\(\int x^2 \cosh (a+b \sqrt [3]{c+d x}) \, dx\) [64]

Optimal result

Integrand size = 18, antiderivative size = 537 \[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {120960 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

[Out]

720*c*cosh(a+b*(d*x+c)^(1/3))/b^6/d^3-120960*(d*x+c)^(1/3)*cosh(a+b*(d*x+c)^(1/3))/b^8/d^3-6*c^2*(d*x+c)^(1/3)
*cosh(a+b*(d*x+c)^(1/3))/b^2/d^3+360*c*(d*x+c)^(2/3)*cosh(a+b*(d*x+c)^(1/3))/b^4/d^3-20160*(d*x+c)*cosh(a+b*(d
*x+c)^(1/3))/b^6/d^3+30*c*(d*x+c)^(4/3)*cosh(a+b*(d*x+c)^(1/3))/b^2/d^3-1008*(d*x+c)^(5/3)*cosh(a+b*(d*x+c)^(1
/3))/b^4/d^3-24*(d*x+c)^(7/3)*cosh(a+b*(d*x+c)^(1/3))/b^2/d^3+120960*sinh(a+b*(d*x+c)^(1/3))/b^9/d^3+6*c^2*sin
h(a+b*(d*x+c)^(1/3))/b^3/d^3-720*c*(d*x+c)^(1/3)*sinh(a+b*(d*x+c)^(1/3))/b^5/d^3+60480*(d*x+c)^(2/3)*sinh(a+b*
(d*x+c)^(1/3))/b^7/d^3+3*c^2*(d*x+c)^(2/3)*sinh(a+b*(d*x+c)^(1/3))/b/d^3-120*c*(d*x+c)*sinh(a+b*(d*x+c)^(1/3))
/b^3/d^3+5040*(d*x+c)^(4/3)*sinh(a+b*(d*x+c)^(1/3))/b^5/d^3-6*c*(d*x+c)^(5/3)*sinh(a+b*(d*x+c)^(1/3))/b/d^3+16
8*(d*x+c)^2*sinh(a+b*(d*x+c)^(1/3))/b^3/d^3+3*(d*x+c)^(8/3)*sinh(a+b*(d*x+c)^(1/3))/b/d^3

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5473, 1607, 5395, 3377, 2717, 2718} \[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {120960 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

[In]

Int[x^2*Cosh[a + b*(c + d*x)^(1/3)],x]

[Out]

(720*c*Cosh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^8*d^3)
- (6*c^2*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)
^(1/3)])/(b^4*d^3) - (20160*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) + (30*c*(c + d*x)^(4/3)*Cosh[a +
b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^4*d^3) - (24*(c + d*x)^(
7/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (120960*Sinh[a + b*(c + d*x)^(1/3)])/(b^9*d^3) + (6*c^2*Sinh[a +
b*(c + d*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*(c + d
*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^7*d^3) + (3*c^2*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^3)
- (120*c*(c + d*x)*Sinh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^(1/3)]
)/(b^5*d^3) - (6*c*(c + d*x)^(5/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^3) + (168*(c + d*x)^2*Sinh[a + b*(c + d*x
)^(1/3)])/(b^3*d^3) + (3*(c + d*x)^(8/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^3)

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2717

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

Rule 2718

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

Rule 3377

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 5395

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 5473

Int[((a_.) + Cosh[(c_.) + (d_.)*(u_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Cosh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (-c+x)^2 \cosh \left (a+b \sqrt [3]{x}\right ) \, dx,x,c+d x\right )}{d^3} \\ & = \frac {3 \text {Subst}\left (\int \left (-c x+x^4\right )^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3} \\ & = \frac {3 \text {Subst}\left (\int x^2 \left (-c+x^3\right )^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3} \\ & = \frac {3 \text {Subst}\left (\int \left (c^2 x^2 \cosh (a+b x)-2 c x^5 \cosh (a+b x)+x^8 \cosh (a+b x)\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3} \\ & = \frac {3 \text {Subst}\left (\int x^8 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}-\frac {(6 c) \text {Subst}\left (\int x^5 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3} \\ & = \frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {24 \text {Subst}\left (\int x^7 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac {(30 c) \text {Subst}\left (\int x^4 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}-\frac {\left (6 c^2\right ) \text {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3} \\ & = -\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 \text {Subst}\left (\int x^6 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {(120 c) \text {Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {\left (6 c^2\right ) \text {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3} \\ & = -\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {1008 \text {Subst}\left (\int x^5 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {(360 c) \text {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3} \\ & = -\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {5040 \text {Subst}\left (\int x^4 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {(720 c) \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3} \\ & = -\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {20160 \text {Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {(720 c) \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3} \\ & = \frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {60480 \text {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d^3} \\ & = \frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120960 \text {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^7 d^3} \\ & = \frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {120960 \text {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^8 d^3} \\ & = \frac {720 c \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac {120960 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac {6 c^2 \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {360 c (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {20160 (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac {30 c (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac {1008 (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac {24 (c+d x)^{7/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac {120960 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac {6 c^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac {720 c \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac {60480 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac {3 c^2 (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac {120 c (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {5040 (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac {6 c (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac {168 (c+d x)^2 \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac {3 (c+d x)^{8/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.66 \[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {e^{-a-b \sqrt [3]{c+d x}} \left (120960 \left (-1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right )-120960 b \left (1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) \sqrt [3]{c+d x}+60480 b^2 \left (-1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) (c+d x)^{2/3}+3 b^8 d^2 \left (-1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) x^2 (c+d x)^{2/3}-6 b^7 d \left (1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) x \sqrt [3]{c+d x} (3 c+4 d x)+720 b^4 \left (-1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) \sqrt [3]{c+d x} (6 c+7 d x)-72 b^5 \left (1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) (c+d x)^{2/3} (9 c+14 d x)-720 b^3 \left (1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) (27 c+28 d x)+6 b^6 \left (-1+e^{2 \left (a+b \sqrt [3]{c+d x}\right )}\right ) \left (9 c^2+36 c d x+28 d^2 x^2\right )\right )}{2 b^9 d^3} \]

[In]

Integrate[x^2*Cosh[a + b*(c + d*x)^(1/3)],x]

[Out]

(E^(-a - b*(c + d*x)^(1/3))*(120960*(-1 + E^(2*(a + b*(c + d*x)^(1/3)))) - 120960*b*(1 + E^(2*(a + b*(c + d*x)
^(1/3))))*(c + d*x)^(1/3) + 60480*b^2*(-1 + E^(2*(a + b*(c + d*x)^(1/3))))*(c + d*x)^(2/3) + 3*b^8*d^2*(-1 + E
^(2*(a + b*(c + d*x)^(1/3))))*x^2*(c + d*x)^(2/3) - 6*b^7*d*(1 + E^(2*(a + b*(c + d*x)^(1/3))))*x*(c + d*x)^(1
/3)*(3*c + 4*d*x) + 720*b^4*(-1 + E^(2*(a + b*(c + d*x)^(1/3))))*(c + d*x)^(1/3)*(6*c + 7*d*x) - 72*b^5*(1 + E
^(2*(a + b*(c + d*x)^(1/3))))*(c + d*x)^(2/3)*(9*c + 14*d*x) - 720*b^3*(1 + E^(2*(a + b*(c + d*x)^(1/3))))*(27
*c + 28*d*x) + 6*b^6*(-1 + E^(2*(a + b*(c + d*x)^(1/3))))*(9*c^2 + 36*c*d*x + 28*d^2*x^2)))/(2*b^9*d^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1814\) vs. \(2(477)=954\).

Time = 0.21 (sec) , antiderivative size = 1815, normalized size of antiderivative = 3.38

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1815\)
default	\(\text {Expression too large to display}\)	\(1815\)
parts	\(\text {Expression too large to display}\)	\(2938\)

[In]

int(x^2*cosh(a+b*(d*x+c)^(1/3)),x,method=_RETURNVERBOSE)

[Out]

3/d^3/b^3*(1/b^6*((a+b*(d*x+c)^(1/3))^8*sinh(a+b*(d*x+c)^(1/3))-8*(a+b*(d*x+c)^(1/3))^7*cosh(a+b*(d*x+c)^(1/3)
)+56*(a+b*(d*x+c)^(1/3))^6*sinh(a+b*(d*x+c)^(1/3))-336*(a+b*(d*x+c)^(1/3))^5*cosh(a+b*(d*x+c)^(1/3))+1680*(a+b
*(d*x+c)^(1/3))^4*sinh(a+b*(d*x+c)^(1/3))-6720*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))+20160*(a+b*(d*x+c
)^(1/3))^2*sinh(a+b*(d*x+c)^(1/3))-40320*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))+40320*sinh(a+b*(d*x+c)^(1
/3)))+1/b^6*a^8*sinh(a+b*(d*x+c)^(1/3))+c^2*a^2*sinh(a+b*(d*x+c)^(1/3))+c^2*((a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d
*x+c)^(1/3))-2*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))+2*sinh(a+b*(d*x+c)^(1/3)))-56/b^6*a^5*((a+b*(d*x+c)
^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))-3*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))+6*(a+b*(d*x+c)^(1/3))*sinh(a
+b*(d*x+c)^(1/3))-6*cosh(a+b*(d*x+c)^(1/3)))+2/b^3*a^5*c*sinh(a+b*(d*x+c)^(1/3))+70/b^6*a^4*((a+b*(d*x+c)^(1/3
))^4*sinh(a+b*(d*x+c)^(1/3))-4*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*sinh(a+b
*(d*x+c)^(1/3))-24*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))+24*sinh(a+b*(d*x+c)^(1/3)))-56/b^6*a^3*((a+b*(d
*x+c)^(1/3))^5*sinh(a+b*(d*x+c)^(1/3))-5*(a+b*(d*x+c)^(1/3))^4*cosh(a+b*(d*x+c)^(1/3))+20*(a+b*(d*x+c)^(1/3))^
3*sinh(a+b*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))+120*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d
*x+c)^(1/3))-120*cosh(a+b*(d*x+c)^(1/3)))-2/b^3*c*((a+b*(d*x+c)^(1/3))^5*sinh(a+b*(d*x+c)^(1/3))-5*(a+b*(d*x+c
)^(1/3))^4*cosh(a+b*(d*x+c)^(1/3))+20*(a+b*(d*x+c)^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*c
osh(a+b*(d*x+c)^(1/3))+120*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))-120*cosh(a+b*(d*x+c)^(1/3)))+28/b^6*a^2
*((a+b*(d*x+c)^(1/3))^6*sinh(a+b*(d*x+c)^(1/3))-6*(a+b*(d*x+c)^(1/3))^5*cosh(a+b*(d*x+c)^(1/3))+30*(a+b*(d*x+c
)^(1/3))^4*sinh(a+b*(d*x+c)^(1/3))-120*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))+360*(a+b*(d*x+c)^(1/3))^2
*sinh(a+b*(d*x+c)^(1/3))-720*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))+720*sinh(a+b*(d*x+c)^(1/3)))-8/b^6*a*
((a+b*(d*x+c)^(1/3))^7*sinh(a+b*(d*x+c)^(1/3))-7*(a+b*(d*x+c)^(1/3))^6*cosh(a+b*(d*x+c)^(1/3))+42*(a+b*(d*x+c)
^(1/3))^5*sinh(a+b*(d*x+c)^(1/3))-210*(a+b*(d*x+c)^(1/3))^4*cosh(a+b*(d*x+c)^(1/3))+840*(a+b*(d*x+c)^(1/3))^3*
sinh(a+b*(d*x+c)^(1/3))-2520*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))+5040*(a+b*(d*x+c)^(1/3))*sinh(a+b*(
d*x+c)^(1/3))-5040*cosh(a+b*(d*x+c)^(1/3)))-8/b^6*a^7*((a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))-cosh(a+b*(d
*x+c)^(1/3)))+28/b^6*a^6*((a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^(1/3))-2*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)
^(1/3))+2*sinh(a+b*(d*x+c)^(1/3)))-2*c^2*a*((a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))-cosh(a+b*(d*x+c)^(1/3)
))-20/b^3*c*a^2*((a+b*(d*x+c)^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))-3*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))
+6*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))-6*cosh(a+b*(d*x+c)^(1/3)))+10/b^3*c*a*((a+b*(d*x+c)^(1/3))^4*si
nh(a+b*(d*x+c)^(1/3))-4*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c
)^(1/3))-24*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))+24*sinh(a+b*(d*x+c)^(1/3)))-10/b^3*a^4*c*((a+b*(d*x+c)
^(1/3))*sinh(a+b*(d*x+c)^(1/3))-cosh(a+b*(d*x+c)^(1/3)))+20/b^3*a^3*c*((a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^
(1/3))-2*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))+2*sinh(a+b*(d*x+c)^(1/3))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.34 \[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (2 \, {\left (3360 \, b^{3} d x + 3240 \, b^{3} c + 12 \, {\left (14 \, b^{5} d x + 9 \, b^{5} c\right )} {\left (d x + c\right )}^{\frac {2}{3}} + {\left (4 \, b^{7} d^{2} x^{2} + 3 \, b^{7} c d x + 20160 \, b\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left (56 \, b^{6} d^{2} x^{2} + 72 \, b^{6} c d x + 18 \, b^{6} c^{2} + {\left (b^{8} d^{2} x^{2} + 20160 \, b^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}} + 240 \, {\left (7 \, b^{4} d x + 6 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} + 40320\right )} \sinh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{9} d^{3}} \]

[In]

integrate(x^2*cosh(a+b*(d*x+c)^(1/3)),x, algorithm="fricas")

[Out]

-3*(2*(3360*b^3*d*x + 3240*b^3*c + 12*(14*b^5*d*x + 9*b^5*c)*(d*x + c)^(2/3) + (4*b^7*d^2*x^2 + 3*b^7*c*d*x +
20160*b)*(d*x + c)^(1/3))*cosh((d*x + c)^(1/3)*b + a) - (56*b^6*d^2*x^2 + 72*b^6*c*d*x + 18*b^6*c^2 + (b^8*d^2
*x^2 + 20160*b^2)*(d*x + c)^(2/3) + 240*(7*b^4*d*x + 6*b^4*c)*(d*x + c)^(1/3) + 40320)*sinh((d*x + c)^(1/3)*b
+ a))/(b^9*d^3)

Sympy [F]

\[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int x^{2} \cosh {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \]

[In]

integrate(x**2*cosh(a+b*(d*x+c)**(1/3)),x)

[Out]

Integral(x**2*cosh(a + b*(c + d*x)**(1/3)), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.20 \[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {2 \, d^{3} x^{3} \cosh \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left (\frac {c^{3} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}}{b} + \frac {c^{3} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b} - \frac {3 \, {\left ({\left (d x + c\right )} b^{3} e^{a} - 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 6 \, e^{a}\right )} c^{2} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{4}} - \frac {3 \, {\left ({\left (d x + c\right )} b^{3} + 3 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 6\right )} c^{2} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{4}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} e^{a} - 120 \, {\left (d x + c\right )} b^{3} e^{a} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} - 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} + 720 \, e^{a}\right )} c e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{7}} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} + 30 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} + 120 \, {\left (d x + c\right )} b^{3} + 360 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 720 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 720\right )} c e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{7}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{9} e^{a} - 9 \, {\left (d x + c\right )}^{\frac {8}{3}} b^{8} e^{a} + 72 \, {\left (d x + c\right )}^{\frac {7}{3}} b^{7} e^{a} - 504 \, {\left (d x + c\right )}^{2} b^{6} e^{a} + 3024 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} e^{a} - 15120 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} e^{a} + 60480 \, {\left (d x + c\right )} b^{3} e^{a} - 181440 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} e^{a} + 362880 \, {\left (d x + c\right )}^{\frac {1}{3}} b e^{a} - 362880 \, e^{a}\right )} e^{\left ({\left (d x + c\right )}^{\frac {1}{3}} b\right )}}{b^{10}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{9} + 9 \, {\left (d x + c\right )}^{\frac {8}{3}} b^{8} + 72 \, {\left (d x + c\right )}^{\frac {7}{3}} b^{7} + 504 \, {\left (d x + c\right )}^{2} b^{6} + 3024 \, {\left (d x + c\right )}^{\frac {5}{3}} b^{5} + 15120 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{4} + 60480 \, {\left (d x + c\right )} b^{3} + 181440 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} + 362880 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 362880\right )} e^{\left (-{\left (d x + c\right )}^{\frac {1}{3}} b - a\right )}}{b^{10}}\right )} b}{6 \, d^{3}} \]

[In]

integrate(x^2*cosh(a+b*(d*x+c)^(1/3)),x, algorithm="maxima")

[Out]

1/6*(2*d^3*x^3*cosh((d*x + c)^(1/3)*b + a) + (c^3*e^((d*x + c)^(1/3)*b + a)/b + c^3*e^(-(d*x + c)^(1/3)*b - a)
/b - 3*((d*x + c)*b^3*e^a - 3*(d*x + c)^(2/3)*b^2*e^a + 6*(d*x + c)^(1/3)*b*e^a - 6*e^a)*c^2*e^((d*x + c)^(1/3
)*b)/b^4 - 3*((d*x + c)*b^3 + 3*(d*x + c)^(2/3)*b^2 + 6*(d*x + c)^(1/3)*b + 6)*c^2*e^(-(d*x + c)^(1/3)*b - a)/
b^4 + 3*((d*x + c)^2*b^6*e^a - 6*(d*x + c)^(5/3)*b^5*e^a + 30*(d*x + c)^(4/3)*b^4*e^a - 120*(d*x + c)*b^3*e^a
+ 360*(d*x + c)^(2/3)*b^2*e^a - 720*(d*x + c)^(1/3)*b*e^a + 720*e^a)*c*e^((d*x + c)^(1/3)*b)/b^7 + 3*((d*x + c
)^2*b^6 + 6*(d*x + c)^(5/3)*b^5 + 30*(d*x + c)^(4/3)*b^4 + 120*(d*x + c)*b^3 + 360*(d*x + c)^(2/3)*b^2 + 720*(
d*x + c)^(1/3)*b + 720)*c*e^(-(d*x + c)^(1/3)*b - a)/b^7 - ((d*x + c)^3*b^9*e^a - 9*(d*x + c)^(8/3)*b^8*e^a +
72*(d*x + c)^(7/3)*b^7*e^a - 504*(d*x + c)^2*b^6*e^a + 3024*(d*x + c)^(5/3)*b^5*e^a - 15120*(d*x + c)^(4/3)*b^
4*e^a + 60480*(d*x + c)*b^3*e^a - 181440*(d*x + c)^(2/3)*b^2*e^a + 362880*(d*x + c)^(1/3)*b*e^a - 362880*e^a)*
e^((d*x + c)^(1/3)*b)/b^10 - ((d*x + c)^3*b^9 + 9*(d*x + c)^(8/3)*b^8 + 72*(d*x + c)^(7/3)*b^7 + 504*(d*x + c)
^2*b^6 + 3024*(d*x + c)^(5/3)*b^5 + 15120*(d*x + c)^(4/3)*b^4 + 60480*(d*x + c)*b^3 + 181440*(d*x + c)^(2/3)*b
^2 + 362880*(d*x + c)^(1/3)*b + 362880)*e^(-(d*x + c)^(1/3)*b - a)/b^10)*b)/d^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2163 vs. \(2 (477) = 954\).

Time = 0.36 (sec) , antiderivative size = 2163, normalized size of antiderivative = 4.03 \[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x^2*cosh(a+b*(d*x+c)^(1/3)),x, algorithm="giac")

[Out]

3/2*((((d*x + c)^(1/3)*b + a)^2*b^6*c^2 - 2*((d*x + c)^(1/3)*b + a)*a*b^6*c^2 + a^2*b^6*c^2 - 2*((d*x + c)^(1/
3)*b + a)^5*b^3*c + 10*((d*x + c)^(1/3)*b + a)^4*a*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3*a^2*b^3*c + 20*((d*x +
c)^(1/3)*b + a)^2*a^3*b^3*c - 10*((d*x + c)^(1/3)*b + a)*a^4*b^3*c + 2*a^5*b^3*c - 2*((d*x + c)^(1/3)*b + a)*
b^6*c^2 + 2*a*b^6*c^2 + ((d*x + c)^(1/3)*b + a)^8 - 8*((d*x + c)^(1/3)*b + a)^7*a + 28*((d*x + c)^(1/3)*b + a)
^6*a^2 - 56*((d*x + c)^(1/3)*b + a)^5*a^3 + 70*((d*x + c)^(1/3)*b + a)^4*a^4 - 56*((d*x + c)^(1/3)*b + a)^3*a^
5 + 28*((d*x + c)^(1/3)*b + a)^2*a^6 - 8*((d*x + c)^(1/3)*b + a)*a^7 + a^8 + 10*((d*x + c)^(1/3)*b + a)^4*b^3*
c - 40*((d*x + c)^(1/3)*b + a)^3*a*b^3*c + 60*((d*x + c)^(1/3)*b + a)^2*a^2*b^3*c - 40*((d*x + c)^(1/3)*b + a)
*a^3*b^3*c + 10*a^4*b^3*c + 2*b^6*c^2 - 8*((d*x + c)^(1/3)*b + a)^7 + 56*((d*x + c)^(1/3)*b + a)^6*a - 168*((d
*x + c)^(1/3)*b + a)^5*a^2 + 280*((d*x + c)^(1/3)*b + a)^4*a^3 - 280*((d*x + c)^(1/3)*b + a)^3*a^4 + 168*((d*x
+ c)^(1/3)*b + a)^2*a^5 - 56*((d*x + c)^(1/3)*b + a)*a^6 + 8*a^7 - 40*((d*x + c)^(1/3)*b + a)^3*b^3*c + 120*(
(d*x + c)^(1/3)*b + a)^2*a*b^3*c - 120*((d*x + c)^(1/3)*b + a)*a^2*b^3*c + 40*a^3*b^3*c + 56*((d*x + c)^(1/3)*
b + a)^6 - 336*((d*x + c)^(1/3)*b + a)^5*a + 840*((d*x + c)^(1/3)*b + a)^4*a^2 - 1120*((d*x + c)^(1/3)*b + a)^
3*a^3 + 840*((d*x + c)^(1/3)*b + a)^2*a^4 - 336*((d*x + c)^(1/3)*b + a)*a^5 + 56*a^6 + 120*((d*x + c)^(1/3)*b
+ a)^2*b^3*c - 240*((d*x + c)^(1/3)*b + a)*a*b^3*c + 120*a^2*b^3*c - 336*((d*x + c)^(1/3)*b + a)^5 + 1680*((d*
x + c)^(1/3)*b + a)^4*a - 3360*((d*x + c)^(1/3)*b + a)^3*a^2 + 3360*((d*x + c)^(1/3)*b + a)^2*a^3 - 1680*((d*x
+ c)^(1/3)*b + a)*a^4 + 336*a^5 - 240*((d*x + c)^(1/3)*b + a)*b^3*c + 240*a*b^3*c + 1680*((d*x + c)^(1/3)*b +
a)^4 - 6720*((d*x + c)^(1/3)*b + a)^3*a + 10080*((d*x + c)^(1/3)*b + a)^2*a^2 - 6720*((d*x + c)^(1/3)*b + a)*
a^3 + 1680*a^4 + 240*b^3*c - 6720*((d*x + c)^(1/3)*b + a)^3 + 20160*((d*x + c)^(1/3)*b + a)^2*a - 20160*((d*x
+ c)^(1/3)*b + a)*a^2 + 6720*a^3 + 20160*((d*x + c)^(1/3)*b + a)^2 - 40320*((d*x + c)^(1/3)*b + a)*a + 20160*a
^2 - 40320*(d*x + c)^(1/3)*b + 40320)*e^((d*x + c)^(1/3)*b + a)/(b^8*d^2) - (((d*x + c)^(1/3)*b + a)^2*b^6*c^2
- 2*((d*x + c)^(1/3)*b + a)*a*b^6*c^2 + a^2*b^6*c^2 - 2*((d*x + c)^(1/3)*b + a)^5*b^3*c + 10*((d*x + c)^(1/3)
*b + a)^4*a*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3*a^2*b^3*c + 20*((d*x + c)^(1/3)*b + a)^2*a^3*b^3*c - 10*((d*x
+ c)^(1/3)*b + a)*a^4*b^3*c + 2*a^5*b^3*c + 2*((d*x + c)^(1/3)*b + a)*b^6*c^2 - 2*a*b^6*c^2 + ((d*x + c)^(1/3
)*b + a)^8 - 8*((d*x + c)^(1/3)*b + a)^7*a + 28*((d*x + c)^(1/3)*b + a)^6*a^2 - 56*((d*x + c)^(1/3)*b + a)^5*a
^3 + 70*((d*x + c)^(1/3)*b + a)^4*a^4 - 56*((d*x + c)^(1/3)*b + a)^3*a^5 + 28*((d*x + c)^(1/3)*b + a)^2*a^6 -
8*((d*x + c)^(1/3)*b + a)*a^7 + a^8 - 10*((d*x + c)^(1/3)*b + a)^4*b^3*c + 40*((d*x + c)^(1/3)*b + a)^3*a*b^3*
c - 60*((d*x + c)^(1/3)*b + a)^2*a^2*b^3*c + 40*((d*x + c)^(1/3)*b + a)*a^3*b^3*c - 10*a^4*b^3*c + 2*b^6*c^2 +
8*((d*x + c)^(1/3)*b + a)^7 - 56*((d*x + c)^(1/3)*b + a)^6*a + 168*((d*x + c)^(1/3)*b + a)^5*a^2 - 280*((d*x
+ c)^(1/3)*b + a)^4*a^3 + 280*((d*x + c)^(1/3)*b + a)^3*a^4 - 168*((d*x + c)^(1/3)*b + a)^2*a^5 + 56*((d*x + c
)^(1/3)*b + a)*a^6 - 8*a^7 - 40*((d*x + c)^(1/3)*b + a)^3*b^3*c + 120*((d*x + c)^(1/3)*b + a)^2*a*b^3*c - 120*
((d*x + c)^(1/3)*b + a)*a^2*b^3*c + 40*a^3*b^3*c + 56*((d*x + c)^(1/3)*b + a)^6 - 336*((d*x + c)^(1/3)*b + a)^
5*a + 840*((d*x + c)^(1/3)*b + a)^4*a^2 - 1120*((d*x + c)^(1/3)*b + a)^3*a^3 + 840*((d*x + c)^(1/3)*b + a)^2*a
^4 - 336*((d*x + c)^(1/3)*b + a)*a^5 + 56*a^6 - 120*((d*x + c)^(1/3)*b + a)^2*b^3*c + 240*((d*x + c)^(1/3)*b +
a)*a*b^3*c - 120*a^2*b^3*c + 336*((d*x + c)^(1/3)*b + a)^5 - 1680*((d*x + c)^(1/3)*b + a)^4*a + 3360*((d*x +
c)^(1/3)*b + a)^3*a^2 - 3360*((d*x + c)^(1/3)*b + a)^2*a^3 + 1680*((d*x + c)^(1/3)*b + a)*a^4 - 336*a^5 - 240*
((d*x + c)^(1/3)*b + a)*b^3*c + 240*a*b^3*c + 1680*((d*x + c)^(1/3)*b + a)^4 - 6720*((d*x + c)^(1/3)*b + a)^3*
a + 10080*((d*x + c)^(1/3)*b + a)^2*a^2 - 6720*((d*x + c)^(1/3)*b + a)*a^3 + 1680*a^4 - 240*b^3*c + 6720*((d*x
+ c)^(1/3)*b + a)^3 - 20160*((d*x + c)^(1/3)*b + a)^2*a + 20160*((d*x + c)^(1/3)*b + a)*a^2 - 6720*a^3 + 2016
0*((d*x + c)^(1/3)*b + a)^2 - 40320*((d*x + c)^(1/3)*b + a)*a + 20160*a^2 + 40320*(d*x + c)^(1/3)*b + 40320)*e
^(-(d*x + c)^(1/3)*b - a)/(b^8*d^2))/(b*d)

Mupad [F(-1)]

Timed out. \[ \int x^2 \cosh \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int x^2\,\mathrm {cosh}\left (a+b\,{\left (c+d\,x\right )}^{1/3}\right ) \,d x \]

[In]

int(x^2*cosh(a + b*(c + d*x)^(1/3)),x)

[Out]

int(x^2*cosh(a + b*(c + d*x)^(1/3)), x)