3.40 \(\int x^3 (a+b x^2) \cosh (c+d x) \, dx\)

Optimal. Leaf size=139 \[ -\frac{3 a x^2 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}-\frac{6 a \cosh (c+d x)}{d^4}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{20 b x^3 \sinh (c+d x)}{d^3}-\frac{5 b x^4 \cosh (c+d x)}{d^2}-\frac{60 b x^2 \cosh (c+d x)}{d^4}+\frac{120 b x \sinh (c+d x)}{d^5}-\frac{120 b \cosh (c+d x)}{d^6}+\frac{b x^5 \sinh (c+d x)}{d} \]

[Out]

(-120*b*Cosh[c + d*x])/d^6 - (6*a*Cosh[c + d*x])/d^4 - (60*b*x^2*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + d*x])/
d^2 - (5*b*x^4*Cosh[c + d*x])/d^2 + (120*b*x*Sinh[c + d*x])/d^5 + (6*a*x*Sinh[c + d*x])/d^3 + (20*b*x^3*Sinh[c
+ d*x])/d^3 + (a*x^3*Sinh[c + d*x])/d + (b*x^5*Sinh[c + d*x])/d

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Rubi [A]  time = 0.255416, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5287, 3296, 2638} \[ -\frac{3 a x^2 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}-\frac{6 a \cosh (c+d x)}{d^4}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{20 b x^3 \sinh (c+d x)}{d^3}-\frac{5 b x^4 \cosh (c+d x)}{d^2}-\frac{60 b x^2 \cosh (c+d x)}{d^4}+\frac{120 b x \sinh (c+d x)}{d^5}-\frac{120 b \cosh (c+d x)}{d^6}+\frac{b x^5 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)*Cosh[c + d*x],x]

[Out]

(-120*b*Cosh[c + d*x])/d^6 - (6*a*Cosh[c + d*x])/d^4 - (60*b*x^2*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + d*x])/
d^2 - (5*b*x^4*Cosh[c + d*x])/d^2 + (120*b*x*Sinh[c + d*x])/d^5 + (6*a*x*Sinh[c + d*x])/d^3 + (20*b*x^3*Sinh[c
+ d*x])/d^3 + (a*x^3*Sinh[c + d*x])/d + (b*x^5*Sinh[c + d*x])/d

Rule 5287

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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 x^3 \left (a+b x^2\right ) \cosh (c+d x) \, dx &=\int \left (a x^3 \cosh (c+d x)+b x^5 \cosh (c+d x)\right ) \, dx\\ &=a \int x^3 \cosh (c+d x) \, dx+b \int x^5 \cosh (c+d x) \, dx\\ &=\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^5 \sinh (c+d x)}{d}-\frac{(3 a) \int x^2 \sinh (c+d x) \, dx}{d}-\frac{(5 b) \int x^4 \sinh (c+d x) \, dx}{d}\\ &=-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{5 b x^4 \cosh (c+d x)}{d^2}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^5 \sinh (c+d x)}{d}+\frac{(6 a) \int x \cosh (c+d x) \, dx}{d^2}+\frac{(20 b) \int x^3 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{5 b x^4 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{20 b x^3 \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^5 \sinh (c+d x)}{d}-\frac{(6 a) \int \sinh (c+d x) \, dx}{d^3}-\frac{(60 b) \int x^2 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{60 b x^2 \cosh (c+d x)}{d^4}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{5 b x^4 \cosh (c+d x)}{d^2}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{20 b x^3 \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^5 \sinh (c+d x)}{d}+\frac{(120 b) \int x \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{6 a \cosh (c+d x)}{d^4}-\frac{60 b x^2 \cosh (c+d x)}{d^4}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{5 b x^4 \cosh (c+d x)}{d^2}+\frac{120 b x \sinh (c+d x)}{d^5}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{20 b x^3 \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^5 \sinh (c+d x)}{d}-\frac{(120 b) \int \sinh (c+d x) \, dx}{d^5}\\ &=-\frac{120 b \cosh (c+d x)}{d^6}-\frac{6 a \cosh (c+d x)}{d^4}-\frac{60 b x^2 \cosh (c+d x)}{d^4}-\frac{3 a x^2 \cosh (c+d x)}{d^2}-\frac{5 b x^4 \cosh (c+d x)}{d^2}+\frac{120 b x \sinh (c+d x)}{d^5}+\frac{6 a x \sinh (c+d x)}{d^3}+\frac{20 b x^3 \sinh (c+d x)}{d^3}+\frac{a x^3 \sinh (c+d x)}{d}+\frac{b x^5 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.149123, size = 92, normalized size = 0.66 \[ \frac{d x \left (a d^2 \left (d^2 x^2+6\right )+b \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \sinh (c+d x)-\left (3 a d^2 \left (d^2 x^2+2\right )+5 b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \cosh (c+d x)}{d^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)*Cosh[c + d*x],x]

[Out]

(-((3*a*d^2*(2 + d^2*x^2) + 5*b*(24 + 12*d^2*x^2 + d^4*x^4))*Cosh[c + d*x]) + d*x*(a*d^2*(6 + d^2*x^2) + b*(12
0 + 20*d^2*x^2 + d^4*x^4))*Sinh[c + d*x])/d^6

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Maple [B]  time = 0.009, size = 447, normalized size = 3.2 \begin{align*}{\frac{1}{{d}^{4}} \left ({\frac{b \left ( \left ( dx+c \right ) ^{5}\sinh \left ( dx+c \right ) -5\, \left ( dx+c \right ) ^{4}\cosh \left ( dx+c \right ) +20\, \left ( dx+c \right ) ^{3}\sinh \left ( dx+c \right ) -60\, \left ( dx+c \right ) ^{2}\cosh \left ( dx+c \right ) +120\, \left ( dx+c \right ) \sinh \left ( dx+c \right ) -120\,\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}-5\,{\frac{cb \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}^{2}}}+10\,{\frac{b{c}^{2} \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}^{2}}}-10\,{\frac{b{c}^{3} \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}^{2}}}+5\,{\frac{b{c}^{4} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}-{\frac{b{c}^{5}\sinh \left ( dx+c \right ) }{{d}^{2}}}+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^2+a)*cosh(d*x+c),x)

[Out]

1/d^4*(1/d^2*b*((d*x+c)^5*sinh(d*x+c)-5*(d*x+c)^4*cosh(d*x+c)+20*(d*x+c)^3*sinh(d*x+c)-60*(d*x+c)^2*cosh(d*x+c
)+120*(d*x+c)*sinh(d*x+c)-120*cosh(d*x+c))-5/d^2*b*c*((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*sinh(d*x+c))+10/d^2*b*c^2*((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))-10/d^2*b*c^3*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+
c))+5/d^2*b*c^4*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-1/d^2*b*c^5*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*sinh(d*x+c))

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Maxima [A]  time = 1.07114, size = 338, normalized size = 2.43 \begin{align*} -\frac{1}{24} \, d{\left (\frac{3 \,{\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{3 \,{\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{2 \,{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{7}} + \frac{2 \,{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b e^{\left (-d x - c\right )}}{d^{7}}\right )} + \frac{1}{12} \,{\left (2 \, b x^{6} + 3 \, 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^2+a)*cosh(d*x+c),x, algorithm="maxima")

[Out]

-1/24*d*(3*(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 + 3*(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 + 2*(d^6*x^6*e^c - 6*d^5*x^5*e^c + 30*d^4*x^4*e^c - 12
0*d^3*x^3*e^c + 360*d^2*x^2*e^c - 720*d*x*e^c + 720*e^c)*b*e^(d*x)/d^7 + 2*(d^6*x^6 + 6*d^5*x^5 + 30*d^4*x^4 +
120*d^3*x^3 + 360*d^2*x^2 + 720*d*x + 720)*b*e^(-d*x - c)/d^7) + 1/12*(2*b*x^6 + 3*a*x^4)*cosh(d*x + c)

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Fricas [A]  time = 2.01632, size = 212, normalized size = 1.53 \begin{align*} -\frac{{\left (5 \, b d^{4} x^{4} + 6 \, a d^{2} + 3 \,{\left (a d^{4} + 20 \, b d^{2}\right )} x^{2} + 120 \, b\right )} \cosh \left (d x + c\right ) -{\left (b d^{5} x^{5} +{\left (a d^{5} + 20 \, b d^{3}\right )} x^{3} + 6 \,{\left (a d^{3} + 20 \, b d\right )} x\right )} \sinh \left (d x + c\right )}{d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)*cosh(d*x+c),x, algorithm="fricas")

[Out]

-((5*b*d^4*x^4 + 6*a*d^2 + 3*(a*d^4 + 20*b*d^2)*x^2 + 120*b)*cosh(d*x + c) - (b*d^5*x^5 + (a*d^5 + 20*b*d^3)*x
^3 + 6*(a*d^3 + 20*b*d)*x)*sinh(d*x + c))/d^6

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Sympy [A]  time = 5.48512, size = 168, normalized size = 1.21 \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^{5} \sinh{\left (c + d x \right )}}{d} - \frac{5 b x^{4} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{20 b x^{3} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{60 b x^{2} \cosh{\left (c + d x \right )}}{d^{4}} + \frac{120 b x \sinh{\left (c + d x \right )}}{d^{5}} - \frac{120 b \cosh{\left (c + d x \right )}}{d^{6}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{4}}{4} + \frac{b x^{6}}{6}\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**2+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**5*sinh(c + d*x)/d - 5*b*x**4*cosh(c + d*x)/d**2 + 20*b*x**3*sinh(c + d*x)/d**3 - 60*b*x**2*cosh(c
+ d*x)/d**4 + 120*b*x*sinh(c + d*x)/d**5 - 120*b*cosh(c + d*x)/d**6, Ne(d, 0)), ((a*x**4/4 + b*x**6/6)*cosh(c
), True))

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Giac [A]  time = 1.20411, size = 235, normalized size = 1.69 \begin{align*} \frac{{\left (b d^{5} x^{5} + a d^{5} x^{3} - 5 \, b d^{4} x^{4} - 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x - 60 \, b d^{2} x^{2} - 6 \, a d^{2} + 120 \, b d x - 120 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac{{\left (b d^{5} x^{5} + a d^{5} x^{3} + 5 \, b d^{4} x^{4} + 3 \, a d^{4} x^{2} + 20 \, b d^{3} x^{3} + 6 \, a d^{3} x + 60 \, b d^{2} x^{2} + 6 \, a d^{2} + 120 \, b d x + 120 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)*cosh(d*x+c),x, algorithm="giac")

[Out]

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