Integrand size = 12, antiderivative size = 127 \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a^6}-\frac {\sqrt {3 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^6}-\frac {5 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{8 a^6} \] Output:
2*x^5*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)-1/2*2^(1/2)*Pi^(1/2)*FresnelC (2*2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^6-1/8*3^(1/2)*Pi^(1/2)*FresnelC(2 *3^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^6-5/8*Pi^(1/2)*FresnelC(2*arccos(a* x)^(1/2)/Pi^(1/2))/a^6
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.78 \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\frac {i \left (5 \sqrt {2} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-2 i \arccos (a x)\right )-5 \sqrt {2} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},2 i \arccos (a x)\right )+8 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-4 i \arccos (a x)\right )-8 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},4 i \arccos (a x)\right )+\sqrt {6} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-6 i \arccos (a x)\right )-\sqrt {6} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},6 i \arccos (a x)\right )-10 i \sin (2 \arccos (a x))-8 i \sin (4 \arccos (a x))-2 i \sin (6 \arccos (a x))\right )}{32 a^6 \sqrt {\arccos (a x)}} \] Input:
Integrate[x^5/ArcCos[a*x]^(3/2),x]
Output:
((I/32)*(5*Sqrt[2]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-2*I)*ArcCos[a*x]] - 5*Sqrt[2]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (2*I)*ArcCos[a*x]] + 8*Sqrt[(-I) *ArcCos[a*x]]*Gamma[1/2, (-4*I)*ArcCos[a*x]] - 8*Sqrt[I*ArcCos[a*x]]*Gamma [1/2, (4*I)*ArcCos[a*x]] + Sqrt[6]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-6*I )*ArcCos[a*x]] - Sqrt[6]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (6*I)*ArcCos[a*x]] - (10*I)*Sin[2*ArcCos[a*x]] - (8*I)*Sin[4*ArcCos[a*x]] - (2*I)*Sin[6*ArcC os[a*x]]))/(a^6*Sqrt[ArcCos[a*x]])
Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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.
\(\displaystyle \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 5143 |
\(\displaystyle \frac {2 \int \left (-\frac {5 \cos (2 \arccos (a x))}{16 \sqrt {\arccos (a x)}}-\frac {\cos (4 \arccos (a x))}{2 \sqrt {\arccos (a x)}}-\frac {3 \cos (6 \arccos (a x))}{16 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^6}+\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {1}{16} \sqrt {3 \pi } \operatorname {FresnelC}\left (2 \sqrt {\frac {3}{\pi }} \sqrt {\arccos (a x)}\right )-\frac {5}{16} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )\right )}{a^6}+\frac {2 x^5 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\) |
Input:
Int[x^5/ArcCos[a*x]^(3/2),x]
Output:
(2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) + (2*(-1/2*(Sqrt[Pi/2]*Fre snelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]]) - (Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi] *Sqrt[ArcCos[a*x]]])/16 - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[ Pi]])/16))/a^6
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]
Time = 0.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {-2 \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}-8 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}-10 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}+5 \sin \left (2 \arccos \left (a x \right )\right )+4 \sin \left (4 \arccos \left (a x \right )\right )+\sin \left (6 \arccos \left (a x \right )\right )}{16 a^{6} \sqrt {\arccos \left (a x \right )}}\) | \(121\) |
Input:
int(x^5/arccos(a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/16/a^6/arccos(a*x)^(1/2)*(-2*Pi^(1/2)*3^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)* 6^(1/2)*arccos(a*x)^(1/2))*arccos(a*x)^(1/2)-8*FresnelC(2*2^(1/2)/Pi^(1/2) *arccos(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2)-10*FresnelC(2*arcco s(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)*arccos(a*x)^(1/2)+5*sin(2*arccos(a*x))+4*s in(4*arccos(a*x))+sin(6*arccos(a*x)))
Exception generated. \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5/arccos(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^{5}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:
integrate(x**5/acos(a*x)**(3/2),x)
Output:
Integral(x**5/acos(a*x)**(3/2), x)
Exception generated. \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^5/arccos(a*x)^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\int { \frac {x^{5}}{\arccos \left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^5/arccos(a*x)^(3/2),x, algorithm="giac")
Output:
integrate(x^5/arccos(a*x)^(3/2), x)
Timed out. \[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\int \frac {x^5}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \] Input:
int(x^5/acos(a*x)^(3/2),x)
Output:
int(x^5/acos(a*x)^(3/2), x)
\[ \int \frac {x^5}{\arccos (a x)^{3/2}} \, dx=\frac {30 \sqrt {\mathit {acos} \left (a x \right )}\, \mathit {acos} \left (a x \right )-48 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{6}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a^{7}+40 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{4}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a^{5}-15 \mathit {acos} \left (a x \right ) \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a +8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a^{5} x^{5}}{4 \mathit {acos} \left (a x \right ) a^{6}} \] Input:
int(x^5/acos(a*x)^(3/2),x)
Output:
(30*sqrt(acos(a*x))*acos(a*x) - 48*acos(a*x)*int((sqrt( - a**2*x**2 + 1)*s qrt(acos(a*x))*x**6)/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a**7 + 40*acos(a *x)*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**4)/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a**5 - 15*acos(a*x)*int((sqrt( - a**2*x**2 + 1)*sqrt(acos (a*x)))/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a + 8*sqrt( - a**2*x**2 + 1)* sqrt(acos(a*x))*a**5*x**5)/(4*acos(a*x)*a**6)