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

Optimal. Leaf size=230 \[ -\frac{d^3 (c x-1)^5 (c x+1)^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 (c x-1)^4 (c x+1)^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{b d^3 x (c x-1)^{9/2} (c x+1)^{9/2}}{100 c^3}+\frac{7 b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{1600 c^3}-\frac{49 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{9600 c^3}+\frac{49 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{7680 c^3}-\frac{49 b d^3 x \sqrt{c x-1} \sqrt{c x+1}}{5120 c^3}+\frac{49 b d^3 \cosh ^{-1}(c x)}{5120 c^4} \]

[Out]

(-49*b*d^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5120*c^3) + (49*b*d^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(7680*c^
3) - (49*b*d^3*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(9600*c^3) + (7*b*d^3*x*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(
1600*c^3) + (b*d^3*x*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(100*c^3) + (49*b*d^3*ArcCosh[c*x])/(5120*c^4) - (d^3*(
-1 + c*x)^4*(1 + c*x)^4*(a + b*ArcCosh[c*x]))/(8*c^4) - (d^3*(-1 + c*x)^5*(1 + c*x)^5*(a + b*ArcCosh[c*x]))/(1
0*c^4)

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Rubi [A]  time = 0.281196, antiderivative size = 328, normalized size of antiderivative = 1.43, number of steps used = 11, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 43, 5731, 12, 566, 21, 388, 195, 217, 206} \[ \frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{49 b d^3 \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{5120 c^4 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(49*b*d^3*x*(1 - c^2*x^2))/(5120*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (49*b*d^3*x*(1 - c^2*x^2)^2)/(7680*c^3*Sq
rt[-1 + c*x]*Sqrt[1 + c*x]) + (49*b*d^3*x*(1 - c^2*x^2)^3)/(9600*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (7*b*d^3*
x*(1 - c^2*x^2)^4)/(1600*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^3*x*(1 - c^2*x^2)^5)/(100*c^3*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcCosh[c*x]))/(8*c^4) + (d^3*(1 - c^2*x^2)^5*(a + b*ArcCosh[c*x
]))/(10*c^4) + (49*b*d^3*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(5120*c^4*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 5731

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] && EqQ[c^2*d + e, 0] && IGtQ[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 566

Int[((e1_) + (f1_.)*(x_)^(n2_.))^(r_.)*((e2_) + (f2_.)*(x_)^(n2_.))^(r_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_)
+ (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[((e1 + f1*x^(n/2))^FracPart[r]*(e2 + f2*x^(n/2))^FracPart[r])/(e1
*e2 + f1*f2*x^n)^FracPart[r], Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x], x] /; FreeQ[{a, b, c,
d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 388

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-(b c) \int \frac{d^3 \left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{40 c^4 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3\right ) \int \frac{\left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{40 c^3}\\ &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{\left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^4}{\sqrt{-1+c^2 x^2}} \, dx}{40 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1-4 c^2 x^2\right ) \left (-1+c^2 x^2\right )^{7/2} \, dx}{40 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (7 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{7/2} \, dx}{200 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{5/2} \, dx}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \left (-1+c^2 x^2\right )^{3/2} \, dx}{1920 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}-\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \sqrt{-1+c^2 x^2} \, dx}{2560 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3 \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 )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )}{5120 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^2}{7680 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^3}{9600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^4}{1600 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 x \left (1-c^2 x^2\right )^5}{100 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \cosh ^{-1}(c x)\right )}{10 c^4}+\frac{49 b d^3 \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{5120 c^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.342736, size = 162, normalized size = 0.7 \[ -\frac{d^3 \left (1920 a c^4 x^4 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right )+b c x \sqrt{c x-1} \sqrt{c x+1} \left (-768 c^8 x^8+2736 c^6 x^6-3208 c^4 x^4+790 c^2 x^2+1185\right )+1920 b c^4 x^4 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right ) \cosh ^{-1}(c x)+2370 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{76800 c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(d^3*(1920*a*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1185 +
790*c^2*x^2 - 3208*c^4*x^4 + 2736*c^6*x^6 - 768*c^8*x^8) + 1920*b*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*
c^6*x^6)*ArcCosh[c*x] + 2370*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(76800*c^4)

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Maple [A]  time = 0.019, size = 284, normalized size = 1.2 \begin{align*} -{\frac{{c}^{6}{d}^{3}a{x}^{10}}{10}}+{\frac{3\,{c}^{4}{d}^{3}a{x}^{8}}{8}}-{\frac{{c}^{2}{d}^{3}a{x}^{6}}{2}}+{\frac{{d}^{3}a{x}^{4}}{4}}-{\frac{{c}^{6}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{10}}{10}}+{\frac{3\,{c}^{4}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{8}}{8}}-{\frac{{c}^{2}{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{6}}{2}}+{\frac{{d}^{3}b{\rm arccosh} \left (cx\right ){x}^{4}}{4}}+{\frac{{d}^{3}b{c}^{5}{x}^{9}}{100}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{57\,{d}^{3}b{c}^{3}{x}^{7}}{1600}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{401\,{d}^{3}bc{x}^{5}}{9600}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{79\,{d}^{3}b{x}^{3}}{7680\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{79\,{d}^{3}bx}{5120\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{79\,{d}^{3}b}{5120\,{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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*c^6*d^3*a*x^10+3/8*c^4*d^3*a*x^8-1/2*c^2*d^3*a*x^6+1/4*d^3*a*x^4-1/10*c^6*d^3*b*arccosh(c*x)*x^10+3/8*c^
4*d^3*b*arccosh(c*x)*x^8-1/2*c^2*d^3*b*arccosh(c*x)*x^6+1/4*d^3*b*arccosh(c*x)*x^4+1/100*c^5*d^3*b*(c*x-1)^(1/
2)*(c*x+1)^(1/2)*x^9-57/1600*c^3*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^7+401/9600*c*d^3*b*(c*x-1)^(1/2)*(c*x+1)^
(1/2)*x^5-79/7680/c*d^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^3-79/5120*b*d^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-79/5
120/c^4*d^3*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))

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Maxima [B]  time = 1.40942, size = 725, normalized size = 3.15 \begin{align*} -\frac{1}{10} \, a c^{6} d^{3} x^{10} + \frac{3}{8} \, a c^{4} d^{3} x^{8} - \frac{1}{2} \, a c^{2} d^{3} x^{6} - \frac{1}{12800} \,{\left (1280 \, x^{10} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{128 \, \sqrt{c^{2} x^{2} - 1} x^{9}}{c^{2}} + \frac{144 \, \sqrt{c^{2} x^{2} - 1} x^{7}}{c^{4}} + \frac{168 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{6}} + \frac{210 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{8}} + \frac{315 \, \sqrt{c^{2} x^{2} - 1} x}{c^{10}} + \frac{315 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{10}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{1024} \,{\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 c^{4} d^{3} + \frac{1}{4} \, a d^{3} x^{4} - \frac{1}{96} \,{\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 c^{2} d^{3} + \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^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*a*c^6*d^3*x^10 + 3/8*a*c^4*d^3*x^8 - 1/2*a*c^2*d^3*x^6 - 1/12800*(1280*x^10*arccosh(c*x) - (128*sqrt(c^2
*x^2 - 1)*x^9/c^2 + 144*sqrt(c^2*x^2 - 1)*x^7/c^4 + 168*sqrt(c^2*x^2 - 1)*x^5/c^6 + 210*sqrt(c^2*x^2 - 1)*x^3/
c^8 + 315*sqrt(c^2*x^2 - 1)*x/c^10 + 315*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^10))*c)*b*c
^6*d^3 + 1/1024*(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*c^4*d^3 + 1/4*a*d^3*x^4 - 1/96*(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*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c
^6))*c)*b*c^2*d^3 + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*lo
g(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d^3

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Fricas [A]  time = 1.85556, size = 475, normalized size = 2.07 \begin{align*} -\frac{7680 \, a c^{10} d^{3} x^{10} - 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} - 19200 \, a c^{4} d^{3} x^{4} + 15 \,{\left (512 \, b c^{10} d^{3} x^{10} - 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} - 1280 \, b c^{4} d^{3} x^{4} + 79 \, b d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (768 \, b c^{9} d^{3} x^{9} - 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} - 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt{c^{2} x^{2} - 1}}{76800 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/76800*(7680*a*c^10*d^3*x^10 - 28800*a*c^8*d^3*x^8 + 38400*a*c^6*d^3*x^6 - 19200*a*c^4*d^3*x^4 + 15*(512*b*c
^10*d^3*x^10 - 1920*b*c^8*d^3*x^8 + 2560*b*c^6*d^3*x^6 - 1280*b*c^4*d^3*x^4 + 79*b*d^3)*log(c*x + sqrt(c^2*x^2
- 1)) - (768*b*c^9*d^3*x^9 - 2736*b*c^7*d^3*x^7 + 3208*b*c^5*d^3*x^5 - 790*b*c^3*d^3*x^3 - 1185*b*c*d^3*x)*sq
rt(c^2*x^2 - 1))/c^4

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Sympy [A]  time = 47.4026, size = 287, normalized size = 1.25 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{10}}{10} + \frac{3 a c^{4} d^{3} x^{8}}{8} - \frac{a c^{2} d^{3} x^{6}}{2} + \frac{a d^{3} x^{4}}{4} - \frac{b c^{6} d^{3} x^{10} \operatorname{acosh}{\left (c x \right )}}{10} + \frac{b c^{5} d^{3} x^{9} \sqrt{c^{2} x^{2} - 1}}{100} + \frac{3 b c^{4} d^{3} x^{8} \operatorname{acosh}{\left (c x \right )}}{8} - \frac{57 b c^{3} d^{3} x^{7} \sqrt{c^{2} x^{2} - 1}}{1600} - \frac{b c^{2} d^{3} x^{6} \operatorname{acosh}{\left (c x \right )}}{2} + \frac{401 b c d^{3} x^{5} \sqrt{c^{2} x^{2} - 1}}{9600} + \frac{b d^{3} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} - \frac{79 b d^{3} x^{3} \sqrt{c^{2} x^{2} - 1}}{7680 c} - \frac{79 b d^{3} x \sqrt{c^{2} x^{2} - 1}}{5120 c^{3}} - \frac{79 b d^{3} \operatorname{acosh}{\left (c x \right )}}{5120 c^{4}} & \text{for}\: c \neq 0 \\\frac{d^{3} x^{4} \left (a + \frac{i \pi b}{2}\right )}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**6*d**3*x**10/10 + 3*a*c**4*d**3*x**8/8 - a*c**2*d**3*x**6/2 + a*d**3*x**4/4 - b*c**6*d**3*x**
10*acosh(c*x)/10 + b*c**5*d**3*x**9*sqrt(c**2*x**2 - 1)/100 + 3*b*c**4*d**3*x**8*acosh(c*x)/8 - 57*b*c**3*d**3
*x**7*sqrt(c**2*x**2 - 1)/1600 - b*c**2*d**3*x**6*acosh(c*x)/2 + 401*b*c*d**3*x**5*sqrt(c**2*x**2 - 1)/9600 +
b*d**3*x**4*acosh(c*x)/4 - 79*b*d**3*x**3*sqrt(c**2*x**2 - 1)/(7680*c) - 79*b*d**3*x*sqrt(c**2*x**2 - 1)/(5120
*c**3) - 79*b*d**3*acosh(c*x)/(5120*c**4), Ne(c, 0)), (d**3*x**4*(a + I*pi*b/2)/4, True))

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Giac [B]  time = 1.75086, size = 633, normalized size = 2.75 \begin{align*} -\frac{1}{10} \, a c^{6} d^{3} x^{10} + \frac{3}{8} \, a c^{4} d^{3} x^{8} - \frac{1}{2} \, a c^{2} d^{3} x^{6} - \frac{1}{12800} \,{\left (1280 \, x^{10} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \,{\left (4 \,{\left (2 \, x^{2}{\left (\frac{8 \, x^{2}}{c^{2}} + \frac{9}{c^{4}}\right )} + \frac{21}{c^{6}}\right )} x^{2} + \frac{105}{c^{8}}\right )} x^{2} + \frac{315}{c^{10}}\right )} x - \frac{315 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{10}{\left | c \right |}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{1024} \,{\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 c^{4} d^{3} + \frac{1}{4} \, a d^{3} x^{4} - \frac{1}{96} \,{\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 c^{2} d^{3} + \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^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*a*c^6*d^3*x^10 + 3/8*a*c^4*d^3*x^8 - 1/2*a*c^2*d^3*x^6 - 1/12800*(1280*x^10*log(c*x + sqrt(c^2*x^2 - 1))
- (sqrt(c^2*x^2 - 1)*(2*(4*(2*x^2*(8*x^2/c^2 + 9/c^4) + 21/c^6)*x^2 + 105/c^8)*x^2 + 315/c^10)*x - 315*log(ab
s(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^10*abs(c)))*c)*b*c^6*d^3 + 1/1024*(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*c^4*d^3 + 1/4*a*d^3*x^4 - 1/96*(48*x^6*log(c*x + sqrt(c^2*x^2 - 1)) - (sqrt(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*c^2*d^3 + 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(abs(
-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^4*abs(c)))*c)*b*d^3