Integrand size = 12, antiderivative size = 121 \[ \int x^4 \sqrt {\arcsin (a x)} \, dx=\frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{8 a^5}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{16 a^5}-\frac {\sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{80 a^5} \] Output:
1/5*x^5*arcsin(a*x)^(1/2)-1/16*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)* arcsin(a*x)^(1/2))/a^5+1/96*6^(1/2)*Pi^(1/2)*FresnelS(6^(1/2)/Pi^(1/2)*arc sin(a*x)^(1/2))/a^5-1/800*10^(1/2)*Pi^(1/2)*FresnelS(10^(1/2)/Pi^(1/2)*arc sin(a*x)^(1/2))/a^5
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.59 \[ \int x^4 \sqrt {\arcsin (a x)} \, dx=\frac {150 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-i \arcsin (a x)\right )+150 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},i \arcsin (a x)\right )-25 \sqrt {3} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-3 i \arcsin (a x)\right )-25 \sqrt {3} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},3 i \arcsin (a x)\right )+3 \sqrt {5} \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-5 i \arcsin (a x)\right )+3 \sqrt {5} \sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},5 i \arcsin (a x)\right )}{2400 a^5 \sqrt {\arcsin (a x)}} \] Input:
Integrate[x^4*Sqrt[ArcSin[a*x]],x]
Output:
(150*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-I)*ArcSin[a*x]] + 150*Sqrt[I*ArcS in[a*x]]*Gamma[3/2, I*ArcSin[a*x]] - 25*Sqrt[3]*Sqrt[(-I)*ArcSin[a*x]]*Gam ma[3/2, (-3*I)*ArcSin[a*x]] - 25*Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (3 *I)*ArcSin[a*x]] + 3*Sqrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-5*I)*ArcS in[a*x]] + 3*Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (5*I)*ArcSin[a*x]])/(2 400*a^5*Sqrt[ArcSin[a*x]])
Time = 0.51 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5140, 5224, 3042, 3793, 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.
\(\displaystyle \int x^4 \sqrt {\arcsin (a x)} \, dx\) |
\(\Big \downarrow \) 5140 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {1}{10} a \int \frac {x^5}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\int \frac {a^5 x^5}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{10 a^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\int \frac {\sin (\arcsin (a x))^5}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{10 a^5}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\int \left (\frac {5 a x}{8 \sqrt {\arcsin (a x)}}-\frac {5 \sin (3 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}+\frac {\sin (5 \arcsin (a x))}{16 \sqrt {\arcsin (a x)}}\right )d\arcsin (a x)}{10 a^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} x^5 \sqrt {\arcsin (a x)}-\frac {\frac {5}{4} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )-\frac {5}{8} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arcsin (a x)}\right )}{10 a^5}\) |
Input:
Int[x^4*Sqrt[ArcSin[a*x]],x]
Output:
(x^5*Sqrt[ArcSin[a*x]])/5 - ((5*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin [a*x]]])/4 - (5*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/8 + (Sq rt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/8)/(10*a^5)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcSin[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18
method | result | size |
default | \(-\frac {3 \sqrt {5}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-25 \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+150 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-300 \arcsin \left (a x \right ) a x +150 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-30 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )}{2400 a^{5} \sqrt {\arcsin \left (a x \right )}}\) | \(143\) |
Input:
int(x^4*arcsin(a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2400/a^5/arcsin(a*x)^(1/2)*(3*5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2 )*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))-25*3^(1/2)*2^(1/2)* arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*arcsin(a*x)^( 1/2))+150*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arc sin(a*x)^(1/2))-300*arcsin(a*x)*a*x+150*arcsin(a*x)*sin(3*arcsin(a*x))-30* arcsin(a*x)*sin(5*arcsin(a*x)))
Exception generated. \[ \int x^4 \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4*arcsin(a*x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^4 \sqrt {\arcsin (a x)} \, dx=\int x^{4} \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \] Input:
integrate(x**4*asin(a*x)**(1/2),x)
Output:
Integral(x**4*sqrt(asin(a*x)), x)
Exception generated. \[ \int x^4 \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^4*arcsin(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 = 247, normalized size of antiderivative = 2.04 \[ \int x^4 \sqrt {\arcsin (a x)} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{3200 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {10} \sqrt {\arcsin \left (a x\right )}\right )}{3200 \, a^{5}} + \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{384 \, a^{5}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{384 \, a^{5}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{5}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{5}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{16 \, a^{5}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{5}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-5 i \, \arcsin \left (a x\right )\right )}}{160 \, a^{5}} \] Input:
integrate(x^4*arcsin(a*x)^(1/2),x, algorithm="giac")
Output:
-(1/3200*I - 1/3200)*sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arc sin(a*x)))/a^5 + (1/3200*I + 1/3200)*sqrt(10)*sqrt(pi)*erf(-(1/2*I + 1/2)* sqrt(10)*sqrt(arcsin(a*x)))/a^5 + (1/384*I - 1/384)*sqrt(6)*sqrt(pi)*erf(( 1/2*I - 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 - (1/384*I + 1/384)*sqrt(6)*sq rt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/a^5 - (1/64*I - 1/64) *sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 + (1/64 *I + 1/64)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/ a^5 - 1/160*I*sqrt(arcsin(a*x))*e^(5*I*arcsin(a*x))/a^5 + 1/32*I*sqrt(arcs in(a*x))*e^(3*I*arcsin(a*x))/a^5 - 1/16*I*sqrt(arcsin(a*x))*e^(I*arcsin(a* x))/a^5 + 1/16*I*sqrt(arcsin(a*x))*e^(-I*arcsin(a*x))/a^5 - 1/32*I*sqrt(ar csin(a*x))*e^(-3*I*arcsin(a*x))/a^5 + 1/160*I*sqrt(arcsin(a*x))*e^(-5*I*ar csin(a*x))/a^5
Timed out. \[ \int x^4 \sqrt {\arcsin (a x)} \, dx=\int x^4\,\sqrt {\mathrm {asin}\left (a\,x\right )} \,d x \] Input:
int(x^4*asin(a*x)^(1/2),x)
Output:
int(x^4*asin(a*x)^(1/2), x)
\[ \int x^4 \sqrt {\arcsin (a x)} \, dx=\int \sqrt {\mathit {asin} \left (a x \right )}\, x^{4}d x \] Input:
int(x^4*asin(a*x)^(1/2),x)
Output:
int(sqrt(asin(a*x))*x**4,x)