Integrand size = 10, antiderivative size = 87 \[ \int \frac {\arccos (a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{6 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \arccos (a x)}{3 x}-\frac {\arccos (a x)^2}{4 x^4}+\frac {1}{3} a^4 \log (x) \]
-1/12*a^2/x^2-1/4*arccos(a*x)^2/x^4+1/3*a^4*ln(x)+1/6*a*arccos(a*x)*(-a^2* x^2+1)^(1/2)/x^3+1/3*a^3*arccos(a*x)*(-a^2*x^2+1)^(1/2)/x
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.79 \[ \int \frac {\arccos (a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}+\frac {a \sqrt {1-a^2 x^2} \left (1+2 a^2 x^2\right ) \arccos (a x)}{6 x^3}-\frac {\arccos (a x)^2}{4 x^4}+\frac {1}{3} a^4 \log (x) \]
-1/12*a^2/x^2 + (a*Sqrt[1 - a^2*x^2]*(1 + 2*a^2*x^2)*ArcCos[a*x])/(6*x^3) - ArcCos[a*x]^2/(4*x^4) + (a^4*Log[x])/3
Time = 0.37 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5139, 5205, 15, 5187, 14}
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.
\(\displaystyle \int \frac {\arccos (a x)^2}{x^5} \, dx\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle -\frac {1}{2} a \int \frac {\arccos (a x)}{x^4 \sqrt {1-a^2 x^2}}dx-\frac {\arccos (a x)^2}{4 x^4}\) |
\(\Big \downarrow \) 5205 |
\(\displaystyle -\frac {1}{2} a \left (\frac {2}{3} a^2 \int \frac {\arccos (a x)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {1}{3} a \int \frac {1}{x^3}dx-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{3 x^3}\right )-\frac {\arccos (a x)^2}{4 x^4}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {1}{2} a \left (\frac {2}{3} a^2 \int \frac {\arccos (a x)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{3 x^3}+\frac {a}{6 x^2}\right )-\frac {\arccos (a x)^2}{4 x^4}\) |
\(\Big \downarrow \) 5187 |
\(\displaystyle -\frac {1}{2} a \left (\frac {2}{3} a^2 \left (-a \int \frac {1}{x}dx-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{3 x^3}+\frac {a}{6 x^2}\right )-\frac {\arccos (a x)^2}{4 x^4}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle -\frac {1}{2} a \left (\frac {2}{3} a^2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{x}-a \log (x)\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{3 x^3}+\frac {a}{6 x^2}\right )-\frac {\arccos (a x)^2}{4 x^4}\) |
-1/4*ArcCos[a*x]^2/x^4 - (a*(a/(6*x^2) - (Sqrt[1 - a^2*x^2]*ArcCos[a*x])/( 3*x^3) + (2*a^2*(-((Sqrt[1 - a^2*x^2]*ArcCos[a*x])/x) - a*Log[x]))/3))/2
3.1.21.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcCos[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x ^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*A rcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.52 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(a^{4} \left (-\frac {\arccos \left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{6 a^{3} x^{3}}-\frac {1}{12 a^{2} x^{2}}+\frac {\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3 x a}+\frac {\ln \left (a x \right )}{3}\right )\) | \(82\) |
default | \(a^{4} \left (-\frac {\arccos \left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{6 a^{3} x^{3}}-\frac {1}{12 a^{2} x^{2}}+\frac {\arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3 x a}+\frac {\ln \left (a x \right )}{3}\right )\) | \(82\) |
a^4*(-1/4*arccos(a*x)^2/a^4/x^4+1/6*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a^3/x^3 -1/12/a^2/x^2+1/3*arccos(a*x)/x/a*(-a^2*x^2+1)^(1/2)+1/3*ln(a*x))
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \frac {\arccos (a x)^2}{x^5} \, dx=\frac {4 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} + 2 \, {\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 3 \, \arccos \left (a x\right )^{2}}{12 \, x^{4}} \]
1/12*(4*a^4*x^4*log(x) - a^2*x^2 + 2*(2*a^3*x^3 + a*x)*sqrt(-a^2*x^2 + 1)* arccos(a*x) - 3*arccos(a*x)^2)/x^4
\[ \int \frac {\arccos (a x)^2}{x^5} \, dx=\int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{x^{5}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \frac {\arccos (a x)^2}{x^5} \, dx=\frac {1}{12} \, {\left (4 \, a^{2} \log \left (x\right ) - \frac {1}{x^{2}}\right )} a^{2} + \frac {1}{6} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} a \arccos \left (a x\right ) - \frac {\arccos \left (a x\right )^{2}}{4 \, x^{4}} \]
1/12*(4*a^2*log(x) - 1/x^2)*a^2 + 1/6*(2*sqrt(-a^2*x^2 + 1)*a^2/x + sqrt(- a^2*x^2 + 1)/x^3)*a*arccos(a*x) - 1/4*arccos(a*x)^2/x^4
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (73) = 146\).
Time = 0.35 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.13 \[ \int \frac {\arccos (a x)^2}{x^5} \, dx=-\frac {1}{48} \, {\left ({\left (\frac {{\left (a^{4} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2} {\left | a \right |}}\right )} \arccos \left (a x\right ) - \frac {4 \, {\left (2 \, a^{4} \log \left (a^{2} x^{2}\right ) - \frac {2 \, {\left (a^{2} x^{2} - 1\right )} a^{4} + 3 \, a^{4}}{a^{2} x^{2}}\right )}}{a}\right )} a - \frac {\arccos \left (a x\right )^{2}}{4 \, x^{4}} \]
-1/48*(((a^4 + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2)*a^6*x^3/((sqrt(-a^ 2*x^2 + 1)*abs(a) + a)^3*abs(a)) - (9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4/ x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/x^3)/(a^2*abs(a)))*arccos(a*x) - 4*( 2*a^4*log(a^2*x^2) - (2*(a^2*x^2 - 1)*a^4 + 3*a^4)/(a^2*x^2))/a)*a - 1/4*a rccos(a*x)^2/x^4
Timed out. \[ \int \frac {\arccos (a x)^2}{x^5} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{x^5} \,d x \]