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

Optimal. Leaf size=234 \[ \frac{2 a^2 \sinh (c+d x)}{d^3}-\frac{2 a^2 x \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}-\frac{48 a b x \cosh (c+d x)}{d^4}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac{720 b^2 \sinh (c+d x)}{d^7}-\frac{720 b^2 x \cosh (c+d x)}{d^6}+\frac{b^2 x^6 \sinh (c+d x)}{d} \]

[Out]

(-720*b^2*x*Cosh[c + d*x])/d^6 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2*a^2*x*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cos
h[c + d*x])/d^4 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7
+ (48*a*b*Sinh[c + d*x])/d^5 + (2*a^2*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (24*a*b*x^2*Sinh[
c + d*x])/d^3 + (a^2*x^2*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c + d*x])/d^3 + (2*a*b*x^4*Sinh[c + d*x])/d + (b^
2*x^6*Sinh[c + d*x])/d

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Rubi [A]  time = 0.388255, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {5287, 3296, 2637} \[ \frac{2 a^2 \sinh (c+d x)}{d^3}-\frac{2 a^2 x \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}-\frac{48 a b x \cosh (c+d x)}{d^4}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}+\frac{720 b^2 \sinh (c+d x)}{d^7}-\frac{720 b^2 x \cosh (c+d x)}{d^6}+\frac{b^2 x^6 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-720*b^2*x*Cosh[c + d*x])/d^6 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2*a^2*x*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cos
h[c + d*x])/d^4 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7
+ (48*a*b*Sinh[c + d*x])/d^5 + (2*a^2*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (24*a*b*x^2*Sinh[
c + d*x])/d^3 + (a^2*x^2*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c + d*x])/d^3 + (2*a*b*x^4*Sinh[c + d*x])/d + (b^
2*x^6*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 2637

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

Rubi steps

\begin{align*} \int x^2 \left (a+b x^2\right )^2 \cosh (c+d x) \, dx &=\int \left (a^2 x^2 \cosh (c+d x)+2 a b x^4 \cosh (c+d x)+b^2 x^6 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int x^2 \cosh (c+d x) \, dx+(2 a b) \int x^4 \cosh (c+d x) \, dx+b^2 \int x^6 \cosh (c+d x) \, dx\\ &=\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}-\frac{\left (2 a^2\right ) \int x \sinh (c+d x) \, dx}{d}-\frac{(8 a b) \int x^3 \sinh (c+d x) \, dx}{d}-\frac{\left (6 b^2\right ) \int x^5 \sinh (c+d x) \, dx}{d}\\ &=-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}+\frac{\left (2 a^2\right ) \int \cosh (c+d x) \, dx}{d^2}+\frac{(24 a b) \int x^2 \cosh (c+d x) \, dx}{d^2}+\frac{\left (30 b^2\right ) \int x^4 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}-\frac{(48 a b) \int x \sinh (c+d x) \, dx}{d^3}-\frac{\left (120 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}+\frac{(48 a b) \int \cosh (c+d x) \, dx}{d^4}+\frac{\left (360 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}-\frac{\left (720 b^2\right ) \int x \sinh (c+d x) \, dx}{d^5}\\ &=-\frac{720 b^2 x \cosh (c+d x)}{d^6}-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{48 a b \sinh (c+d x)}{d^5}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}+\frac{\left (720 b^2\right ) \int \cosh (c+d x) \, dx}{d^6}\\ &=-\frac{720 b^2 x \cosh (c+d x)}{d^6}-\frac{48 a b x \cosh (c+d x)}{d^4}-\frac{2 a^2 x \cosh (c+d x)}{d^2}-\frac{120 b^2 x^3 \cosh (c+d x)}{d^4}-\frac{8 a b x^3 \cosh (c+d x)}{d^2}-\frac{6 b^2 x^5 \cosh (c+d x)}{d^2}+\frac{720 b^2 \sinh (c+d x)}{d^7}+\frac{48 a b \sinh (c+d x)}{d^5}+\frac{2 a^2 \sinh (c+d x)}{d^3}+\frac{360 b^2 x^2 \sinh (c+d x)}{d^5}+\frac{24 a b x^2 \sinh (c+d x)}{d^3}+\frac{a^2 x^2 \sinh (c+d x)}{d}+\frac{30 b^2 x^4 \sinh (c+d x)}{d^3}+\frac{2 a b x^4 \sinh (c+d x)}{d}+\frac{b^2 x^6 \sinh (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.320592, size = 138, normalized size = 0.59 \[ \frac{\left (a^2 d^4 \left (d^2 x^2+2\right )+2 a b d^2 \left (d^4 x^4+12 d^2 x^2+24\right )+b^2 \left (d^6 x^6+30 d^4 x^4+360 d^2 x^2+720\right )\right ) \sinh (c+d x)-2 d x \left (a^2 d^4+4 a b d^2 \left (d^2 x^2+6\right )+3 b^2 \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \cosh (c+d x)}{d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*x*(a^2*d^4 + 4*a*b*d^2*(6 + d^2*x^2) + 3*b^2*(120 + 20*d^2*x^2 + d^4*x^4))*Cosh[c + d*x] + (a^2*d^4*(2 +
d^2*x^2) + 2*a*b*d^2*(24 + 12*d^2*x^2 + d^4*x^4) + b^2*(720 + 360*d^2*x^2 + 30*d^4*x^4 + d^6*x^6))*Sinh[c + d
*x])/d^7

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Maple [B]  time = 0.009, size = 738, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^3*(1/d^4*b^2*((d*x+c)^6*sinh(d*x+c)-6*(d*x+c)^5*cosh(d*x+c)+30*(d*x+c)^4*sinh(d*x+c)-120*(d*x+c)^3*cosh(d*
x+c)+360*(d*x+c)^2*sinh(d*x+c)-720*(d*x+c)*cosh(d*x+c)+720*sinh(d*x+c))+1/d^4*b^2*c^6*sinh(d*x+c)-2*a^2*c*((d*
x+c)*sinh(d*x+c)-cosh(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))-8/d^2*b*c^3*a*((
d*x+c)*sinh(d*x+c)-cosh(d*x+c))-8/d^2*b*a*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))+12/d^2*b*c^2*a*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))+2/d^2*b*c^4*a*sin
h(d*x+c)-6/d^4*b^2*c^5*((d*x+c)*sinh(d*x+c)-cosh(d*x+c))-20/d^4*b^2*c^3*((d*x+c)^3*sinh(d*x+c)-3*(d*x+c)^2*cos
h(d*x+c)+6*(d*x+c)*sinh(d*x+c)-6*cosh(d*x+c))+15/d^4*b^2*c^4*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*si
nh(d*x+c))+2/d^2*b*a*((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))-6/d^4*b^2*c*((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))+15/d^4*b^2*c^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))+a^2*c^2*sinh(d*x+c))

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Maxima [A]  time = 1.05534, size = 517, normalized size = 2.21 \begin{align*} -\frac{1}{210} \, d{\left (\frac{35 \,{\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{35 \,{\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{42 \,{\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 )} a b e^{\left (d x\right )}}{d^{6}} + \frac{42 \,{\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 )} a b e^{\left (-d x - c\right )}}{d^{6}} + \frac{15 \,{\left (d^{7} x^{7} e^{c} - 7 \, d^{6} x^{6} e^{c} + 42 \, d^{5} x^{5} e^{c} - 210 \, d^{4} x^{4} e^{c} + 840 \, d^{3} x^{3} e^{c} - 2520 \, d^{2} x^{2} e^{c} + 5040 \, d x e^{c} - 5040 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{8}} + \frac{15 \,{\left (d^{7} x^{7} + 7 \, d^{6} x^{6} + 42 \, d^{5} x^{5} + 210 \, d^{4} x^{4} + 840 \, d^{3} x^{3} + 2520 \, d^{2} x^{2} + 5040 \, d x + 5040\right )} b^{2} e^{\left (-d x - c\right )}}{d^{8}}\right )} + \frac{1}{105} \,{\left (15 \, b^{2} x^{7} + 42 \, a b x^{5} + 35 \, 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^2+a)^2*cosh(d*x+c),x, algorithm="maxima")

[Out]

-1/210*d*(35*(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 + 35*(d^3*x^3 + 3*d^2*x^2 + 6*d
*x + 6)*a^2*e^(-d*x - c)/d^4 + 42*(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)*a*b*e^(d*x)/d^6 + 42*(d^5*x^5 + 5*d^4*x^4 + 20*d^3*x^3 + 60*d^2*x^2 + 120*d*x + 120)*a*b*e^(-d*x -
c)/d^6 + 15*(d^7*x^7*e^c - 7*d^6*x^6*e^c + 42*d^5*x^5*e^c - 210*d^4*x^4*e^c + 840*d^3*x^3*e^c - 2520*d^2*x^2*
e^c + 5040*d*x*e^c - 5040*e^c)*b^2*e^(d*x)/d^8 + 15*(d^7*x^7 + 7*d^6*x^6 + 42*d^5*x^5 + 210*d^4*x^4 + 840*d^3*
x^3 + 2520*d^2*x^2 + 5040*d*x + 5040)*b^2*e^(-d*x - c)/d^8) + 1/105*(15*b^2*x^7 + 42*a*b*x^5 + 35*a^2*x^3)*cos
h(d*x + c)

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Fricas [A]  time = 2.05895, size = 336, normalized size = 1.44 \begin{align*} -\frac{2 \,{\left (3 \, b^{2} d^{5} x^{5} + 4 \,{\left (a b d^{5} + 15 \, b^{2} d^{3}\right )} x^{3} +{\left (a^{2} d^{5} + 24 \, a b d^{3} + 360 \, b^{2} d\right )} x\right )} \cosh \left (d x + c\right ) -{\left (b^{2} d^{6} x^{6} + 2 \, a^{2} d^{4} + 2 \,{\left (a b d^{6} + 15 \, b^{2} d^{4}\right )} x^{4} + 48 \, a b d^{2} +{\left (a^{2} d^{6} + 24 \, a b d^{4} + 360 \, b^{2} d^{2}\right )} x^{2} + 720 \, b^{2}\right )} \sinh \left (d x + c\right )}{d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(3*b^2*d^5*x^5 + 4*(a*b*d^5 + 15*b^2*d^3)*x^3 + (a^2*d^5 + 24*a*b*d^3 + 360*b^2*d)*x)*cosh(d*x + c) - (b^2
*d^6*x^6 + 2*a^2*d^4 + 2*(a*b*d^6 + 15*b^2*d^4)*x^4 + 48*a*b*d^2 + (a^2*d^6 + 24*a*b*d^4 + 360*b^2*d^2)*x^2 +
720*b^2)*sinh(d*x + c))/d^7

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Sympy [A]  time = 10.1603, size = 286, normalized size = 1.22 \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^{4} \sinh{\left (c + d x \right )}}{d} - \frac{8 a b x^{3} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{24 a b x^{2} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{48 a b x \cosh{\left (c + d x \right )}}{d^{4}} + \frac{48 a b \sinh{\left (c + d x \right )}}{d^{5}} + \frac{b^{2} x^{6} \sinh{\left (c + d x \right )}}{d} - \frac{6 b^{2} x^{5} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{30 b^{2} x^{4} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{120 b^{2} x^{3} \cosh{\left (c + d x \right )}}{d^{4}} + \frac{360 b^{2} x^{2} \sinh{\left (c + d x \right )}}{d^{5}} - \frac{720 b^{2} x \cosh{\left (c + d x \right )}}{d^{6}} + \frac{720 b^{2} \sinh{\left (c + d x \right )}}{d^{7}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{7}}{7}\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**2+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**4*si
nh(c + d*x)/d - 8*a*b*x**3*cosh(c + d*x)/d**2 + 24*a*b*x**2*sinh(c + d*x)/d**3 - 48*a*b*x*cosh(c + d*x)/d**4 +
48*a*b*sinh(c + d*x)/d**5 + b**2*x**6*sinh(c + d*x)/d - 6*b**2*x**5*cosh(c + d*x)/d**2 + 30*b**2*x**4*sinh(c
+ d*x)/d**3 - 120*b**2*x**3*cosh(c + d*x)/d**4 + 360*b**2*x**2*sinh(c + d*x)/d**5 - 720*b**2*x*cosh(c + d*x)/d
**6 + 720*b**2*sinh(c + d*x)/d**7, Ne(d, 0)), ((a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7)*cosh(c), True))

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Giac [A]  time = 1.21762, size = 410, normalized size = 1.75 \begin{align*} \frac{{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} - 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} - 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} - 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} - 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} - 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} - 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac{{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{4} + 6 \, b^{2} d^{5} x^{5} + a^{2} d^{6} x^{2} + 8 \, a b d^{5} x^{3} + 30 \, b^{2} d^{4} x^{4} + 2 \, a^{2} d^{5} x + 24 \, a b d^{4} x^{2} + 120 \, b^{2} d^{3} x^{3} + 2 \, a^{2} d^{4} + 48 \, a b d^{3} x + 360 \, b^{2} d^{2} x^{2} + 48 \, a b d^{2} + 720 \, b^{2} d x + 720 \, b^{2}\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*d^6*x^6 + 2*a*b*d^6*x^4 - 6*b^2*d^5*x^5 + a^2*d^6*x^2 - 8*a*b*d^5*x^3 + 30*b^2*d^4*x^4 - 2*a^2*d^5*x
+ 24*a*b*d^4*x^2 - 120*b^2*d^3*x^3 + 2*a^2*d^4 - 48*a*b*d^3*x + 360*b^2*d^2*x^2 + 48*a*b*d^2 - 720*b^2*d*x + 7
20*b^2)*e^(d*x + c)/d^7 - 1/2*(b^2*d^6*x^6 + 2*a*b*d^6*x^4 + 6*b^2*d^5*x^5 + a^2*d^6*x^2 + 8*a*b*d^5*x^3 + 30*
b^2*d^4*x^4 + 2*a^2*d^5*x + 24*a*b*d^4*x^2 + 120*b^2*d^3*x^3 + 2*a^2*d^4 + 48*a*b*d^3*x + 360*b^2*d^2*x^2 + 48
*a*b*d^2 + 720*b^2*d*x + 720*b^2)*e^(-d*x - c)/d^7