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

Optimal. Leaf size=154 \[ -\frac {6 a \cosh (c+d x)}{d^4}+\frac {6 a x \sinh (c+d x)}{d^3}-\frac {3 a x^2 \cosh (c+d x)}{d^2}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {720 b \sinh (c+d x)}{d^7}-\frac {720 b x \cosh (c+d x)}{d^6}+\frac {360 b x^2 \sinh (c+d x)}{d^5}-\frac {120 b x^3 \cosh (c+d x)}{d^4}+\frac {30 b x^4 \sinh (c+d x)}{d^3}-\frac {6 b x^5 \cosh (c+d x)}{d^2}+\frac {b x^6 \sinh (c+d x)}{d} \]

[Out]

-6*a*cosh(d*x+c)/d^4-720*b*x*cosh(d*x+c)/d^6-3*a*x^2*cosh(d*x+c)/d^2-120*b*x^3*cosh(d*x+c)/d^4-6*b*x^5*cosh(d*
x+c)/d^2+720*b*sinh(d*x+c)/d^7+6*a*x*sinh(d*x+c)/d^3+360*b*x^2*sinh(d*x+c)/d^5+a*x^3*sinh(d*x+c)/d+30*b*x^4*si
nh(d*x+c)/d^3+b*x^6*sinh(d*x+c)/d

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Rubi [A]  time = 0.30, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5287, 3296, 2638, 2637} \[ -\frac {3 a x^2 \cosh (c+d x)}{d^2}+\frac {6 a x \sinh (c+d x)}{d^3}-\frac {6 a \cosh (c+d x)}{d^4}+\frac {a x^3 \sinh (c+d x)}{d}+\frac {30 b x^4 \sinh (c+d x)}{d^3}+\frac {360 b x^2 \sinh (c+d x)}{d^5}-\frac {6 b x^5 \cosh (c+d x)}{d^2}-\frac {120 b x^3 \cosh (c+d x)}{d^4}+\frac {720 b \sinh (c+d x)}{d^7}-\frac {720 b x \cosh (c+d x)}{d^6}+\frac {b x^6 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^3)*Cosh[c + d*x],x]

[Out]

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

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Mathematica [A]  time = 0.19, size = 100, normalized size = 0.65 \[ \frac {\left (a d^4 x \left (d^2 x^2+6\right )+b \left (d^6 x^6+30 d^4 x^4+360 d^2 x^2+720\right )\right ) \sinh (c+d x)-3 d \left (a d^2 \left (d^2 x^2+2\right )+2 b x \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \cosh (c+d x)}{d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 105, normalized size = 0.68 \[ -\frac {3 \, {\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} + 40 \, b d^{3} x^{3} + 2 \, a d^{3} + 240 \, b d x\right )} \cosh \left (d x + c\right ) - {\left (b d^{6} x^{6} + a d^{6} x^{3} + 30 \, b d^{4} x^{4} + 6 \, a d^{4} x + 360 \, b d^{2} x^{2} + 720 \, b\right )} \sinh \left (d x + c\right )}{d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(2*b*d^5*x^5 + a*d^5*x^2 + 40*b*d^3*x^3 + 2*a*d^3 + 240*b*d*x)*cosh(d*x + c) - (b*d^6*x^6 + a*d^6*x^3 + 30
*b*d^4*x^4 + 6*a*d^4*x + 360*b*d^2*x^2 + 720*b)*sinh(d*x + c))/d^7

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giac [A]  time = 0.12, size = 192, normalized size = 1.25 \[ \frac {{\left (b d^{6} x^{6} - 6 \, b d^{5} x^{5} + a d^{6} x^{3} + 30 \, b d^{4} x^{4} - 3 \, a d^{5} x^{2} - 120 \, b d^{3} x^{3} + 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 6 \, a d^{3} - 720 \, b d x + 720 \, b\right )} e^{\left (d x + c\right )}}{2 \, d^{7}} - \frac {{\left (b d^{6} x^{6} + 6 \, b d^{5} x^{5} + a d^{6} x^{3} + 30 \, b d^{4} x^{4} + 3 \, a d^{5} x^{2} + 120 \, b d^{3} x^{3} + 6 \, a d^{4} x + 360 \, b d^{2} x^{2} + 6 \, a d^{3} + 720 \, b d x + 720 \, b\right )} e^{\left (-d x - c\right )}}{2 \, d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 551, normalized size = 3.58 \[ \frac {\frac {b \left (\left (d x +c \right )^{6} \sinh \left (d x +c \right )-6 \left (d x +c \right )^{5} \cosh \left (d x +c \right )+30 \left (d x +c \right )^{4} \sinh \left (d x +c \right )-120 \left (d x +c \right )^{3} \cosh \left (d x +c \right )+360 \left (d x +c \right )^{2} \sinh \left (d x +c \right )-720 \left (d x +c \right ) \cosh \left (d x +c \right )+720 \sinh \left (d x +c \right )\right )}{d^{3}}-\frac {6 b c \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 {15 b \,c^{2} \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 {20 b \,c^{3} \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 {15 b \,c^{4} \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 {6 b \,c^{5} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d^{3}}+\frac {b \,c^{6} \sinh \left (d x +c \right )}{d^{3}}+a \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 )-3 a c \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 )+3 a \,c^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-a \,c^{3} \sinh \left (d x +c \right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(1/d^3*b*((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))-6/d^3*b*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^3*b*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*s
inh(d*x+c))-20/d^3*b*c^3*((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))+1
5/d^3*b*c^4*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))-6/d^3*b*c^5*((d*x+c)*sinh(d*x+c)-cosh(
d*x+c))+1/d^3*b*c^6*sinh(d*x+c)+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))-3*a*c*((d*x+c)^2*sinh(d*x+c)-2*(d*x+c)*cosh(d*x+c)+2*sinh(d*x+c))+3*a*c^2*((d*x+c)*sinh(d*x+c)-cosh(d*
x+c))-a*c^3*sinh(d*x+c))

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maxima [A]  time = 1.09, size = 268, normalized size = 1.74 \[ -\frac {1}{56} \, d {\left (\frac {7 \, {\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} a e^{\left (d x\right )}}{d^{5}} + \frac {7 \, {\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} a e^{\left (-d x - c\right )}}{d^{5}} + \frac {4 \, {\left (d^{7} x^{7} e^{c} - 7 \, d^{6} x^{6} e^{c} + 42 \, d^{5} x^{5} e^{c} - 210 \, d^{4} x^{4} e^{c} + 840 \, d^{3} x^{3} e^{c} - 2520 \, d^{2} x^{2} e^{c} + 5040 \, d x e^{c} - 5040 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{8}} + \frac {4 \, {\left (d^{7} x^{7} + 7 \, d^{6} x^{6} + 42 \, d^{5} x^{5} + 210 \, d^{4} x^{4} + 840 \, d^{3} x^{3} + 2520 \, d^{2} x^{2} + 5040 \, d x + 5040\right )} b e^{\left (-d x - c\right )}}{d^{8}}\right )} + \frac {1}{28} \, {\left (4 \, b x^{7} + 7 \, a x^{4}\right )} \cosh \left (d x + c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*d*(7*(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*e^(d*x)/d^5 + 7*(d^4*x^4 + 4
*d^3*x^3 + 12*d^2*x^2 + 24*d*x + 24)*a*e^(-d*x - c)/d^5 + 4*(d^7*x^7*e^c - 7*d^6*x^6*e^c + 42*d^5*x^5*e^c - 21
0*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*e^(d*x)/d^8 + 4*(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*e^(-d*x - c)/d^8) + 1/28*(
4*b*x^7 + 7*a*x^4)*cosh(d*x + c)

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mupad [B]  time = 1.05, size = 152, normalized size = 0.99 \[ \frac {30\,b\,x^4\,\mathrm {sinh}\left (c+d\,x\right )+6\,a\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d^3}-\frac {3\,a\,x^2\,\mathrm {cosh}\left (c+d\,x\right )+6\,b\,x^5\,\mathrm {cosh}\left (c+d\,x\right )}{d^2}+\frac {a\,x^3\,\mathrm {sinh}\left (c+d\,x\right )+b\,x^6\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {6\,a\,\mathrm {cosh}\left (c+d\,x\right )+120\,b\,x^3\,\mathrm {cosh}\left (c+d\,x\right )}{d^4}+\frac {720\,b\,\mathrm {sinh}\left (c+d\,x\right )}{d^7}-\frac {720\,b\,x\,\mathrm {cosh}\left (c+d\,x\right )}{d^6}+\frac {360\,b\,x^2\,\mathrm {sinh}\left (c+d\,x\right )}{d^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(c + d*x)*(a + b*x^3),x)

[Out]

(30*b*x^4*sinh(c + d*x) + 6*a*x*sinh(c + d*x))/d^3 - (3*a*x^2*cosh(c + d*x) + 6*b*x^5*cosh(c + d*x))/d^2 + (a*
x^3*sinh(c + d*x) + b*x^6*sinh(c + d*x))/d - (6*a*cosh(c + d*x) + 120*b*x^3*cosh(c + d*x))/d^4 + (720*b*sinh(c
+ d*x))/d^7 - (720*b*x*cosh(c + d*x))/d^6 + (360*b*x^2*sinh(c + d*x))/d^5

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sympy [A]  time = 5.71, size = 185, normalized size = 1.20 \[ \begin {cases} \frac {a x^{3} \sinh {\left (c + d x \right )}}{d} - \frac {3 a x^{2} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \sinh {\left (c + d x \right )}}{d^{3}} - \frac {6 a \cosh {\left (c + d x \right )}}{d^{4}} + \frac {b x^{6} \sinh {\left (c + d x \right )}}{d} - \frac {6 b x^{5} \cosh {\left (c + d x \right )}}{d^{2}} + \frac {30 b x^{4} \sinh {\left (c + d x \right )}}{d^{3}} - \frac {120 b x^{3} \cosh {\left (c + d x \right )}}{d^{4}} + \frac {360 b x^{2} \sinh {\left (c + d x \right )}}{d^{5}} - \frac {720 b x \cosh {\left (c + d x \right )}}{d^{6}} + \frac {720 b \sinh {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{7}}{7}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**3+a)*cosh(d*x+c),x)

[Out]

Piecewise((a*x**3*sinh(c + d*x)/d - 3*a*x**2*cosh(c + d*x)/d**2 + 6*a*x*sinh(c + d*x)/d**3 - 6*a*cosh(c + d*x)
/d**4 + b*x**6*sinh(c + d*x)/d - 6*b*x**5*cosh(c + d*x)/d**2 + 30*b*x**4*sinh(c + d*x)/d**3 - 120*b*x**3*cosh(
c + d*x)/d**4 + 360*b*x**2*sinh(c + d*x)/d**5 - 720*b*x*cosh(c + d*x)/d**6 + 720*b*sinh(c + d*x)/d**7, Ne(d, 0
)), ((a*x**4/4 + b*x**7/7)*cosh(c), True))

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