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

Optimal. Leaf size=184 \[ -\frac{a^2 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac{120 b^2 x \sinh (c+d x)}{d^5}-\frac{120 b^2 \cosh (c+d x)}{d^6}+\frac{b^2 x^5 \sinh (c+d x)}{d} \]

[Out]

(-120*b^2*Cosh[c + d*x])/d^6 - (12*a*b*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x
])/d^4 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (5*b^2*x^4*Cosh[c + d*x])/d^2 + (120*b^2*x*Sinh[c + d*x])/d^5 + (12*a
*b*x*Sinh[c + d*x])/d^3 + (a^2*x*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x])/d^3 + (2*a*b*x^3*Sinh[c + d*x])
/d + (b^2*x^5*Sinh[c + d*x])/d

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Rubi [A]  time = 0.27713, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5287, 3296, 2638} \[ -\frac{a^2 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}-\frac{12 a b \cosh (c+d x)}{d^4}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac{120 b^2 x \sinh (c+d x)}{d^5}-\frac{120 b^2 \cosh (c+d x)}{d^6}+\frac{b^2 x^5 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*Cosh[c + d*x],x]

[Out]

(-120*b^2*Cosh[c + d*x])/d^6 - (12*a*b*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x
])/d^4 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (5*b^2*x^4*Cosh[c + d*x])/d^2 + (120*b^2*x*Sinh[c + d*x])/d^5 + (12*a
*b*x*Sinh[c + d*x])/d^3 + (a^2*x*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x])/d^3 + (2*a*b*x^3*Sinh[c + d*x])
/d + (b^2*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 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx &=\int \left (a^2 x \cosh (c+d x)+2 a b x^3 \cosh (c+d x)+b^2 x^5 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int x \cosh (c+d x) \, dx+(2 a b) \int x^3 \cosh (c+d x) \, dx+b^2 \int x^5 \cosh (c+d x) \, dx\\ &=\frac{a^2 x \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}-\frac{a^2 \int \sinh (c+d x) \, dx}{d}-\frac{(6 a b) \int x^2 \sinh (c+d x) \, dx}{d}-\frac{\left (5 b^2\right ) \int x^4 \sinh (c+d x) \, dx}{d}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}+\frac{(12 a b) \int x \cosh (c+d x) \, dx}{d^2}+\frac{\left (20 b^2\right ) \int x^3 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}-\frac{(12 a b) \int \sinh (c+d x) \, dx}{d^3}-\frac{\left (60 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}+\frac{\left (120 b^2\right ) \int x \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac{120 b^2 x \sinh (c+d x)}{d^5}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}-\frac{\left (120 b^2\right ) \int \sinh (c+d x) \, dx}{d^5}\\ &=-\frac{120 b^2 \cosh (c+d x)}{d^6}-\frac{12 a b \cosh (c+d x)}{d^4}-\frac{a^2 \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac{6 a b x^2 \cosh (c+d x)}{d^2}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+\frac{120 b^2 x \sinh (c+d x)}{d^5}+\frac{12 a b x \sinh (c+d x)}{d^3}+\frac{a^2 x \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{2 a b x^3 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.230315, size = 113, normalized size = 0.61 \[ \frac{d x \left (a^2 d^4+2 a b d^2 \left (d^2 x^2+6\right )+b^2 \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \sinh (c+d x)-\left (a^2 d^4+6 a b d^2 \left (d^2 x^2+2\right )+5 b^2 \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \cosh (c+d x)}{d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a^2*d^4 + 6*a*b*d^2*(2 + d^2*x^2) + 5*b^2*(24 + 12*d^2*x^2 + d^4*x^4))*Cosh[c + d*x]) + d*x*(a^2*d^4 + 2*a
*b*d^2*(6 + d^2*x^2) + b^2*(120 + 20*d^2*x^2 + d^4*x^4))*Sinh[c + d*x])/d^6

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Maple [B]  time = 0.008, size = 513, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(1/d^4*b^2*((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^4*b^2*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^4*b^2*c^2*((d*x+c)^3*sinh(d*x+c)-3*(d*x+c)^2*co
sh(d*x+c)+6*(d*x+c)*sinh(d*x+c)-6*cosh(d*x+c))+2/d^2*b*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))-10/d^4*b^2*c^3*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-6/d^
2*b*c*a*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))+5/d^4*b^2*c^4*((d*x+c)*sinh(d*x+c)-cosh(d*
x+c))+6/d^2*b*c^2*a*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))+a^2*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-1/d^4*b^2*c^5*sinh
(d*x+c)-2/d^2*b*c^3*a*sinh(d*x+c)-c*a^2*sinh(d*x+c))

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Maxima [A]  time = 1.07846, size = 477, normalized size = 2.59 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{3} \cosh \left (d x + c\right )}{6 \, b} - \frac{{\left (\frac{a^{3} e^{\left (d x + c\right )}}{d} + \frac{a^{3} e^{\left (-d x - c\right )}}{d} + \frac{3 \,{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a^{2} b e^{\left (d x\right )}}{d^{3}} + \frac{3 \,{\left (d^{2} x^{2} + 2 \, d x + 2\right )} a^{2} b e^{\left (-d x - c\right )}}{d^{3}} + \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 b^{2} 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 b^{2} e^{\left (-d x - c\right )}}{d^{5}} + \frac{{\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^{3} e^{\left (d x\right )}}{d^{7}} + \frac{{\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^{3} e^{\left (-d x - c\right )}}{d^{7}}\right )} d}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2 + a)^3*cosh(d*x + c)/b - 1/12*(a^3*e^(d*x + c)/d + a^3*e^(-d*x - c)/d + 3*(d^2*x^2*e^c - 2*d*x*e^c
+ 2*e^c)*a^2*b*e^(d*x)/d^3 + 3*(d^2*x^2 + 2*d*x + 2)*a^2*b*e^(-d*x - c)/d^3 + 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*b^2*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*b^2*e^(-d*x - c)/d^5 + (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 - 7
20*d*x*e^c + 720*e^c)*b^3*e^(d*x)/d^7 + (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^3*e^(-d*x - c)/d^7)*d/b

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Fricas [A]  time = 2.08417, size = 273, normalized size = 1.48 \begin{align*} -\frac{{\left (5 \, b^{2} d^{4} x^{4} + a^{2} d^{4} + 12 \, a b d^{2} + 6 \,{\left (a b d^{4} + 10 \, b^{2} d^{2}\right )} x^{2} + 120 \, b^{2}\right )} \cosh \left (d x + c\right ) -{\left (b^{2} d^{5} x^{5} + 2 \,{\left (a b d^{5} + 10 \, b^{2} d^{3}\right )} x^{3} +{\left (a^{2} d^{5} + 12 \, a b d^{3} + 120 \, b^{2} 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*(b*x^2+a)^2*cosh(d*x+c),x, algorithm="fricas")

[Out]

-((5*b^2*d^4*x^4 + a^2*d^4 + 12*a*b*d^2 + 6*(a*b*d^4 + 10*b^2*d^2)*x^2 + 120*b^2)*cosh(d*x + c) - (b^2*d^5*x^5
+ 2*(a*b*d^5 + 10*b^2*d^3)*x^3 + (a^2*d^5 + 12*a*b*d^3 + 120*b^2*d)*x)*sinh(d*x + c))/d^6

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Sympy [A]  time = 5.25899, size = 226, normalized size = 1.23 \begin{align*} \begin{cases} \frac{a^{2} x \sinh{\left (c + d x \right )}}{d} - \frac{a^{2} \cosh{\left (c + d x \right )}}{d^{2}} + \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^{5} \sinh{\left (c + d x \right )}}{d} - \frac{5 b^{2} x^{4} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{20 b^{2} x^{3} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{60 b^{2} x^{2} \cosh{\left (c + d x \right )}}{d^{4}} + \frac{120 b^{2} x \sinh{\left (c + d x \right )}}{d^{5}} - \frac{120 b^{2} \cosh{\left (c + d x \right )}}{d^{6}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} 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*(b*x**2+a)**2*cosh(d*x+c),x)

[Out]

Piecewise((a**2*x*sinh(c + d*x)/d - a**2*cosh(c + d*x)/d**2 + 2*a*b*x**3*sinh(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**5*sinh(c + d*x)/d - 5*b**2*x**4
*cosh(c + d*x)/d**2 + 20*b**2*x**3*sinh(c + d*x)/d**3 - 60*b**2*x**2*cosh(c + d*x)/d**4 + 120*b**2*x*sinh(c +
d*x)/d**5 - 120*b**2*cosh(c + d*x)/d**6, Ne(d, 0)), ((a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)*cosh(c), True))

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Giac [A]  time = 1.17159, size = 323, normalized size = 1.76 \begin{align*} \frac{{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} - 5 \, b^{2} d^{4} x^{4} + a^{2} d^{5} x - 6 \, a b d^{4} x^{2} + 20 \, b^{2} d^{3} x^{3} - a^{2} d^{4} + 12 \, a b d^{3} x - 60 \, b^{2} d^{2} x^{2} - 12 \, a b d^{2} + 120 \, b^{2} d x - 120 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac{{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} + 5 \, b^{2} d^{4} x^{4} + a^{2} d^{5} x + 6 \, a b d^{4} x^{2} + 20 \, b^{2} d^{3} x^{3} + a^{2} d^{4} + 12 \, a b d^{3} x + 60 \, b^{2} d^{2} x^{2} + 12 \, a b d^{2} + 120 \, b^{2} d x + 120 \, b^{2}\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*(b*x^2+a)^2*cosh(d*x+c),x, algorithm="giac")

[Out]

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