Integrand size = 10, antiderivative size = 144 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {2 x^4}{3 \arcsin (a x)^2}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)}-\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{3 a^4}+\frac {4 \operatorname {CosIntegral}(4 \arcsin (a x))}{3 a^4} \]
-1/2*x^2/a^2/arcsin(a*x)^2+2/3*x^4/arcsin(a*x)^2-1/3*Ci(2*arcsin(a*x))/a^4 +4/3*Ci(4*arcsin(a*x))/a^4-1/3*x^3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^3-x*(- a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)+8/3*x^3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)
Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.74 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\frac {\frac {a x \left (-2 a^2 x^2 \sqrt {1-a^2 x^2}+a x \left (-3+4 a^2 x^2\right ) \arcsin (a x)+2 \sqrt {1-a^2 x^2} \left (-3+8 a^2 x^2\right ) \arcsin (a x)^2\right )}{\arcsin (a x)^3}-2 \operatorname {CosIntegral}(2 \arcsin (a x))+8 \operatorname {CosIntegral}(4 \arcsin (a x))}{6 a^4} \]
((a*x*(-2*a^2*x^2*Sqrt[1 - a^2*x^2] + a*x*(-3 + 4*a^2*x^2)*ArcSin[a*x] + 2 *Sqrt[1 - a^2*x^2]*(-3 + 8*a^2*x^2)*ArcSin[a*x]^2))/ArcSin[a*x]^3 - 2*CosI ntegral[2*ArcSin[a*x]] + 8*CosIntegral[4*ArcSin[a*x]])/(6*a^4)
Time = 0.70 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5144, 5222, 5142, 2009, 3042, 3783}
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.
\(\displaystyle \int \frac {x^3}{\arcsin (a x)^4} \, dx\) |
\(\Big \downarrow \) 5144 |
\(\displaystyle \frac {\int \frac {x^2}{\sqrt {1-a^2 x^2} \arcsin (a x)^3}dx}{a}-\frac {4}{3} a \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)^3}dx-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 5222 |
\(\displaystyle \frac {\frac {\int \frac {x}{\arcsin (a x)^2}dx}{a}-\frac {x^2}{2 a \arcsin (a x)^2}}{a}-\frac {4}{3} a \left (\frac {2 \int \frac {x^3}{\arcsin (a x)^2}dx}{a}-\frac {x^4}{2 a \arcsin (a x)^2}\right )-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 5142 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\cos (2 \arcsin (a x))}{\arcsin (a x)}d\arcsin (a x)}{a^2}-\frac {x \sqrt {1-a^2 x^2}}{a \arcsin (a x)}}{a}-\frac {x^2}{2 a \arcsin (a x)^2}}{a}-\frac {4}{3} a \left (\frac {2 \left (\frac {\int \left (\frac {\cos (2 \arcsin (a x))}{2 \arcsin (a x)}-\frac {\cos (4 \arcsin (a x))}{2 \arcsin (a x)}\right )d\arcsin (a x)}{a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{a}-\frac {x^4}{2 a \arcsin (a x)^2}\right )-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\cos (2 \arcsin (a x))}{\arcsin (a x)}d\arcsin (a x)}{a^2}-\frac {x \sqrt {1-a^2 x^2}}{a \arcsin (a x)}}{a}-\frac {x^2}{2 a \arcsin (a x)^2}}{a}-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {4}{3} a \left (\frac {2 \left (\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arcsin (a x))-\frac {1}{2} \operatorname {CosIntegral}(4 \arcsin (a x))}{a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{a}-\frac {x^4}{2 a \arcsin (a x)^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\int \frac {\sin \left (2 \arcsin (a x)+\frac {\pi }{2}\right )}{\arcsin (a x)}d\arcsin (a x)}{a^2}-\frac {x \sqrt {1-a^2 x^2}}{a \arcsin (a x)}}{a}-\frac {x^2}{2 a \arcsin (a x)^2}}{a}-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {4}{3} a \left (\frac {2 \left (\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arcsin (a x))-\frac {1}{2} \operatorname {CosIntegral}(4 \arcsin (a x))}{a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{a}-\frac {x^4}{2 a \arcsin (a x)^2}\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {\frac {\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{a^2}-\frac {x \sqrt {1-a^2 x^2}}{a \arcsin (a x)}}{a}-\frac {x^2}{2 a \arcsin (a x)^2}}{a}-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {4}{3} a \left (\frac {2 \left (\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arcsin (a x))-\frac {1}{2} \operatorname {CosIntegral}(4 \arcsin (a x))}{a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\right )}{a}-\frac {x^4}{2 a \arcsin (a x)^2}\right )\) |
-1/3*(x^3*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^3) + (-1/2*x^2/(a*ArcSin[a*x]^ 2) + (-((x*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + CosIntegral[2*ArcSin[a*x] ]/a^2)/a)/a - (4*a*(-1/2*x^4/(a*ArcSin[a*x]^2) + (2*(-((x^3*Sqrt[1 - a^2*x ^2])/(a*ArcSin[a*x])) + (CosIntegral[2*ArcSin[a*x]]/2 - CosIntegral[4*ArcS in[a*x]]/2)/a^4))/a))/3
3.1.68.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp [1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c* x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Sim p[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt [1 - c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcSi n[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[ m, 0] && LtQ[n, -2]
Int[(((a_.) + ArcSin[(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*ArcSin[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* ArcSin[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 = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}+\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{3}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{24 \arcsin \left (a x \right )^{3}}+\frac {\cos \left (4 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}-\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{3 \arcsin \left (a x \right )}+\frac {4 \,\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{3}}{a^{4}}\) | \(114\) |
default | \(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}+\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{3}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{24 \arcsin \left (a x \right )^{3}}+\frac {\cos \left (4 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}-\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{3 \arcsin \left (a x \right )}+\frac {4 \,\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{3}}{a^{4}}\) | \(114\) |
1/a^4*(-1/12/arcsin(a*x)^3*sin(2*arcsin(a*x))-1/12/arcsin(a*x)^2*cos(2*arc sin(a*x))+1/6/arcsin(a*x)*sin(2*arcsin(a*x))-1/3*Ci(2*arcsin(a*x))+1/24/ar csin(a*x)^3*sin(4*arcsin(a*x))+1/12/arcsin(a*x)^2*cos(4*arcsin(a*x))-1/3/a rcsin(a*x)*sin(4*arcsin(a*x))+4/3*Ci(4*arcsin(a*x)))
\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{4}} \,d x } \]
\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int \frac {x^{3}}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{4}} \,d x } \]
-1/6*(6*a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3*integrate(1/3*(32 *a^4*x^4 - 30*a^2*x^2 + 3)*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^5*x^2 - a^3)*a rctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + 2*(a^2*x^3 - (8*a^2*x^3 - 3*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a* x + 1) - (4*a^3*x^4 - 3*a*x^2)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))) /(a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=-\frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{3 \, a^{3} \arcsin \left (a x\right )} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x}{3 \, a^{3} \arcsin \left (a x\right )} + \frac {4 \, \operatorname {Ci}\left (4 \, \arcsin \left (a x\right )\right )}{3 \, a^{4}} - \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{3 \, a^{4}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{3 \, a^{3} \arcsin \left (a x\right )^{3}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{3 \, a^{4} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{3 \, a^{3} \arcsin \left (a x\right )^{3}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )}}{6 \, a^{4} \arcsin \left (a x\right )^{2}} + \frac {1}{6 \, a^{4} \arcsin \left (a x\right )^{2}} \]
-8/3*(-a^2*x^2 + 1)^(3/2)*x/(a^3*arcsin(a*x)) + 5/3*sqrt(-a^2*x^2 + 1)*x/( a^3*arcsin(a*x)) + 4/3*cos_integral(4*arcsin(a*x))/a^4 - 1/3*cos_integral( 2*arcsin(a*x))/a^4 + 1/3*(-a^2*x^2 + 1)^(3/2)*x/(a^3*arcsin(a*x)^3) + 2/3* (a^2*x^2 - 1)^2/(a^4*arcsin(a*x)^2) - 1/3*sqrt(-a^2*x^2 + 1)*x/(a^3*arcsin (a*x)^3) + 5/6*(a^2*x^2 - 1)/(a^4*arcsin(a*x)^2) + 1/6/(a^4*arcsin(a*x)^2)
Timed out. \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \]