Integrand size = 8, antiderivative size = 105 \[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}+\frac {4 x}{15 \arcsin (a x)^{3/2}}+\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arcsin (a x)}}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{15 a} \] Output:
-2/5*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(5/2)+4/15*x/arcsin(a*x)^(3/2)+8/15* (-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^(1/2)+8/15*2^(1/2)*Pi^(1/2)*FresnelS(2^(1 /2)/Pi^(1/2)*arcsin(a*x)^(1/2))/a
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=\frac {2 e^{i \arcsin (a x)} \left (-3-2 i \arcsin (a x)+4 \arcsin (a x)^2\right )-8 \sqrt {-i \arcsin (a x)} \arcsin (a x)^2 \Gamma \left (\frac {1}{2},-i \arcsin (a x)\right )+e^{-i \arcsin (a x)} \left (-6+4 i \arcsin (a x)+8 \arcsin (a x)^2+8 e^{i \arcsin (a x)} (i \arcsin (a x))^{5/2} \Gamma \left (\frac {1}{2},i \arcsin (a x)\right )\right )}{30 a \arcsin (a x)^{5/2}} \] Input:
Integrate[ArcSin[a*x]^(-7/2),x]
Output:
(2*E^(I*ArcSin[a*x])*(-3 - (2*I)*ArcSin[a*x] + 4*ArcSin[a*x]^2) - 8*Sqrt[( -I)*ArcSin[a*x]]*ArcSin[a*x]^2*Gamma[1/2, (-I)*ArcSin[a*x]] + (-6 + (4*I)* ArcSin[a*x] + 8*ArcSin[a*x]^2 + 8*E^(I*ArcSin[a*x])*(I*ArcSin[a*x])^(5/2)* Gamma[1/2, I*ArcSin[a*x]])/E^(I*ArcSin[a*x]))/(30*a*ArcSin[a*x]^(5/2))
Time = 0.60 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5132, 5222, 5132, 5224, 3042, 3786, 3832}
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.
\(\displaystyle \int \frac {1}{\arcsin (a x)^{7/2}} \, dx\) |
\(\Big \downarrow \) 5132 |
\(\displaystyle -\frac {2}{5} a \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^{5/2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\) |
\(\Big \downarrow \) 5222 |
\(\displaystyle -\frac {2}{5} a \left (\frac {2 \int \frac {1}{\arcsin (a x)^{3/2}}dx}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\) |
\(\Big \downarrow \) 5132 |
\(\displaystyle -\frac {2}{5} a \left (\frac {2 \left (-2 a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {2}{5} a \left (\frac {2 \left (-\frac {2 \int \frac {a x}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2}{5} a \left (\frac {2 \left (-\frac {2 \int \frac {\sin (\arcsin (a x))}{\sqrt {\arcsin (a x)}}d\arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle -\frac {2}{5} a \left (\frac {2 \left (-\frac {4 \int a xd\sqrt {\arcsin (a x)}}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -\frac {2}{5} a \left (\frac {2 \left (-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)}}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a}\right )}{3 a}-\frac {2 x}{3 a \arcsin (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a^2 x^2}}{5 a \arcsin (a x)^{5/2}}\) |
Input:
Int[ArcSin[a*x]^(-7/2),x]
Output:
(-2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2)) - (2*a*((-2*x)/(3*a*ArcSin[ a*x]^(3/2)) + (2*((-2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]]) - (2*Sqrt[2 *Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a))/(3*a)))/5
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[((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 = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\sqrt {2}\, \left (8 \arcsin \left (a x \right )^{3} \pi \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+4 \arcsin \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+2 \arcsin \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x -3 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{15 a \sqrt {\pi }\, \arcsin \left (a x \right )^{3}}\) | \(110\) |
Input:
int(1/arcsin(a*x)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/15/a*2^(1/2)/Pi^(1/2)/arcsin(a*x)^3*(8*arcsin(a*x)^3*Pi*FresnelS(2^(1/2) /Pi^(1/2)*arcsin(a*x)^(1/2))+4*arcsin(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(-a^2*x^ 2+1)^(1/2)+2*arcsin(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*a*x-3*2^(1/2)*arcsin(a*x)^ (1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2))
Exception generated. \[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/arcsin(a*x)^(7/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=\int \frac {1}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \] Input:
integrate(1/asin(a*x)**(7/2),x)
Output:
Integral(asin(a*x)**(-7/2), x)
Exception generated. \[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/arcsin(a*x)^(7/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/arcsin(a*x)^(7/2),x, algorithm="giac")
Output:
integrate(arcsin(a*x)^(-7/2), x)
Timed out. \[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \] Input:
int(1/asin(a*x)^(7/2),x)
Output:
int(1/asin(a*x)^(7/2), x)
\[ \int \frac {1}{\arcsin (a x)^{7/2}} \, dx=\frac {\frac {2 \mathit {asin} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}\, x}{\mathit {asin} \left (a x \right )^{3} a^{2} x^{2}-\mathit {asin} \left (a x \right )^{3}}d x \right ) a^{2}}{5}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {asin} \left (a x \right )}}{5}}{\mathit {asin} \left (a x \right )^{3} a} \] Input:
int(1/asin(a*x)^(7/2),x)
Output:
(2*(asin(a*x)**3*int((sqrt( - a**2*x**2 + 1)*sqrt(asin(a*x))*x)/(asin(a*x) **3*a**2*x**2 - asin(a*x)**3),x)*a**2 - sqrt( - a**2*x**2 + 1)*sqrt(asin(a *x))))/(5*asin(a*x)**3*a)