Integrand size = 12, antiderivative size = 264 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5} \]
-16/15*x^3/a^2/arccos(a*x)^(3/2)+4/3*x^5/arccos(a*x)^(3/2)+9/10*FresnelC(6 ^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*6^(1/2)*Pi^(1/2)/a^5+1/15*FresnelC(2^(1 /2)/Pi^(1/2)*arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^5+5/6*FresnelC(10^(1/2) /Pi^(1/2)*arccos(a*x)^(1/2))*10^(1/2)*Pi^(1/2)/a^5+2/5*x^4*(-a^2*x^2+1)^(1 /2)/a/arccos(a*x)^(5/2)+32/5*x^2*(-a^2*x^2+1)^(1/2)/a^3/arccos(a*x)^(1/2)- 40/3*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.37 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=-\frac {2 \left (-6 \sqrt {1-a^2 x^2}-2 i e^{i \arccos (a x)} \arccos (a x) (-i+2 \arccos (a x))-4 (-i \arccos (a x))^{3/2} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+e^{-i \arccos (a x)} \arccos (a x) \left (-2+4 i \arccos (a x)-4 e^{i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )\right )-5 \arccos (a x) \left (2 e^{5 i \arccos (a x)} (1+10 i \arccos (a x))+20 \sqrt {5} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )+e^{-5 i \arccos (a x)} \left (2-20 i \arccos (a x)+20 \sqrt {5} e^{5 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )\right )\right )+9 \left (-2 \arccos (a x) \left (e^{3 i \arccos (a x)} (1+6 i \arccos (a x))+6 \sqrt {3} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )+e^{-3 i \arccos (a x)} \left (1-6 i \arccos (a x)+6 \sqrt {3} e^{3 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )\right )-2 \sin (3 \arccos (a x))\right )-6 \sin (5 \arccos (a x))}{240 a^5 \arccos (a x)^{5/2}} \]
-1/240*(2*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2*ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)* ArcCos[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])* (I*ArcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x])) - 5*A rcCos[a*x]*(2*E^((5*I)*ArcCos[a*x])*(1 + (10*I)*ArcCos[a*x]) + 20*Sqrt[5]* ((-I)*ArcCos[a*x])^(3/2)*Gamma[1/2, (-5*I)*ArcCos[a*x]] + (2 - (20*I)*ArcC os[a*x] + 20*Sqrt[5]*E^((5*I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2 , (5*I)*ArcCos[a*x]])/E^((5*I)*ArcCos[a*x])) + 9*(-2*ArcCos[a*x]*(E^((3*I) *ArcCos[a*x])*(1 + (6*I)*ArcCos[a*x]) + 6*Sqrt[3]*((-I)*ArcCos[a*x])^(3/2) *Gamma[1/2, (-3*I)*ArcCos[a*x]] + (1 - (6*I)*ArcCos[a*x] + 6*Sqrt[3]*E^((3 *I)*ArcCos[a*x])*(I*ArcCos[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcCos[a*x]])/E^(( 3*I)*ArcCos[a*x])) - 2*Sin[3*ArcCos[a*x]]) - 6*Sin[5*ArcCos[a*x]])/(a^5*Ar cCos[a*x]^(5/2))
Time = 0.69 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5145, 5223, 5143, 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.
\(\displaystyle \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 5145 |
\(\displaystyle 2 a \int \frac {x^5}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}}dx-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}}dx}{5 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
\(\Big \downarrow \) 5223 |
\(\displaystyle 2 a \left (\frac {2 x^5}{3 a \arccos (a x)^{3/2}}-\frac {10 \int \frac {x^4}{\arccos (a x)^{3/2}}dx}{3 a}\right )-\frac {8 \left (\frac {2 x^3}{3 a \arccos (a x)^{3/2}}-\frac {2 \int \frac {x^2}{\arccos (a x)^{3/2}}dx}{a}\right )}{5 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
\(\Big \downarrow \) 5143 |
\(\displaystyle 2 a \left (\frac {2 x^5}{3 a \arccos (a x)^{3/2}}-\frac {10 \left (\frac {2 \int \left (-\frac {a x}{8 \sqrt {\arccos (a x)}}-\frac {9 \cos (3 \arccos (a x))}{16 \sqrt {\arccos (a x)}}-\frac {5 \cos (5 \arccos (a x))}{16 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^5}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\right )}{3 a}\right )-\frac {8 \left (\frac {2 x^3}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (\frac {2 \int \left (-\frac {a x}{4 \sqrt {\arccos (a x)}}-\frac {3 \cos (3 \arccos (a x))}{4 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\right )}{a}\right )}{5 a}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+2 a \left (\frac {2 x^5}{3 a \arccos (a x)^{3/2}}-\frac {10 \left (\frac {2 \left (-\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {3}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {1}{8} \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )\right )}{a^5}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\right )}{3 a}\right )-\frac {8 \left (\frac {2 x^3}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {1}{2} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )\right )}{a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\right )}{a}\right )}{5 a}\) |
(2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (8*((2*x^3)/(3*a*ArcCo s[a*x]^(3/2)) - (2*((2*x^2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) + (2*( -1/2*(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]]) - (Sqrt[(3*Pi)/2] *FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/2))/a^3))/a))/(5*a) + 2*a*((2*x^5 )/(3*a*ArcCos[a*x]^(3/2)) - (10*((2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[ a*x]]) + (2*(-1/4*(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]]) - (3 *Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/8 - (Sqrt[(5*Pi)/2 ]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/8))/a^5))/(3*a))
3.2.13.3.1 Defintions of rubi rules used
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] - S imp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cos[- a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos [c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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_)*((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 = 1.07 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+8 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+100 \arccos \left (a x \right )^{2} \sin \left (5 \arccos \left (a x \right )\right )+108 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )-4 \arccos \left (a x \right ) a x -10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )-18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )-6 \sqrt {-a^{2} x^{2}+1}-3 \sin \left (5 \arccos \left (a x \right )\right )-9 \sin \left (3 \arccos \left (a x \right )\right )}{120 a^{5} \arccos \left (a x \right )^{\frac {5}{2}}}\) | \(225\) |
-1/120/a^5*(-100*2^(1/2)*Pi^(1/2)*5^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2 )*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-108*2^(1/2)*Pi^(1/2)*3^(1/2)*Fresne lC(2^(1/2)/Pi^(1/2)*3^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)-8*2^(1/2) *Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(5/2)+8 *arccos(a*x)^2*(-a^2*x^2+1)^(1/2)+100*arccos(a*x)^2*sin(5*arccos(a*x))+108 *arccos(a*x)^2*sin(3*arccos(a*x))-4*arccos(a*x)*a*x-10*arccos(a*x)*cos(5*a rccos(a*x))-18*arccos(a*x)*cos(3*arccos(a*x))-6*(-a^2*x^2+1)^(1/2)-3*sin(5 *arccos(a*x))-9*sin(3*arccos(a*x)))/arccos(a*x)^(5/2)
Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]