3.5.71 \(\int x^3 (d+e x^2)^2 (a+b \cosh ^{-1}(c x)) \, dx\) [471]

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

[Out]

1/4*d^2*x^4*(a+b*arccosh(c*x))+1/3*d*e*x^6*(a+b*arccosh(c*x))+1/8*e^2*x^8*(a+b*arccosh(c*x))+1/3072*b*(288*c^4
*d^2+320*c^2*d*e+105*e^2)*x*(-c^2*x^2+1)/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/4608*b*(288*c^4*d^2+320*c^2*d*e+105
*e^2)*x^3*(-c^2*x^2+1)/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/1152*b*e*(64*c^2*d+21*e)*x^5*(-c^2*x^2+1)/c^3/(c*x-1)
^(1/2)/(c*x+1)^(1/2)+1/64*b*e^2*x^7*(-c^2*x^2+1)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3072*b*(288*c^4*d^2+320*c^2*d
*e+105*e^2)*arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c^8/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 341, normalized size of antiderivative = 1.00, 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 = {272, 45, 5958, 12, 534, 1281, 470, 327, 223, 212} \begin {gather*} \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 e^2 x^7 \left (1-c^2 x^2\right )}{64 c \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 \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 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 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}} \end {gather*}

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 12

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

Rule 45

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 212

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

Rule 223

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 272

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 327

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 470

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 534

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 1281

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 5958

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]))

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 ) \text {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*}

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Mathematica [A]
time = 0.20, size = 220, normalized size = 0.65 \begin {gather*} \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 {-1+c x} \sqrt {1+c x} \left (315 e^2+30 c^2 e \left (32 d+7 e x^2\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )\right )+384 b c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \cosh ^{-1}(c x)-3 b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{9216 c^8} \end {gather*}

Antiderivative was successfully verified.

[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] - 3*b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*Log[c*x + Sq
rt[-1 + c*x]*Sqrt[1 + c*x]])/(9216*c^8)

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Maple [A]
time = 2.87, size = 525, normalized size = 1.54

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} c^{8} e^{2} x^{8}\right )}{c^{4}}-\frac {b \,c^{4} \mathrm {arccosh}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\mathrm {arccosh}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \mathrm {arccosh}\left (c x \right ) x^{8}}{8}+\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{24 e^{2} \sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} c^{3} x^{3}}{16}-\frac {b \,c^{3} e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{5}}{18}-\frac {b \,c^{3} e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{7}}{64}-\frac {3 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {5 b c e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{3}}{72}-\frac {7 b c \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{384}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d^{2}}{32 \sqrt {c^{2} x^{2}-1}}-\frac {5 b d e x \sqrt {c x -1}\, \sqrt {c x +1}}{48 c}-\frac {35 b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{1536 c}-\frac {5 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{48 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {35 b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{1024 c^{3}}-\frac {35 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{1024 c^{4} \sqrt {c^{2} x^{2}-1}}}{c^{4}}\)	\(525\)
default	\(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} c^{8} e^{2} x^{8}\right )}{c^{4}}-\frac {b \,c^{4} \mathrm {arccosh}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\mathrm {arccosh}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \mathrm {arccosh}\left (c x \right ) x^{8}}{8}+\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, d^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{24 e^{2} \sqrt {c^{2} x^{2}-1}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} c^{3} x^{3}}{16}-\frac {b \,c^{3} e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{5}}{18}-\frac {b \,c^{3} e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{7}}{64}-\frac {3 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {5 b c e \sqrt {c x -1}\, \sqrt {c x +1}\, d \,x^{3}}{72}-\frac {7 b c \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{5}}{384}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d^{2}}{32 \sqrt {c^{2} x^{2}-1}}-\frac {5 b d e x \sqrt {c x -1}\, \sqrt {c x +1}}{48 c}-\frac {35 b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{1536 c}-\frac {5 b e \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{48 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {35 b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{1024 c^{3}}-\frac {35 b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{1024 c^{4} \sqrt {c^{2} x^{2}-1}}}{c^{4}}\)	\(525\)

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,method=_RETURNVERBOSE)

[Out]

1/c^4*(a/c^4*(1/4*c^8*d^2*x^4+1/3*c^8*d*e*x^6+1/8*c^8*e^2*x^8)-1/24*b*c^4/e^2*arccosh(c*x)*d^4+1/4*b*arccosh(c
*x)*d^2*c^4*x^4+1/3*b*c^4*e*arccosh(c*x)*d*x^6+1/8*b*c^4*e^2*arccosh(c*x)*x^8+1/24*b*c^4/e^2*(c*x-1)^(1/2)*(c*
x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d^4*ln(c*x+(c^2*x^2-1)^(1/2))-1/16*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d^2*c^3*x^3-1/18
*b*c^3*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d*x^5-1/64*b*c^3*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^7-3/32*b*c*d^2*x*(c*x-
1)^(1/2)*(c*x+1)^(1/2)-5/72*b*c*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d*x^3-7/384*b*c*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*
x^5-3/32*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))*d^2-5/48*b*d*e*x*(c*x-1)^(1
/2)*(c*x+1)^(1/2)/c-35/1536*b*e^2*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-5/48*b/c^2*e*(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))*d-35/1024*b*e^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-35/1024*b/c^4*e^2
*(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)))

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Maxima [A]
time = 0.28, size = 332, normalized size = 0.97 \begin {gather*} \frac {1}{8} \, a x^{8} e^{2} + \frac {1}{3} \, a d x^{6} e + \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} c\right )}{c^{5}}\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} c\right )}{c^{7}}\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} c\right )}{c^{9}}\right )} c\right )} b e^{2} \end {gather*}

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*x^8*e^2 + 1/3*a*d*x^6*e + 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)*c)/c^5)*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 + 15*log(2*c^2*x + 2*sq
rt(c^2*x^2 - 1)*c)/c^7)*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)*c)/c^9)*c)*b*e^2

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Fricas [A]
time = 0.39, size = 438, normalized size = 1.28 \begin {gather*} \frac {1152 \, a c^{8} x^{8} \cosh \left (1\right )^{2} + 1152 \, a c^{8} x^{8} \sinh \left (1\right )^{2} + 3072 \, a c^{8} d x^{6} \cosh \left (1\right ) + 2304 \, a c^{8} d^{2} x^{4} + 3 \, {\left (768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} + 3 \, {\left (128 \, b c^{8} x^{8} - 35 \, b\right )} \cosh \left (1\right )^{2} + 3 \, {\left (128 \, b c^{8} x^{8} - 35 \, b\right )} \sinh \left (1\right )^{2} + 64 \, {\left (16 \, b c^{8} d x^{6} - 5 \, b c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (512 \, b c^{8} d x^{6} - 160 \, b c^{2} d + 3 \, {\left (128 \, b c^{8} x^{8} - 35 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 768 \, {\left (3 \, a c^{8} x^{8} \cosh \left (1\right ) + 4 \, a c^{8} d x^{6}\right )} \sinh \left (1\right ) - {\left (576 \, b c^{7} d^{2} x^{3} + 864 \, b c^{5} d^{2} x + 3 \, {\left (48 \, b c^{7} x^{7} + 56 \, b c^{5} x^{5} + 70 \, b c^{3} x^{3} + 105 \, b c x\right )} \cosh \left (1\right )^{2} + 3 \, {\left (48 \, b c^{7} x^{7} + 56 \, b c^{5} x^{5} + 70 \, b c^{3} x^{3} + 105 \, b c x\right )} \sinh \left (1\right )^{2} + 64 \, {\left (8 \, b c^{7} d x^{5} + 10 \, b c^{5} d x^{3} + 15 \, b c^{3} d x\right )} \cosh \left (1\right ) + 2 \, {\left (256 \, b c^{7} d x^{5} + 320 \, b c^{5} d x^{3} + 480 \, b c^{3} d x + 3 \, {\left (48 \, b c^{7} x^{7} + 56 \, b c^{5} x^{5} + 70 \, b c^{3} x^{3} + 105 \, b c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{9216 \, c^{8}} \end {gather*}

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*x^8*cosh(1)^2 + 1152*a*c^8*x^8*sinh(1)^2 + 3072*a*c^8*d*x^6*cosh(1) + 2304*a*c^8*d^2*x^4 +
3*(768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 + 3*(128*b*c^8*x^8 - 35*b)*cosh(1)^2 + 3*(128*b*c^8*x^8 - 35*b)*sinh(1)^2
+ 64*(16*b*c^8*d*x^6 - 5*b*c^2*d)*cosh(1) + 2*(512*b*c^8*d*x^6 - 160*b*c^2*d + 3*(128*b*c^8*x^8 - 35*b)*cosh(
1))*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) + 768*(3*a*c^8*x^8*cosh(1) + 4*a*c^8*d*x^6)*sinh(1) - (576*b*c^7*d^2
*x^3 + 864*b*c^5*d^2*x + 3*(48*b*c^7*x^7 + 56*b*c^5*x^5 + 70*b*c^3*x^3 + 105*b*c*x)*cosh(1)^2 + 3*(48*b*c^7*x^
7 + 56*b*c^5*x^5 + 70*b*c^3*x^3 + 105*b*c*x)*sinh(1)^2 + 64*(8*b*c^7*d*x^5 + 10*b*c^5*d*x^3 + 15*b*c^3*d*x)*co
sh(1) + 2*(256*b*c^7*d*x^5 + 320*b*c^5*d*x^3 + 480*b*c^3*d*x + 3*(48*b*c^7*x^7 + 56*b*c^5*x^5 + 70*b*c^3*x^3 +
105*b*c*x)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))/c^8

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Sympy [C] Result contains complex when optimal does not.
time = 1.08, size = 389, normalized size = 1.14 \begin {gather*} \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 {gather*}

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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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