size = 2.9 \begin{align*}{\frac{1}{{d}^{4}} \left ({\frac{b \left ( \left ( dx+c \right ) ^{4}\sinh \left ( dx+c \right ) -4\, \left ( dx+c \right ) ^{3}\cosh \left ( dx+c \right ) +12\, \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -24\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +24\,\sinh \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{cb \left ( \left ( dx+c \right ) ^{3}\sinh \left ( dx+c \right ) -3\, \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) +6\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) -6\,\cosh \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{b{c}^{2} \left ( \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +2\,\sinh \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{b{c}^{3} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}}+{\frac{b{c}^{4}\sinh \left ( dx+c \right ) }{d}}+a \left ( \left ( dx+c \right ) ^{3}\sinh \left ( dx+c \right ) -3\, \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) +6\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) -6\,\cosh \left ( dx+c \right ) \right ) -3\,ac \left ( \left ( dx+c \right ) ^{2}\sinh \left ( dx+c \right ) -2\, \left ( dx+c \right ) \cosh \left ( dx+c \right ) +2\,\sinh \left ( dx+c \right ) \right ) +3\,a{c}^{2} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) -a{c}^{3}\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(b/d*((d*x+c)^4*sinh(d*x+c)-4*(d*x+c)^3*cosh(d*x+c)+12*(d*x+c)^2*sinh(d*x+c)-24*(d*x+c)*cosh(d*x+c)+24*s
inh(d*x+c))-4*b*c/d*((d*x+c)^3*sinh(d*x+c)-3*(d*x+c)^2*cosh(d*x+c)+6*(d*x+c)*sinh(d*x+c)-6*cosh(d*x+c))+6*b/d*
c^2*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-4*b/d*c^3*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))+b*
c^4/d*sinh(d*x+c)+a*((d*x+c)^3*sinh(d*x+c)-3*(d*x+c)^2*cosh(d*x+c)+6*(d*x+c)*sinh(d*x+c)-6*cosh(d*x+c))-3*a*c*
((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))+3*a*c^2*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-a*c^3*si
nh(d*x+c))

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Maxima [A]  time = 1.14939, size = 313, normalized size = 2.52 \begin{align*} -\frac{1}{40} \, d{\left (\frac{5 \,{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{5}} + \frac{5 \,{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a e^{\left (-d x - c\right )}}{d^{5}} + \frac{4 \,{\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{6}} + \frac{4 \,{\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac{1}{20} \,{\left (4 \, b x^{5} + 5 \, a x^{4}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*d*(5*(d^4*x^4*e^c - 4*d^3*x^3*e^c + 12*d^2*x^2*e^c - 24*d*x*e^c + 24*e^c)*a*e^(d*x)/d^5 + 5*(d^4*x^4 + 4
*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*a*e^(-d*x - c)/d^5 + 4*(d^5*x^5*e^c - 5*d^4*x^4*e^c + 20*d^3*x^3*e^c - 60
*d^2*x^2*e^c + 120*d*x*e^c - 120*e^c)*b*e^(d*x)/d^6 + 4*(d^5*x^5 + 5*d^4*x^4 + 20*d^3*x^3 + 60*d^2*x^2 + 120*d
*x + 120)*b*e^(-d*x - c)/d^6) + 1/20*(4*b*x^5 + 5*a*x^4)*cosh(d*x + c)

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Fricas [A]  time = 1.94238, size = 193, normalized size = 1.56 \begin{align*} -\frac{{\left (4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} + 24 \, b d x + 6 \, a d\right )} \cosh \left (d x + c\right ) -{\left (b d^{4} x^{4} + a d^{4} x^{3} + 12 \, b d^{2} x^{2} + 6 \, a d^{2} x + 24 \, b\right )} \sinh \left (d x + c\right )}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((4*b*d^3*x^3 + 3*a*d^3*x^2 + 24*b*d*x + 6*a*d)*cosh(d*x + c) - (b*d^4*x^4 + a*d^4*x^3 + 12*b*d^2*x^2 + 6*a*d
^2*x + 24*b)*sinh(d*x + c))/d^5

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Sympy [A]  time = 3.63855, size = 151, normalized size = 1.22 \begin{align*} \begin{cases} \frac{a x^{3} \sinh{\left (c + d x \right )}}{d} - \frac{3 a x^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{6 a x \sinh{\left (c + d x \right )}}{d^{3}} - \frac{6 a \cosh{\left (c + d x \right )}}{d^{4}} + \frac{b x^{4} \sinh{\left (c + d x \right )}}{d} - \frac{4 b x^{3} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{12 b x^{2} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{24 b x \cosh{\left (c + d x \right )}}{d^{4}} + \frac{24 b \sinh{\left (c + d x \right )}}{d^{5}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{4}}{4} + \frac{b x^{5}}{5}\right ) \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x**3*sinh(c + d*x)/d - 3*a*x**2*cosh(c + d*x)/d**2 + 6*a*x*sinh(c + d*x)/d**3 - 6*a*cosh(c + d*x)
/d**4 + b*x**4*sinh(c + d*x)/d - 4*b*x**3*cosh(c + d*x)/d**2 + 12*b*x**2*sinh(c + d*x)/d**3 - 24*b*x*cosh(c +
d*x)/d**4 + 24*b*sinh(c + d*x)/d**5, Ne(d, 0)), ((a*x**4/4 + b*x**5/5)*cosh(c), True))

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Giac [A]  time = 1.21328, size = 205, normalized size = 1.65 \begin{align*} \frac{{\left (b d^{4} x^{4} + a d^{4} x^{3} - 4 \, b d^{3} x^{3} - 3 \, a d^{3} x^{2} + 12 \, b d^{2} x^{2} + 6 \, a d^{2} x - 24 \, b d x - 6 \, a d + 24 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac{{\left (b d^{4} x^{4} + a d^{4} x^{3} + 4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} + 12 \, b d^{2} x^{2} + 6 \, a d^{2} x + 24 \, b d x + 6 \, a d + 24 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d^4*x^4 + a*d^4*x^3 - 4*b*d^3*x^3 - 3*a*d^3*x^2 + 12*b*d^2*x^2 + 6*a*d^2*x - 24*b*d*x - 6*a*d + 24*b)*e
^(d*x + c)/d^5 - 1/2*(b*d^4*x^4 + a*d^4*x^3 + 4*b*d^3*x^3 + 3*a*d^3*x^2 + 12*b*d^2*x^2 + 6*a*d^2*x + 24*b*d*x
+ 6*a*d + 24*b)*e^(-d*x - c)/d^5