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

Optimal. Leaf size=124 \[ \frac {2 a \sinh (c+d x)}{d^3}-\frac {2 a x \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}-\frac {120 b \cosh (c+d x)}{d^6}+\frac {120 b x \sinh (c+d x)}{d^5}-\frac {60 b x^2 \cosh (c+d x)}{d^4}+\frac {20 b x^3 \sinh (c+d x)}{d^3}-\frac {5 b x^4 \cosh (c+d x)}{d^2}+\frac {b x^5 \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5287, 3296, 2637, 2638} \[ \frac {2 a \sinh (c+d x)}{d^3}-\frac {2 a x \cosh (c+d x)}{d^2}+\frac {a x^2 \sinh (c+d x)}{d}+\frac {20 b x^3 \sinh (c+d x)}{d^3}-\frac {5 b x^4 \cosh (c+d x)}{d^2}-\frac {60 b x^2 \cosh (c+d x)}{d^4}+\frac {120 b x \sinh (c+d x)}{d^5}-\frac {120 b \cosh (c+d x)}{d^6}+\frac {b x^5 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)*Cosh[c + d*x],x]

[Out]

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

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]

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 84, normalized size = 0.68 \[ \frac {d \left (a d^2 \left (d^2 x^2+2\right )+b x \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \sinh (c+d x)-\left (2 a d^4 x+5 b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \cosh (c+d x)}{d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 87, normalized size = 0.70 \[ -\frac {{\left (5 \, b d^{4} x^{4} + 2 \, a d^{4} x + 60 \, b d^{2} x^{2} + 120 \, b\right )} \cosh \left (d x + c\right ) - {\left (b d^{5} x^{5} + a d^{5} x^{2} + 20 \, b d^{3} x^{3} + 2 \, a d^{3} + 120 \, b d x\right )} \sinh \left (d x + c\right )}{d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 156, normalized size = 1.26 \[ \frac {{\left (b d^{5} x^{5} - 5 \, b d^{4} x^{4} + a d^{5} x^{2} + 20 \, b d^{3} x^{3} - 2 \, a d^{4} x - 60 \, b d^{2} x^{2} + 2 \, a d^{3} + 120 \, b d x - 120 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{6}} - \frac {{\left (b d^{5} x^{5} + 5 \, b d^{4} x^{4} + a d^{5} x^{2} + 20 \, b d^{3} x^{3} + 2 \, a d^{4} x + 60 \, b d^{2} x^{2} + 2 \, a d^{3} + 120 \, b d x + 120 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 389, normalized size = 3.14 \[ \frac {\frac {b \left (\left (d x +c \right )^{5} \sinh \left (d x +c \right )-5 \left (d x +c \right )^{4} \cosh \left (d x +c \right )+20 \left (d x +c \right )^{3} \sinh \left (d x +c \right )-60 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+120 \left (d x +c \right ) \sinh \left (d x +c \right )-120 \cosh \left (d x +c \right )\right )}{d^{3}}-\frac {5 b c \left (\left (d x +c \right )^{4} \sinh \left (d x +c \right )-4 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+12 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-24 \left (d x +c \right ) \cosh \left (d x +c \right )+24 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {10 b \,c^{2} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{3}}-\frac {10 b \,c^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )}{d^{3}}+\frac {5 b \,c^{4} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{3}}-\frac {b \,c^{5} \sinh \left (d x +c \right )}{d^{3}}+a \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-2 a c \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+a \,c^{2} \sinh \left (d x +c \right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)*cosh(d*x+c),x)

[Out]

1/d^3*(1/d^3*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^3*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^3*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^3*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^3*b*c^4*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-1/d^3*b*c^5*sinh(d*x+c)+a*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*c
osh(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 = 0.34, size = 267, normalized size = 2.15 \[ \frac {{\left (b x^{3} + a\right )}^{2} \cosh \left (d x + c\right )}{6 \, b} - \frac {{\left (\frac {a^{2} e^{\left (d x + c\right )}}{d} + \frac {a^{2} e^{\left (-d x - c\right )}}{d} + \frac {2 \, {\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{4}} + \frac {2 \, {\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} a b e^{\left (-d x - c\right )}}{d^{4}} + \frac {{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{7}} + \frac {{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} b^{2} e^{\left (-d x - c\right )}}{d^{7}}\right )} d}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^3 + a)^2*cosh(d*x + c)/b - 1/12*(a^2*e^(d*x + c)/d + a^2*e^(-d*x - c)/d + 2*(d^3*x^3*e^c - 3*d^2*x^2*
e^c + 6*d*x*e^c - 6*e^c)*a*b*e^(d*x)/d^4 + 2*(d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6)*a*b*e^(-d*x - c)/d^4 + (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)*b^2*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^2*e^(-d*x - c)/d^7)*d/
b

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mupad [B]  time = 0.96, size = 122, normalized size = 0.98 \[ \frac {a\,x^2\,\mathrm {sinh}\left (c+d\,x\right )+b\,x^5\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {2\,a\,x\,\mathrm {cosh}\left (c+d\,x\right )+5\,b\,x^4\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {2\,a\,\mathrm {sinh}\left (c+d\,x\right )+20\,b\,x^3\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}-\frac {120\,b\,\mathrm {cosh}\left (c+d\,x\right )}{d^6}+\frac {120\,b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^5}-\frac {60\,b\,x^2\,\mathrm {cosh}\left (c+d\,x\right )}{d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(c + d*x)*(a + b*x^3),x)

[Out]

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

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sympy [A]  time = 3.25, size = 151, normalized size = 1.22 \[ \begin {cases} \frac {a x^{2} \sinh {\left (c + d x \right )}}{d} - \frac {2 a x \cosh {\left (c + d x \right )}}{d^{2}} + \frac {2 a \sinh {\left (c + d x \right )}}{d^{3}} + \frac {b x^{5} \sinh {\left (c + d x \right )}}{d} - \frac {5 b x^{4} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {20 b x^{3} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {60 b x^{2} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {120 b x \sinh {\left (c + d x \right )}}{d^{5}} - \frac {120 b \cosh {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{3}}{3} + \frac {b x^{6}}{6}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**3+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**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**3/3 + b*x**6/6)*cosh(c), True))

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