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

Optimal. Leaf size=94 \[ \frac{2 a \sinh (c+d x)}{d^3}-\frac{2 a x \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{6 b x \sinh (c+d x)}{d^3}-\frac{6 b \cosh (c+d x)}{d^4}+\frac{b x^3 \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.216928, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {6742, 3296, 2637, 2638} \[ \frac{2 a \sinh (c+d x)}{d^3}-\frac{2 a x \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{6 b x \sinh (c+d x)}{d^3}-\frac{6 b \cosh (c+d x)}{d^4}+\frac{b x^3 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)*Cosh[c + d*x],x]

[Out]

(-6*b*Cosh[c + d*x])/d^4 - (2*a*x*Cosh[c + d*x])/d^2 - (3*b*x^2*Cosh[c + d*x])/d^2 + (2*a*Sinh[c + d*x])/d^3 +
(6*b*x*Sinh[c + d*x])/d^3 + (a*x^2*Sinh[c + d*x])/d + (b*x^3*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) \cosh (c+d x) \, dx &=\int \left (a x^2 \cosh (c+d x)+b x^3 \cosh (c+d x)\right ) \, dx\\ &=a \int x^2 \cosh (c+d x) \, dx+b \int x^3 \cosh (c+d x) \, dx\\ &=\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}-\frac{(2 a) \int x \sinh (c+d x) \, dx}{d}-\frac{(3 b) \int x^2 \sinh (c+d x) \, dx}{d}\\ &=-\frac{2 a x \cosh (c+d x)}{d^2}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}+\frac{(2 a) \int \cosh (c+d x) \, dx}{d^2}+\frac{(6 b) \int x \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{2 a x \cosh (c+d x)}{d^2}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{2 a \sinh (c+d x)}{d^3}+\frac{6 b x \sinh (c+d x)}{d^3}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}-\frac{(6 b) \int \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{6 b \cosh (c+d x)}{d^4}-\frac{2 a x \cosh (c+d x)}{d^2}-\frac{3 b x^2 \cosh (c+d x)}{d^2}+\frac{2 a \sinh (c+d x)}{d^3}+\frac{6 b x \sinh (c+d x)}{d^3}+\frac{a x^2 \sinh (c+d x)}{d}+\frac{b x^3 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.117328, size = 65, normalized size = 0.69 \[ \frac{d \left (a \left (d^2 x^2+2\right )+b x \left (d^2 x^2+6\right )\right ) \sinh (c+d x)-\left (2 a d^2 x+3 b \left (d^2 x^2+2\right )\right ) \cosh (c+d x)}{d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 224, normalized size = 2.4 \begin{align*}{\frac{1}{{d}^{3}} \left ({\frac{b \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}}-3\,{\frac{cb \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}}+3\,{\frac{b{c}^{2} \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}}-{\frac{b{c}^{3}\sinh \left ( dx+c \right ) }{d}}+a \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\,ac \left ( \left ( dx+c \right ) \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) +a{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)*cosh(d*x+c),x)

[Out]

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

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Maxima [B]  time = 1.1237, size = 265, normalized size = 2.82 \begin{align*} -\frac{1}{24} \, d{\left (\frac{4 \,{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{4}} + \frac{4 \,{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a e^{\left (-d x - c\right )}}{d^{4}} + \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 )} b 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 )} b e^{\left (-d x - c\right )}}{d^{5}}\right )} + \frac{1}{12} \,{\left (3 \, b x^{4} + 4 \, a 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)*cosh(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92279, size = 151, normalized size = 1.61 \begin{align*} -\frac{{\left (3 \, b d^{2} x^{2} + 2 \, a d^{2} x + 6 \, b\right )} \cosh \left (d x + c\right ) -{\left (b d^{3} x^{3} + a d^{3} x^{2} + 6 \, b d x + 2 \, a d\right )} \sinh \left (d x + c\right )}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.24197, size = 117, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 a x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 a \sinh{\left (c + d x \right )}}{d^{3}} + \frac{b x^{3} \sinh{\left (c + d x \right )}}{d} - \frac{3 b x^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{6 b x \sinh{\left (c + d x \right )}}{d^{3}} - \frac{6 b \cosh{\left (c + d x \right )}}{d^{4}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{3}}{3} + \frac{b x^{4}}{4}\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)*cosh(d*x+c),x)

[Out]

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

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Giac [A]  time = 1.16419, size = 157, normalized size = 1.67 \begin{align*} \frac{{\left (b d^{3} x^{3} + a d^{3} x^{2} - 3 \, b d^{2} x^{2} - 2 \, a d^{2} x + 6 \, b d x + 2 \, a d - 6 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{4}} - \frac{{\left (b d^{3} x^{3} + a d^{3} x^{2} + 3 \, b d^{2} x^{2} + 2 \, a d^{2} x + 6 \, b d x + 2 \, a d + 6 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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