Integrand size = 10, antiderivative size = 89 \[ \int x \arcsin (a x)^{3/2} \, dx=\frac {3 x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{8 a}-\frac {\arcsin (a x)^{3/2}}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{32 a^2} \]
-1/4*arcsin(a*x)^(3/2)/a^2+1/2*x^2*arcsin(a*x)^(3/2)-3/32*FresnelS(2*arcsi n(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^2+3/8*x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^( 1/2)/a
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int x \arcsin (a x)^{3/2} \, dx=\frac {\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {5}{2},-2 i \arcsin (a x)\right )+\sqrt {i \arcsin (a x)} \Gamma \left (\frac {5}{2},2 i \arcsin (a x)\right )}{16 \sqrt {2} a^2 \sqrt {\arcsin (a x)}} \]
(Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (-2*I)*ArcSin[a*x]] + Sqrt[I*ArcSin[a*x ]]*Gamma[5/2, (2*I)*ArcSin[a*x]])/(16*Sqrt[2]*a^2*Sqrt[ArcSin[a*x]])
Time = 0.74 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5140, 5210, 5146, 4906, 27, 3042, 3786, 3832, 5152}
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.
\(\displaystyle \int x \arcsin (a x)^{3/2} \, dx\) |
\(\Big \downarrow \) 5140 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 5210 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {x}{\sqrt {\arcsin (a x)}}dx}{4 a}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
\(\Big \downarrow \) 5146 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int \frac {a x \sqrt {1-a^2 x^2}}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{2 \sqrt {\arcsin (a x)}}d\arcsin (a x)}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sin (2 \arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{8 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \sin (2 \arcsin (a x))d\sqrt {\arcsin (a x)}}{4 a^3}+\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\int \frac {\sqrt {\arcsin (a x)}}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {1}{2} x^2 \arcsin (a x)^{3/2}-\frac {3}{4} a \left (\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin (a x)}}{\sqrt {\pi }}\right )}{8 a^3}+\frac {\arcsin (a x)^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}{2 a^2}\right )\) |
(x^2*ArcSin[a*x]^(3/2))/2 - (3*a*(-1/2*(x*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a* x]])/a^2 + ArcSin[a*x]^(3/2)/(3*a^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a *x]])/Sqrt[Pi]])/(8*a^3)))/4
3.1.83.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + 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_.), x_Symbol] :> Simp[1 /(b*c^(m + 1)) Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {8 \arcsin \left (a x \right )^{2} \cos \left (2 \arcsin \left (a x \right )\right )+3 \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )-6 \arcsin \left (a x \right ) \sin \left (2 \arcsin \left (a x \right )\right )}{32 a^{2} \sqrt {\arcsin \left (a x \right )}}\) | \(64\) |
-1/32/a^2*(8*arcsin(a*x)^2*cos(2*arcsin(a*x))+3*arcsin(a*x)^(1/2)*Pi^(1/2) *FresnelS(2*arcsin(a*x)^(1/2)/Pi^(1/2))-6*arcsin(a*x)*sin(2*arcsin(a*x)))/ arcsin(a*x)^(1/2)
Exception generated. \[ \int x \arcsin (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x \arcsin (a x)^{3/2} \, dx=\int x \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Exception generated. \[ \int x \arcsin (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.20 \[ \int x \arcsin (a x)^{3/2} \, dx=-\frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{8 \, a^{2}} - \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{2}} + \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{128 \, a^{2}} - \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} + \frac {3 i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-2 i \, \arcsin \left (a x\right )\right )}}{32 \, a^{2}} \]
-1/8*arcsin(a*x)^(3/2)*e^(2*I*arcsin(a*x))/a^2 - 1/8*arcsin(a*x)^(3/2)*e^( -2*I*arcsin(a*x))/a^2 - (3/128*I - 3/128)*sqrt(pi)*erf((I - 1)*sqrt(arcsin (a*x)))/a^2 + (3/128*I + 3/128)*sqrt(pi)*erf(-(I + 1)*sqrt(arcsin(a*x)))/a ^2 - 3/32*I*sqrt(arcsin(a*x))*e^(2*I*arcsin(a*x))/a^2 + 3/32*I*sqrt(arcsin (a*x))*e^(-2*I*arcsin(a*x))/a^2
Timed out. \[ \int x \arcsin (a x)^{3/2} \, dx=\int x\,{\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]