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

Optimal. Leaf size=184 \[ \frac{2 a^2 \sinh (c+d x)}{d^3}-\frac{2 a^2 x \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}-\frac{24 b^2 x \cosh (c+d x)}{d^4}+\frac{b^2 x^4 \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.346497, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6742, 3296, 2637, 2638} \[ \frac{2 a^2 \sinh (c+d x)}{d^3}-\frac{2 a^2 x \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}-\frac{24 b^2 x \cosh (c+d x)}{d^4}+\frac{b^2 x^4 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^2*Cosh[c + d*x],x]

[Out]

(-12*a*b*Cosh[c + d*x])/d^4 - (24*b^2*x*Cosh[c + d*x])/d^4 - (2*a^2*x*Cosh[c + d*x])/d^2 - (6*a*b*x^2*Cosh[c +
d*x])/d^2 - (4*b^2*x^3*Cosh[c + d*x])/d^2 + (24*b^2*Sinh[c + d*x])/d^5 + (2*a^2*Sinh[c + d*x])/d^3 + (12*a*b*
x*Sinh[c + d*x])/d^3 + (12*b^2*x^2*Sinh[c + d*x])/d^3 + (a^2*x^2*Sinh[c + d*x])/d + (2*a*b*x^3*Sinh[c + d*x])/
d + (b^2*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 2637

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

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

Mathematica [A]  time = 0.241922, size = 100, normalized size = 0.54 \[ \frac{\left (a^2 d^2 \left (d^2 x^2+2\right )+2 a b d^2 x \left (d^2 x^2+6\right )+b^2 \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \sinh (c+d x)-2 d (a+2 b x) \left (a d^2 x+b \left (d^2 x^2+6\right )\right ) \cosh (c+d x)}{d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 463, normalized size = 2.5 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{{b}^{2} \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}}}-4\,{\frac{c{b}^{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}}}+6\,{\frac{{c}^{2}{b}^{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}^{2}}}-4\,{\frac{{b}^{2}{c}^{3} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{{d}^{2}}}+2\,{\frac{ab \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{cba \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}}+6\,{\frac{ab{c}^{2} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2}{c}^{4}\sinh \left ( dx+c \right ) }{{d}^{2}}}-2\,{\frac{b{c}^{3}a\sinh \left ( dx+c \right ) }{d}}+{a}^{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 ) -2\,{a}^{2}c \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) +{a}^{2}{c}^{2}\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x+a)^2*cosh(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.18353, size = 444, normalized size = 2.41 \begin{align*} -\frac{1}{60} \, d{\left (\frac{10 \,{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a^{2} e^{\left (d x\right )}}{d^{4}} + \frac{10 \,{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a^{2} e^{\left (-d x - c\right )}}{d^{4}} + \frac{15 \,{\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 b e^{\left (d x\right )}}{d^{5}} + \frac{15 \,{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a b e^{\left (-d x - c\right )}}{d^{5}} + \frac{6 \,{\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^{2} e^{\left (d x\right )}}{d^{6}} + \frac{6 \,{\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^{2} e^{\left (-d x - c\right )}}{d^{6}}\right )} + \frac{1}{30} \,{\left (6 \, b^{2} x^{5} + 15 \, a b x^{4} + 10 \, a^{2} x^{3}\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*d*(10*(d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*a^2*e^(d*x)/d^4 + 10*(d^3*x^3 + 3*d^2*x^2 + 6*d*
x + 6)*a^2*e^(-d*x - c)/d^4 + 15*(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*b*e^(d
*x)/d^5 + 15*(d^4*x^4 + 4*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*a*b*e^(-d*x - c)/d^5 + 6*(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^2*e^(d*x)/d^6 + 6*(d^5*x^5 + 5*d^4*x^4 + 20
*d^3*x^3 + 60*d^2*x^2 + 120*d*x + 120)*b^2*e^(-d*x - c)/d^6) + 1/30*(6*b^2*x^5 + 15*a*b*x^4 + 10*a^2*x^3)*cosh
(d*x + c)

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Fricas [A]  time = 2.01887, size = 273, normalized size = 1.48 \begin{align*} -\frac{2 \,{\left (2 \, b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 6 \, a b d +{\left (a^{2} d^{3} + 12 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) -{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} +{\left (a^{2} d^{4} + 12 \, b^{2} d^{2}\right )} x^{2} + 24 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.53062, size = 228, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a^{2} x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 a^{2} x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 a^{2} \sinh{\left (c + d x \right )}}{d^{3}} + \frac{2 a b x^{3} \sinh{\left (c + d x \right )}}{d} - \frac{6 a b x^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{12 a b x \sinh{\left (c + d x \right )}}{d^{3}} - \frac{12 a b \cosh{\left (c + d x \right )}}{d^{4}} + \frac{b^{2} x^{4} \sinh{\left (c + d x \right )}}{d} - \frac{4 b^{2} x^{3} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{12 b^{2} x^{2} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{24 b^{2} x \cosh{\left (c + d x \right )}}{d^{4}} + \frac{24 b^{2} \sinh{\left (c + d x \right )}}{d^{5}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b^{2} 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**2*(b*x+a)**2*cosh(d*x+c),x)

[Out]

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

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Giac [A]  time = 1.17721, size = 319, normalized size = 1.73 \begin{align*} \frac{{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} - 4 \, b^{2} d^{3} x^{3} - 6 \, a b d^{3} x^{2} - 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} - 24 \, b^{2} d x - 12 \, a b d + 24 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{5}} - \frac{{\left (b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{3} + a^{2} d^{4} x^{2} + 4 \, b^{2} d^{3} x^{3} + 6 \, a b d^{3} x^{2} + 2 \, a^{2} d^{3} x + 12 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} x + 2 \, a^{2} d^{2} + 24 \, b^{2} d x + 12 \, a b d + 24 \, b^{2}\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^2*(b*x+a)^2*cosh(d*x+c),x, algorithm="giac")

[Out]

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