Integrand size = 6, antiderivative size = 78 \[ \int \frac {1}{\arcsin (a x)^4} \, dx=-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {x}{6 \arcsin (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{6 a} \]
1/6*x/arcsin(a*x)^2+1/6*Si(arcsin(a*x))/a-1/3*(-a^2*x^2+1)^(1/2)/a/arcsin( a*x)^3+1/6*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\arcsin (a x)^4} \, dx=\frac {-2 \sqrt {1-a^2 x^2}+a x \arcsin (a x)+\sqrt {1-a^2 x^2} \arcsin (a x)^2+\arcsin (a x)^3 \text {Si}(\arcsin (a x))}{6 a \arcsin (a x)^3} \]
(-2*Sqrt[1 - a^2*x^2] + a*x*ArcSin[a*x] + Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + ArcSin[a*x]^3*SinIntegral[ArcSin[a*x]])/(6*a*ArcSin[a*x]^3)
Time = 0.52 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5132, 5222, 5132, 5224, 3042, 3780}
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.
\(\displaystyle \int \frac {1}{\arcsin (a x)^4} \, dx\) |
\(\Big \downarrow \) 5132 |
\(\displaystyle -\frac {1}{3} a \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^3}dx-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 5222 |
\(\displaystyle -\frac {1}{3} a \left (\frac {\int \frac {1}{\arcsin (a x)^2}dx}{2 a}-\frac {x}{2 a \arcsin (a x)^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 5132 |
\(\displaystyle -\frac {1}{3} a \left (\frac {-a \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)}dx-\frac {\sqrt {1-a^2 x^2}}{a \arcsin (a x)}}{2 a}-\frac {x}{2 a \arcsin (a x)^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {1}{3} a \left (\frac {-\frac {\int \frac {a x}{\arcsin (a x)}d\arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a \arcsin (a x)}}{2 a}-\frac {x}{2 a \arcsin (a x)^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{3} a \left (\frac {-\frac {\int \frac {\sin (\arcsin (a x))}{\arcsin (a x)}d\arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a \arcsin (a x)}}{2 a}-\frac {x}{2 a \arcsin (a x)^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle -\frac {1}{3} a \left (\frac {-\frac {\sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {\text {Si}(\arcsin (a x))}{a}}{2 a}-\frac {x}{2 a \arcsin (a x)^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}\) |
-1/3*Sqrt[1 - a^2*x^2]/(a*ArcSin[a*x]^3) - (a*(-1/2*x/(a*ArcSin[a*x]^2) + (-(Sqrt[1 - a^2*x^2]/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/a)/(2*a)) )/3
3.1.71.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2 *x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + Simp[c/(b*(n + 1)) Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a , b, c}, x] && LtQ[n, -1]
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]
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 = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arcsin \left (a x \right )^{3}}+\frac {a x}{6 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{6}}{a}\) | \(63\) |
default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arcsin \left (a x \right )^{3}}+\frac {a x}{6 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{6}}{a}\) | \(63\) |
1/a*(-1/3/arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)+1/6*a*x/arcsin(a*x)^2+1/6/arcsi n(a*x)*(-a^2*x^2+1)^(1/2)+1/6*Si(arcsin(a*x)))
\[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{4}} \,d x } \]
\[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int \frac {1}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{4}} \,d x } \]
-1/6*(6*a^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3*integrate(1/6*sqr t(a*x + 1)*sqrt(-a*x + 1)*x/((a^2*x^2 - 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt (-a*x + 1))), x) - a*x*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)) - sqrt(a *x + 1)*sqrt(-a*x + 1)*(arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2 - 2)) /(a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\arcsin (a x)^4} \, dx=\frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{6 \, a} + \frac {x}{6 \, \arcsin \left (a x\right )^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{6 \, a \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a \arcsin \left (a x\right )^{3}} \]
1/6*sin_integral(arcsin(a*x))/a + 1/6*x/arcsin(a*x)^2 + 1/6*sqrt(-a^2*x^2 + 1)/(a*arcsin(a*x)) - 1/3*sqrt(-a^2*x^2 + 1)/(a*arcsin(a*x)^3)
Timed out. \[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \]