Integrand size = 10, antiderivative size = 103 \[ \int \frac {\arccos (a+b x)}{x^3} \, dx=\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\arccos (a+b x)}{2 x^2}+\frac {a b^2 \text {arctanh}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )^{3/2}} \]
-1/2*arccos(b*x+a)/x^2+1/2*a*b^2*arctanh((1-a*(b*x+a))/(-a^2+1)^(1/2)/(1-( b*x+a)^2)^(1/2))/(-a^2+1)^(3/2)+1/2*b*(1-(b*x+a)^2)^(1/2)/(-a^2+1)/x
Time = 0.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.22 \[ \int \frac {\arccos (a+b x)}{x^3} \, dx=-\frac {\arccos (a+b x)-\frac {b x \left (\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}-a b x \log (x)+a b x \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}\right )\right )}{\left (1-a^2\right )^{3/2}}}{2 x^2} \]
-1/2*(ArcCos[a + b*x] - (b*x*(Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x ^2] - a*b*x*Log[x] + a*b*x*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[1 - a^ 2 - 2*a*b*x - b^2*x^2]]))/(1 - a^2)^(3/2))/x^2
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5305, 25, 27, 5243, 491, 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.
\(\displaystyle \int \frac {\arccos (a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 5305 |
\(\displaystyle \frac {\int \frac {\arccos (a+b x)}{x^3}d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -\frac {\arccos (a+b x)}{x^3}d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -b^2 \int -\frac {\arccos (a+b x)}{b^3 x^3}d(a+b x)\) |
\(\Big \downarrow \) 5243 |
\(\displaystyle -b^2 \left (\frac {1}{2} \int \frac {1}{b^2 x^2 \sqrt {1-(a+b x)^2}}d(a+b x)+\frac {\arccos (a+b x)}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 491 |
\(\displaystyle -b^2 \left (\frac {1}{2} \left (-\frac {a \int -\frac {1}{b x \sqrt {1-(a+b x)^2}}d(a+b x)}{1-a^2}-\frac {\sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}\right )+\frac {\arccos (a+b x)}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {a \int \frac {1}{-a^2-\frac {(a (a+b x)-1)^2}{1-(a+b x)^2}+1}d\frac {a (a+b x)-1}{\sqrt {1-(a+b x)^2}}}{1-a^2}-\frac {\sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}\right )+\frac {\arccos (a+b x)}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {a \text {arctanh}\left (\frac {a (a+b x)-1}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{\left (1-a^2\right )^{3/2}}-\frac {\sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}\right )+\frac {\arccos (a+b x)}{2 b^2 x^2}\right )\) |
-(b^2*(ArcCos[a + b*x]/(2*b^2*x^2) + (-(Sqrt[1 - (a + b*x)^2]/((1 - a^2)*b *x)) + (a*ArcTanh[(-1 + a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2]) ])/(1 - a^2)^(3/2))/2))
3.1.30.3.1 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*(c/(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 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 = 0.86 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.13
method | result | size |
parts | \(-\frac {\arccos \left (b x +a \right )}{2 x^{2}}-\frac {b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2}\) | \(116\) |
derivativedivides | \(b^{2} \left (-\frac {\arccos \left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(124\) |
default | \(b^{2} \left (-\frac {\arccos \left (b x +a \right )}{2 b^{2} x^{2}}+\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(124\) |
-1/2*arccos(b*x+a)/x^2-1/2*b*(-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))
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (87) = 174\).
Time = 0.31 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.68 \[ \int \frac {\arccos (a+b x)}{x^3} \, dx=\left [-\frac {\sqrt {-a^{2} + 1} a b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} b x + 2 \, {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \arccos \left (b x + a\right )}{4 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}, \frac {\sqrt {a^{2} - 1} a b^{2} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} b x - {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \arccos \left (b x + a\right )}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}\right ] \]
[-1/4*(sqrt(-a^2 + 1)*a*b^2*x^2*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) + 2*(a^4 - 2*a^2 + 1)*x^2*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*sqrt(-b^2* x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1)*b*x + 2*(a^4 - (a^4 - 2*a^2 + 1)*x^2 - 2*a^2 + 1)*arccos(b*x + a))/((a^4 - 2*a^2 + 1)*x^2), 1/2*(sqrt(a^2 - 1)*a* b^2*x^2*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)) - (a^4 - 2*a^2 + 1)*x^2*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)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1)*b*x - (a^4 - (a^4 - 2*a^2 + 1)*x^2 - 2*a^2 + 1)*arccos(b*x + a))/((a^4 - 2*a^ 2 + 1)*x^2)]
\[ \int \frac {\arccos (a+b x)}{x^3} \, dx=\int \frac {\operatorname {acos}{\left (a + b x \right )}}{x^{3}}\, dx \]
Exception generated. \[ \int \frac {\arccos (a+b x)}{x^3} \, dx=\text {Exception raised: ValueError} \]
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. 242 vs. \(2 (87) = 174\).
Time = 0.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.35 \[ \int \frac {\arccos (a+b x)}{x^3} \, dx={\left (\frac {a b^{2} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{{\left (a^{2} {\left | b \right |} - {\left | b \right |}\right )} \sqrt {a^{2} - 1}} - \frac {a b^{2} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b}}{{\left (a^{3} {\left | b \right |} - a {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}}\right )} b - \frac {\arccos \left (b x + a\right )}{2 \, x^{2}} \]
(a*b^2*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^2*abs(b) - abs(b))*sqrt(a^2 - 1)) - (a*b^2 - (sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*b^2/(b^2*x + a*b))/((a^3*a bs(b) - a*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))))*b - 1/2*arccos(b*x + a)/x^2
Timed out. \[ \int \frac {\arccos (a+b x)}{x^3} \, dx=\int \frac {\mathrm {acos}\left (a+b\,x\right )}{x^3} \,d x \]