Integrand size = 12, antiderivative size = 106 \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{4 a^5}-\frac {\sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5}-\frac {\sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{8 a^5} \] Output:
-1/8*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5-1/1 6*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5-1/80*1 0^(1/2)*Pi^(1/2)*FresnelS(10^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2))/a^5
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.81 \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=-\frac {-10 \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )-10 \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )-5 \sqrt {3} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )-5 \sqrt {3} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )-\sqrt {5} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )-\sqrt {5} \sqrt {i \arccos (a x)} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )}{160 a^5 \sqrt {\arccos (a x)}} \] Input:
Integrate[x^4/Sqrt[ArcCos[a*x]],x]
Output:
-1/160*(-10*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-I)*ArcCos[a*x]] - 10*Sqrt[ I*ArcCos[a*x]]*Gamma[1/2, I*ArcCos[a*x]] - 5*Sqrt[3]*Sqrt[(-I)*ArcCos[a*x] ]*Gamma[1/2, (-3*I)*ArcCos[a*x]] - 5*Sqrt[3]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2 , (3*I)*ArcCos[a*x]] - Sqrt[5]*Sqrt[(-I)*ArcCos[a*x]]*Gamma[1/2, (-5*I)*Ar cCos[a*x]] - Sqrt[5]*Sqrt[I*ArcCos[a*x]]*Gamma[1/2, (5*I)*ArcCos[a*x]])/(a ^5*Sqrt[ArcCos[a*x]])
Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.
\(\displaystyle \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle -\frac {\int \frac {a^4 x^4 \sqrt {1-a^2 x^2}}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a^5}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {\int \left (\frac {3 \sin (3 \arccos (a x))}{16 \sqrt {\arccos (a x)}}+\frac {\sin (5 \arccos (a x))}{16 \sqrt {\arccos (a x)}}+\frac {\sqrt {1-a^2 x^2}}{8 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{a^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}\) |
Input:
Int[x^4/Sqrt[ArcCos[a*x]],x]
Output:
-(((Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/4 + (Sqrt[(3*Pi)/2] *FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/8 + (Sqrt[Pi/10]*FresnelS[Sqrt[10 /Pi]*Sqrt[ArcCos[a*x]]])/8)/a^5)
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_)^(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]
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.68
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (5 \sqrt {3}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+\sqrt {5}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+10 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{80 a^{5}}\) | \(72\) |
Input:
int(x^4/arccos(a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/80/a^5*2^(1/2)*Pi^(1/2)*(5*3^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*ar ccos(a*x)^(1/2))+5^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arccos(a*x)^(1/ 2))+10*FresnelS(2^(1/2)/Pi^(1/2)*arccos(a*x)^(1/2)))
Exception generated. \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4/arccos(a*x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x^{4}}{\sqrt {\operatorname {acos}{\left (a x \right )}}}\, dx \] Input:
integrate(x**4/acos(a*x)**(1/2),x)
Output:
Integral(x**4/sqrt(acos(a*x)), x)
Exception generated. \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4/arccos(a*x)^(1/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.31 \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{320 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arccos \left (a x\right )}\right )}{320 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32 \, a^{5}} \] Input:
integrate(x^4/arccos(a*x)^(1/2),x, algorithm="giac")
Output:
-(1/320*I - 1/320)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arcco s(a*x)))/a^5 + (1/320*I + 1/320)*sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt (10)*sqrt(arccos(a*x)))/a^5 - (1/64*I - 1/64)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 + (1/64*I + 1/64)*sqrt(6)*sqrt(pi)*e rf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arccos(a*x)))/a^5 - (1/32*I - 1/32)*sqrt(2) *sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5 + (1/32*I + 1/3 2)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arccos(a*x)))/a^5
Timed out. \[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\int \frac {x^4}{\sqrt {\mathrm {acos}\left (a\,x\right )}} \,d x \] Input:
int(x^4/acos(a*x)^(1/2),x)
Output:
int(x^4/acos(a*x)^(1/2), x)
\[ \int \frac {x^4}{\sqrt {\arccos (a x)}} \, dx=\frac {-2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, a^{4} x^{4}+\frac {8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}}{3}-\frac {4 \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a}{3}+\frac {4 \left (\int \frac {\sqrt {\mathit {acos} \left (a x \right )}\, x^{2}}{\mathit {acos} \left (a x \right ) a^{2} x^{2}-\mathit {acos} \left (a x \right )}d x \right ) a^{3}}{3}+10 \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{5}}{a^{2} x^{2}-1}d x \right ) a^{6}-8 \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x^{3}}{a^{2} x^{2}-1}d x \right ) a^{4}-\frac {8 \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x}{a^{2} x^{2}-1}d x \right ) a^{2}}{3}}{a^{5}} \] Input:
int(x^4/acos(a*x)^(1/2),x)
Output:
(2*( - 3*sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*a**4*x**4 + 4*sqrt( - a**2 *x**2 + 1)*sqrt(acos(a*x)) - 2*int(sqrt(acos(a*x))/(acos(a*x)*a**2*x**2 - acos(a*x)),x)*a + 2*int((sqrt(acos(a*x))*x**2)/(acos(a*x)*a**2*x**2 - acos (a*x)),x)*a**3 + 15*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**5)/(a** 2*x**2 - 1),x)*a**6 - 12*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x**3) /(a**2*x**2 - 1),x)*a**4 - 4*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x )/(a**2*x**2 - 1),x)*a**2))/(3*a**5)