Integrand size = 12, antiderivative size = 125 \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}-\frac {8 x}{3 a^2 \sqrt {\arccos (a x)}}+\frac {4 x^3}{\sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^3}+\frac {\sqrt {6 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^3} \]
1/3*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^3+Fres nelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^3+2/3*x^2*(-a^ 2*x^2+1)^(1/2)/a/arccos(a*x)^(3/2)-8/3*x/a^2/arccos(a*x)^(1/2)+4*x^3/arcco s(a*x)^(1/2)
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.76 \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=-\frac {-\sqrt {1-a^2 x^2}-e^{-i \arccos (a x)} \arccos (a x)-e^{i \arccos (a x)} \arccos (a x)+\sqrt {-i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \arccos (a x) \Gamma \left (\frac {1}{2},i \arccos (a x)\right )-3 \arccos (a x) \left (e^{-3 i \arccos (a x)}+e^{3 i \arccos (a x)}-\sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-\sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )-\sin (3 \arccos (a x))}{6 a^3 \arccos (a x)^{3/2}} \]
-1/6*(-Sqrt[1 - a^2*x^2] - ArcCos[a*x]/E^(I*ArcCos[a*x]) - E^(I*ArcCos[a*x ])*ArcCos[a*x] + Sqrt[(-I)*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, (-I)*ArcCos [a*x]] + Sqrt[I*ArcCos[a*x]]*ArcCos[a*x]*Gamma[1/2, I*ArcCos[a*x]] - 3*Arc Cos[a*x]*(E^((-3*I)*ArcCos[a*x]) + E^((3*I)*ArcCos[a*x]) - Sqrt[3]*Sqrt[(- I)*ArcCos[a*x]]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - Sqrt[3]*Sqrt[I*ArcCos[a*x ]]*Gamma[1/2, (3*I)*ArcCos[a*x]]) - Sin[3*ArcCos[a*x]])/(a^3*ArcCos[a*x]^( 3/2))
Time = 0.87 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.41, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5145, 5223, 5135, 3042, 3786, 3832, 5147, 4906, 2009}
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.
\(\displaystyle \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 5145 |
\(\displaystyle -\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}dx}{3 a}+2 a \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^{3/2}}dx+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 5223 |
\(\displaystyle 2 a \left (\frac {2 x^3}{a \sqrt {\arccos (a x)}}-\frac {6 \int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{a}\right )-\frac {4 \left (\frac {2 x}{a \sqrt {\arccos (a x)}}-\frac {2 \int \frac {1}{\sqrt {\arccos (a x)}}dx}{a}\right )}{3 a}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 5135 |
\(\displaystyle -\frac {4 \left (\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a^2}+\frac {2 x}{a \sqrt {\arccos (a x)}}\right )}{3 a}+2 a \left (\frac {2 x^3}{a \sqrt {\arccos (a x)}}-\frac {6 \int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{a}\right )+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {4 \left (\frac {2 \int \frac {\sin (\arccos (a x))}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a^2}+\frac {2 x}{a \sqrt {\arccos (a x)}}\right )}{3 a}+2 a \left (\frac {2 x^3}{a \sqrt {\arccos (a x)}}-\frac {6 \int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{a}\right )+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle -\frac {4 \left (\frac {4 \int \sqrt {1-a^2 x^2}d\sqrt {\arccos (a x)}}{a^2}+\frac {2 x}{a \sqrt {\arccos (a x)}}\right )}{3 a}+2 a \left (\frac {2 x^3}{a \sqrt {\arccos (a x)}}-\frac {6 \int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{a}\right )+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle 2 a \left (\frac {2 x^3}{a \sqrt {\arccos (a x)}}-\frac {6 \int \frac {x^2}{\sqrt {\arccos (a x)}}dx}{a}\right )-\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^2}+\frac {2 x}{a \sqrt {\arccos (a x)}}\right )}{3 a}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle 2 a \left (\frac {6 \int \frac {a^2 x^2 \sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a^4}+\frac {2 x^3}{a \sqrt {\arccos (a x)}}\right )-\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^2}+\frac {2 x}{a \sqrt {\arccos (a x)}}\right )}{3 a}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle 2 a \left (\frac {6 \int \left (\frac {\sin (3 \arccos (a x))}{4 \sqrt {\arccos (a x)}}+\frac {\sqrt {1-a^2 x^2}}{4 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^4}+\frac {2 x^3}{a \sqrt {\arccos (a x)}}\right )-\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^2}+\frac {2 x}{a \sqrt {\arccos (a x)}}\right )}{3 a}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a \left (\frac {6 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )\right )}{a^4}+\frac {2 x^3}{a \sqrt {\arccos (a x)}}\right )-\frac {4 \left (\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^2}+\frac {2 x}{a \sqrt {\arccos (a x)}}\right )}{3 a}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^{3/2}}\) |
(2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (4*((2*x)/(a*Sqrt[ArcC os[a*x]]) + (2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^2))/(3 *a) + 2*a*((2*x^3)/(a*Sqrt[ArcCos[a*x]]) + (6*((Sqrt[Pi/2]*FresnelS[Sqrt[2 /Pi]*Sqrt[ArcCos[a*x]]])/2 + (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a *x]]])/2))/a^4)
3.2.9.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-(b*c)^(-1) Subst[Int[x^n*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[( -x^m)*Sqrt[1 - c^2*x^2]*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] + ( -Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n + 1)/ Sqrt[1 - c^2*x^2]), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && I GtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (b*c^(m + 1))^(-1) Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x , a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Simp[f*(m/(b*c*( n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b *ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2 *d + e, 0] && LtQ[n, -1]
Time = 0.86 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {6 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+2 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+2 \arccos \left (a x \right ) a x +6 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )+\sqrt {-a^{2} x^{2}+1}+\sin \left (3 \arccos \left (a x \right )\right )}{6 a^{3} \arccos \left (a x \right )^{\frac {3}{2}}}\) | \(115\) |
1/6/a^3*(6*2^(1/2)*Pi^(1/2)*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcc os(a*x)^(1/2))*arccos(a*x)^(3/2)+2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1 /2)*arccos(a*x)^(1/2))*arccos(a*x)^(3/2)+2*arccos(a*x)*a*x+6*arccos(a*x)*c os(3*arccos(a*x))+(-a^2*x^2+1)^(1/2)+sin(3*arccos(a*x)))/arccos(a*x)^(3/2)
Exception generated. \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^{2}}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
Exception generated. \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\arccos (a x)^{5/2}} \, dx=\int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \]