Integrand size = 10, antiderivative size = 144 \[ \int \frac {\arccos (a+b x)}{x^4} \, dx=\frac {b \sqrt {1-(a+b x)^2}}{6 \left (1-a^2\right ) x^2}+\frac {a b^2 \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right )^2 x}-\frac {\arccos (a+b x)}{3 x^3}+\frac {\left (1+2 a^2\right ) b^3 \text {arctanh}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{6 \left (1-a^2\right )^{5/2}} \]
-1/3*arccos(b*x+a)/x^3+1/6*(2*a^2+1)*b^3*arctanh((1-a*(b*x+a))/(-a^2+1)^(1 /2)/(1-(b*x+a)^2)^(1/2))/(-a^2+1)^(5/2)+1/6*b*(1-(b*x+a)^2)^(1/2)/(-a^2+1) /x^2+1/2*a*b^2*(1-(b*x+a)^2)^(1/2)/(-a^2+1)^2/x
Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {\arccos (a+b x)}{x^4} \, dx=\frac {\sqrt {1-a^2} b x \left (1-a^2+3 a b x\right ) \sqrt {1-a^2-2 a b x-b^2 x^2}-2 \left (1-a^2\right )^{5/2} \arccos (a+b x)-\left (1+2 a^2\right ) b^3 x^3 \log (x)+\left (1+2 a^2\right ) b^3 x^3 \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{6 \left (1-a^2\right )^{5/2} x^3} \]
(Sqrt[1 - a^2]*b*x*(1 - a^2 + 3*a*b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] - 2*(1 - a^2)^(5/2)*ArcCos[a + b*x] - (1 + 2*a^2)*b^3*x^3*Log[x] + (1 + 2*a ^2)*b^3*x^3*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b ^2*x^2]])/(6*(1 - a^2)^(5/2)*x^3)
Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5305, 27, 5243, 498, 25, 679, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arccos (a+b x)}{x^4} \, dx\) |
| \(\Big \downarrow \) 5305 |
| \(\displaystyle \frac {\int \frac {\arccos (a+b x)}{x^4}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b^3 \int \frac {\arccos (a+b x)}{b^4 x^4}d(a+b x)\) |
| \(\Big \downarrow \) 5243 |
| \(\displaystyle b^3 \left (\frac {1}{3} \int -\frac {1}{b^3 x^3 \sqrt {1-(a+b x)^2}}d(a+b x)-\frac {\arccos (a+b x)}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle b^3 \left (\frac {1}{3} \left (\frac {\int -\frac {3 a+b x}{b^2 x^2 \sqrt {1-(a+b x)^2}}d(a+b x)}{2 \left (1-a^2\right )}+\frac {\sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}\right )-\frac {\arccos (a+b x)}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b^3 \left (\frac {1}{3} \left (\frac {\sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {\int \frac {3 a+b x}{b^2 x^2 \sqrt {1-(a+b x)^2}}d(a+b x)}{2 \left (1-a^2\right )}\right )-\frac {\arccos (a+b x)}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle b^3 \left (\frac {1}{3} \left (\frac {\sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {-\frac {\left (2 a^2+1\right ) \int -\frac {1}{b x \sqrt {1-(a+b x)^2}}d(a+b x)}{1-a^2}-\frac {3 a \sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}}{2 \left (1-a^2\right )}\right )-\frac {\arccos (a+b x)}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle b^3 \left (\frac {1}{3} \left (\frac {\sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {\frac {\left (2 a^2+1\right ) \int \frac {1}{-a^2-\frac {(a (a+b x)-1)^2}{1-(a+b x)^2}+1}d\frac {a (a+b x)-1}{\sqrt {1-(a+b x)^2}}}{1-a^2}-\frac {3 a \sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}}{2 \left (1-a^2\right )}\right )-\frac {\arccos (a+b x)}{3 b^3 x^3}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle b^3 \left (\frac {1}{3} \left (\frac {\sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {\frac {\left (2 a^2+1\right ) \text {arctanh}\left (\frac {a (a+b x)-1}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{\left (1-a^2\right )^{3/2}}-\frac {3 a \sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}}{2 \left (1-a^2\right )}\right )-\frac {\arccos (a+b x)}{3 b^3 x^3}\right )\) |
b^3*(-1/3*ArcCos[a + b*x]/(b^3*x^3) + (Sqrt[1 - (a + b*x)^2]/(2*(1 - a^2)* b^2*x^2) - ((-3*a*Sqrt[1 - (a + b*x)^2])/((1 - a^2)*b*x) + ((1 + 2*a^2)*Ar cTanh[(-1 + a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(1 - a^2) ^(3/2))/(2*(1 - a^2)))/3)
3.1.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 1))), x] + Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.83 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.60
| method | result | size |
| parts | \(-\frac {\arccos \left (b x +a \right )}{3 x^{3}}-\frac {b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) x^{2}}+\frac {3 a b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{3}\) | \(230\) |
| derivativedivides | \(b^{3} \left (-\frac {\arccos \left (b x +a \right )}{3 b^{3} x^{3}}+\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{6 \left (-a^{2}+1\right ) b^{2} x^{2}}-\frac {a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}+\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{6 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(240\) |
| default | \(b^{3} \left (-\frac {\arccos \left (b x +a \right )}{3 b^{3} x^{3}}+\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{6 \left (-a^{2}+1\right ) b^{2} x^{2}}-\frac {a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}+\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{6 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(240\) |
-1/3*arccos(b*x+a)/x^3-1/3*b*(-1/2/(-a^2+1)/x^2*(-b^2*x^2-2*a*b*x-a^2+1)^( 1/2)+3/2*a*b/(-a^2+1)*(-1/(-a^2+1)/x*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a*b/(- a^2+1)^(3/2)*ln((-2*a^2+2-2*a*b*x+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1 )^(1/2))/x))-1/2*b^2/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*a*b*x+2*(-a^2+1)^(1/2)* (-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x))
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (122) = 244\).
Time = 0.32 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.03 \[ \int \frac {\arccos (a+b x)}{x^4} \, dx=\left [-\frac {{\left (2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 4 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 4 \, {\left (a^{6} - 3 \, a^{4} - {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \arccos \left (b x + a\right ) - 2 \, {\left (3 \, {\left (a^{3} - a\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}, -\frac {{\left (2 \, a^{2} + 1\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 2 \, {\left (a^{6} - 3 \, a^{4} - {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \arccos \left (b x + a\right ) - {\left (3 \, {\left (a^{3} - a\right )} b^{2} x^{2} - {\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}\right ] \]
[-1/12*((2*a^2 + 1)*sqrt(-a^2 + 1)*b^3*x^3*log(((2*a^2 - 1)*b^2*x^2 + 2*a^ 4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 4*(a^6 - 3*a^4 + 3*a^2 - 1)*x^3*arct an(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 4*(a^6 - 3*a^4 - (a^6 - 3*a^4 + 3*a^2 - 1)*x^3 + 3*a^2 - 1)*arccos( b*x + a) - 2*(3*(a^3 - a)*b^2*x^2 - (a^4 - 2*a^2 + 1)*b*x)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^6 - 3*a^4 + 3*a^2 - 1)*x^3), -1/6*((2*a^2 + 1)*sq rt(a^2 - 1)*b^3*x^3*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1 )) + 2*(a^6 - 3*a^4 + 3*a^2 - 1)*x^3*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 2*(a^6 - 3*a^4 - (a^6 - 3* a^4 + 3*a^2 - 1)*x^3 + 3*a^2 - 1)*arccos(b*x + a) - (3*(a^3 - a)*b^2*x^2 - (a^4 - 2*a^2 + 1)*b*x)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^6 - 3*a^4 + 3*a^2 - 1)*x^3)]
\[ \int \frac {\arccos (a+b x)}{x^4} \, dx=\int \frac {\operatorname {acos}{\left (a + b x \right )}}{x^{4}}\, dx \]
Exception generated. \[ \int \frac {\arccos (a+b x)}{x^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (122) = 244\).
Time = 0.29 (sec) , antiderivative size = 557, normalized size of antiderivative = 3.87 \[ \int \frac {\arccos (a+b x)}{x^4} \, dx=-\frac {1}{3} \, b {\left (\frac {{\left (2 \, a^{2} b^{3} + b^{3}\right )} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{{\left (a^{4} {\left | b \right |} - 2 \, a^{2} {\left | b \right |} + {\left | b \right |}\right )} \sqrt {a^{2} - 1}} - \frac {\frac {4 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{4} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + 4 \, a^{4} b^{3} - \frac {11 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{3} b^{3}}{b^{2} x + a b} - \frac {5 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} + \frac {7 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} - a^{2} b^{3} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a b^{3}}{b^{2} x + a b} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}}}{{\left (a^{6} {\left | b \right |} - 2 \, a^{4} {\left | b \right |} + a^{2} {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}^{2}}\right )} - \frac {\arccos \left (b x + a\right )}{3 \, x^{3}} \]
-1/3*b*((2*a^2*b^3 + b^3)*arctan(((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs( b) + b)*a/(b^2*x + a*b) - 1)/sqrt(a^2 - 1))/((a^4*abs(b) - 2*a^2*abs(b) + abs(b))*sqrt(a^2 - 1)) - (4*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b )^2*a^4*b^3/(b^2*x + a*b)^2 + 4*a^4*b^3 - 11*(sqrt(-b^2*x^2 - 2*a*b*x - a^ 2 + 1)*abs(b) + b)*a^3*b^3/(b^2*x + a*b) - 5*(sqrt(-b^2*x^2 - 2*a*b*x - a^ 2 + 1)*abs(b) + b)^3*a^3*b^3/(b^2*x + a*b)^3 + 7*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^3/(b^2*x + a*b)^2 - a^2*b^3 + 2*(sqrt(-b^2* x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a*b^3/(b^2*x + a*b) + 2*(sqrt(-b^2*x^ 2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a*b^3/(b^2*x + a*b)^3 - 2*(sqrt(-b^2* x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*b^3/(b^2*x + a*b)^2)/((a^6*abs(b) - 2*a^4*abs(b) + a^2*abs(b))*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + a - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b))^2)) - 1/3*arccos(b*x + a)/x^3
Timed out. \[ \int \frac {\arccos (a+b x)}{x^4} \, dx=\int \frac {\mathrm {acos}\left (a+b\,x\right )}{x^4} \,d x \]