3.471 \(\int x^3 (d+e x^2)^2 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=341 \[ \frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b x^3 \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{4608 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{3072 c^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e x^5 \left (1-c^2 x^2\right ) \left (64 c^2 d+21 e\right )}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x*(1 - c^2*x^2))/(3072*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(288*c^
4*d^2 + 320*c^2*d*e + 105*e^2)*x^3*(1 - c^2*x^2))/(4608*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e*(64*c^2*d + 2
1*e)*x^5*(1 - c^2*x^2))/(1152*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*x^7*(1 - c^2*x^2))/(64*c*Sqrt[-1 + c*
x]*Sqrt[1 + c*x]) + (d^2*x^4*(a + b*ArcCosh[c*x]))/4 + (d*e*x^6*(a + b*ArcCosh[c*x]))/3 + (e^2*x^8*(a + b*ArcC
osh[c*x]))/8 - (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/
(3072*c^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Rubi [A]  time = 0.360308, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {266, 43, 5790, 12, 520, 1267, 459, 321, 217, 206} \[ \frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b x^3 \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{4608 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left (1-c^2 x^2\right ) \left (288 c^4 d^2+320 c^2 d e+105 e^2\right )}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{3072 c^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e x^5 \left (1-c^2 x^2\right ) \left (64 c^2 d+21 e\right )}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]

[Out]

(b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x*(1 - c^2*x^2))/(3072*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(288*c^
4*d^2 + 320*c^2*d*e + 105*e^2)*x^3*(1 - c^2*x^2))/(4608*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e*(64*c^2*d + 2
1*e)*x^5*(1 - c^2*x^2))/(1152*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*x^7*(1 - c^2*x^2))/(64*c*Sqrt[-1 + c*
x]*Sqrt[1 + c*x]) + (d^2*x^4*(a + b*ArcCosh[c*x]))/4 + (d*e*x^6*(a + b*ArcCosh[c*x]))/3 + (e^2*x^8*(a + b*ArcC
osh[c*x]))/8 - (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/
(3072*c^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5790

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[
1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] &
& (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rule 459

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{24} (b c) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{24 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4 \left (48 c^2 d^2+e \left (64 c^2 d+21 e\right ) x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{192 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{x^4}{\sqrt{-1+c^2 x^2}} \, dx}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{1536 c^5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )+\frac{\left (b \left (-288 c^4 d^2-5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x \left (1-c^2 x^2\right )}{3072 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) x^3 \left (1-c^2 x^2\right )}{4608 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (64 c^2 d+21 e\right ) x^5 \left (1-c^2 x^2\right )}{1152 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 x^7 \left (1-c^2 x^2\right )}{64 c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{4} d^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d e x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{8} e^2 x^8 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{3072 c^8 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.335491, size = 214, normalized size = 0.63 \[ \frac{384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-b c x \sqrt{c x-1} \sqrt{c x+1} \left (16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+30 c^2 e \left (32 d+7 e x^2\right )+315 e^2\right )+384 b c^8 x^4 \cosh ^{-1}(c x) \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-6 b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )}{9216 c^8} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(315*e^2 + 30*c^2*e*(32*d
+ 7*e*x^2) + 8*c^4*(108*d^2 + 80*d*e*x^2 + 21*e^2*x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 384*b
*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x] - 6*b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*ArcTanh[Sqrt
[(-1 + c*x)/(1 + c*x)]])/(9216*c^8)

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Maple [A]  time = 0.019, size = 440, normalized size = 1.3 \begin{align*}{\frac{a{e}^{2}{x}^{8}}{8}}+{\frac{ade{x}^{6}}{3}}+{\frac{{d}^{2}a{x}^{4}}{4}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{2}{x}^{8}}{8}}+{\frac{b{\rm arccosh} \left (cx\right )de{x}^{6}}{3}}+{\frac{{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}-{\frac{b{e}^{2}{x}^{7}}{64\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{x}^{5}de}{18\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{2}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{7\,b{e}^{2}{x}^{5}}{384\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bde{x}^{3}}{72\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{2}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{2}}{32\,{c}^{4}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{35\,b{e}^{2}{x}^{3}}{1536\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bdex}{48\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,bde}{48\,{c}^{6}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{35\,b{e}^{2}x}{1024\,{c}^{7}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{35\,b{e}^{2}}{1024\,{c}^{8}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8+1/3*a*d*e*x^6+1/4*d^2*a*x^4+1/8*b*arccosh(c*x)*e^2*x^8+1/3*b*arccosh(c*x)*d*e*x^6+1/4*d^2*b*arcc
osh(c*x)*x^4-1/64/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2*x^7-1/18/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^5*d*e-1/16*b*
d^2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-7/384/c^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2*x^5-5/72/c^3*b*(c*x-1)^(1/2)
*(c*x+1)^(1/2)*d*e*x^3-3/32*b*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-3/32/c^4*d^2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/(c^2*x^2-1)^(1/2)*ln(c*x+(c^2*x^2-1)^(1/2))-35/1536/c^5*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2*x^3-5/48/c^5*b*(c*x
-1)^(1/2)*(c*x+1)^(1/2)*d*e*x-5/48/c^6*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d*e*ln(c*x+(c^2*x^2-1)^
(1/2))-35/1024/c^7*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2*x-35/1024/c^8*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(
1/2)*e^2*ln(c*x+(c^2*x^2-1)^(1/2))

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Maxima [A]  time = 1.14553, size = 485, normalized size = 1.42 \begin{align*} \frac{1}{8} \, a e^{2} x^{8} + \frac{1}{3} \, a d e x^{6} + \frac{1}{4} \, a d^{2} x^{4} + \frac{1}{32} \,{\left (8 \, x^{4} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{2 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} x}{c^{4}} + \frac{3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d^{2} + \frac{1}{144} \,{\left (48 \, x^{6} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} - 1} x}{c^{6}} + \frac{15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b d e + \frac{1}{3072} \,{\left (384 \, x^{8} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{c^{2} x^{2} - 1} x}{c^{8}} + \frac{105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sq
rt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d^2 + 1/144*(48*x
^6*arccosh(c*x) - (8*sqrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c^6 + 1
5*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^6))*c)*b*d*e + 1/3072*(384*x^8*arccosh(c*x) - (48*
sqrt(c^2*x^2 - 1)*x^7/c^2 + 56*sqrt(c^2*x^2 - 1)*x^5/c^4 + 70*sqrt(c^2*x^2 - 1)*x^3/c^6 + 105*sqrt(c^2*x^2 - 1
)*x/c^8 + 105*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^8))*c)*b*e^2

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Fricas [A]  time = 2.54593, size = 539, normalized size = 1.58 \begin{align*} \frac{1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \,{\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (144 \, b c^{7} e^{2} x^{7} + 8 \,{\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \,{\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \,{\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{9216 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9216*(1152*a*c^8*e^2*x^8 + 3072*a*c^8*d*e*x^6 + 2304*a*c^8*d^2*x^4 + 3*(384*b*c^8*e^2*x^8 + 1024*b*c^8*d*e*x
^6 + 768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 - 320*b*c^2*d*e - 105*b*e^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (144*b*c^7*
e^2*x^7 + 8*(64*b*c^7*d*e + 21*b*c^5*e^2)*x^5 + 2*(288*b*c^7*d^2 + 320*b*c^5*d*e + 105*b*c^3*e^2)*x^3 + 3*(288
*b*c^5*d^2 + 320*b*c^3*d*e + 105*b*c*e^2)*x)*sqrt(c^2*x^2 - 1))/c^8

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Sympy [A]  time = 19.7484, size = 389, normalized size = 1.14 \begin{align*} \begin{cases} \frac{a d^{2} x^{4}}{4} + \frac{a d e x^{6}}{3} + \frac{a e^{2} x^{8}}{8} + \frac{b d^{2} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} + \frac{b d e x^{6} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{b e^{2} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} - \frac{b d^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{16 c} - \frac{b d e x^{5} \sqrt{c^{2} x^{2} - 1}}{18 c} - \frac{b e^{2} x^{7} \sqrt{c^{2} x^{2} - 1}}{64 c} - \frac{3 b d^{2} x \sqrt{c^{2} x^{2} - 1}}{32 c^{3}} - \frac{5 b d e x^{3} \sqrt{c^{2} x^{2} - 1}}{72 c^{3}} - \frac{7 b e^{2} x^{5} \sqrt{c^{2} x^{2} - 1}}{384 c^{3}} - \frac{3 b d^{2} \operatorname{acosh}{\left (c x \right )}}{32 c^{4}} - \frac{5 b d e x \sqrt{c^{2} x^{2} - 1}}{48 c^{5}} - \frac{35 b e^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{1536 c^{5}} - \frac{5 b d e \operatorname{acosh}{\left (c x \right )}}{48 c^{6}} - \frac{35 b e^{2} x \sqrt{c^{2} x^{2} - 1}}{1024 c^{7}} - \frac{35 b e^{2} \operatorname{acosh}{\left (c x \right )}}{1024 c^{8}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d^{2} x^{4}}{4} + \frac{d e x^{6}}{3} + \frac{e^{2} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)**2*(a+b*acosh(c*x)),x)

[Out]

Piecewise((a*d**2*x**4/4 + a*d*e*x**6/3 + a*e**2*x**8/8 + b*d**2*x**4*acosh(c*x)/4 + b*d*e*x**6*acosh(c*x)/3 +
b*e**2*x**8*acosh(c*x)/8 - b*d**2*x**3*sqrt(c**2*x**2 - 1)/(16*c) - b*d*e*x**5*sqrt(c**2*x**2 - 1)/(18*c) - b
*e**2*x**7*sqrt(c**2*x**2 - 1)/(64*c) - 3*b*d**2*x*sqrt(c**2*x**2 - 1)/(32*c**3) - 5*b*d*e*x**3*sqrt(c**2*x**2
- 1)/(72*c**3) - 7*b*e**2*x**5*sqrt(c**2*x**2 - 1)/(384*c**3) - 3*b*d**2*acosh(c*x)/(32*c**4) - 5*b*d*e*x*sqr
t(c**2*x**2 - 1)/(48*c**5) - 35*b*e**2*x**3*sqrt(c**2*x**2 - 1)/(1536*c**5) - 5*b*d*e*acosh(c*x)/(48*c**6) - 3
5*b*e**2*x*sqrt(c**2*x**2 - 1)/(1024*c**7) - 35*b*e**2*acosh(c*x)/(1024*c**8), Ne(c, 0)), ((a + I*pi*b/2)*(d**
2*x**4/4 + d*e*x**6/3 + e**2*x**8/8), True))

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Giac [A]  time = 1.44553, size = 432, normalized size = 1.27 \begin{align*} \frac{1}{4} \, a d^{2} x^{4} + \frac{1}{32} \,{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{c^{2}} + \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b d^{2} + \frac{1}{3072} \,{\left (384 \, a x^{8} +{\left (384 \, x^{8} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \, x^{2}{\left (\frac{6 \, x^{2}}{c^{2}} + \frac{7}{c^{4}}\right )} + \frac{35}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x - \frac{105 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{8}{\left | c \right |}}\right )} c\right )} b\right )} e^{2} + \frac{1}{144} \,{\left (48 \, a d x^{6} +{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} + \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x - \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b d\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*d^2*x^4 + 1/32*(8*x^4*log(c*x + sqrt(c^2*x^2 - 1)) - (sqrt(c^2*x^2 - 1)*x*(2*x^2/c^2 + 3/c^4) - 3*log(ab
s(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^4*abs(c)))*c)*b*d^2 + 1/3072*(384*a*x^8 + (384*x^8*log(c*x + sqrt(c^2*x^2
- 1)) - (sqrt(c^2*x^2 - 1)*(2*(4*x^2*(6*x^2/c^2 + 7/c^4) + 35/c^6)*x^2 + 105/c^8)*x - 105*log(abs(-x*abs(c) +
sqrt(c^2*x^2 - 1)))/(c^8*abs(c)))*c)*b)*e^2 + 1/144*(48*a*d*x^6 + (48*x^6*log(c*x + sqrt(c^2*x^2 - 1)) - (sqr
t(c^2*x^2 - 1)*(2*x^2*(4*x^2/c^2 + 5/c^4) + 15/c^6)*x - 15*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^6*abs(c)
))*c)*b*d)*e