Integrand size = 12, antiderivative size = 137 \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c} \]
-2*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*2^( 1/2)*Pi^(1/2)/b^(3/2)/c+2*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2 )/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/b^(3/2)/c-2*(-c^2*x^2+1)^(1/2)/b/c/(a +b*arcsin(c*x))^(1/2)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {e^{-\frac {i (a+b \arcsin (c x))}{b}} \left (e^{i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+e^{\frac {i a}{b}} \left (-1-e^{2 i \arcsin (c x)}+e^{\frac {i (a+b \arcsin (c x))}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )\right )\right )}{b c \sqrt {a+b \arcsin (c x)}} \]
(E^(I*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^((I*a)/b)*(-1 - E^((2*I)*ArcSin[c*x]) + E^((I*(a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b* ArcSin[c*x]))/b]))/(b*c*E^((I*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c* x]])
Time = 0.71 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5132, 5224, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}
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.
\(\displaystyle \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 5132 |
\(\displaystyle -\frac {2 c \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {2 \int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle -\frac {2 \left (-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle -\frac {2 \left (\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle -\frac {2 \left (2 \cos \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle -\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-2 \sin \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c x)}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -\frac {2 \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c}-\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
(-2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (2*(Sqrt[b]*Sqrt[2* Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] - Sqrt [b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[ a/b]))/(b^2*c)
3.8.1.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[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[(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[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[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[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[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_.)*(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 = 158, normalized size of antiderivative = 1.15
method | result | size |
default | \(-\frac {2 \left (-\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }-\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }+\cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )\right )}{c b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(158\) |
-2/c/b/(a+b*arcsin(c*x))^(1/2)*(-2^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)* FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^( 1/2)*Pi^(1/2)-2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi ^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)+cos(- (a+b*arcsin(c*x))/b+a/b))
Exception generated. \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]