Integrand size = 14, antiderivative size = 130 \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} c^2} \]
2*cos(2*a/b)*FresnelC(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2) /b^(3/2)/c^2+2*FresnelS(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2* a/b)*Pi^(1/2)/b^(3/2)/c^2-2*x*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))^(1/ 2)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.19 \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {i e^{-\frac {2 i a}{b}} \left (-\sqrt {2} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+\sqrt {2} e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )+2 i e^{\frac {2 i a}{b}} \sin (2 \arcsin (c x))\right )}{2 b c^2 \sqrt {a+b \arcsin (c x)}} \]
((I/2)*(-(Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b]) + Sqrt[2]*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x ]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] + (2*I)*E^(((2*I)*a)/b)*S in[2*ArcSin[c*x]]))/(b*c^2*E^(((2*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
Time = 0.57 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5142, 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.
\(\displaystyle \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 5142 |
\(\displaystyle \frac {2 \int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3787 |
\(\displaystyle \frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))-\sin \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))\right )}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle \frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c x))}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {2 \left (2 \sin \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arcsin (c x))}{b}\right )d\sqrt {a+b \arcsin (c x)}+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c x))}{b}\right )d\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {2 \left (2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c x))}{b}\right )d\sqrt {a+b \arcsin (c x)}+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2 c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
(-2*x*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (2*(Sqrt[b]*Sqrt[ Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])] + Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi]) ]*Sin[(2*a)/b]))/(b^2*c^2)
3.2.94.3.1 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_)^(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]
Time = 0.07 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {2 \sqrt {-\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }-2 \sqrt {-\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }+\sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )}{c^{2} b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(156\) |
1/c^2/b*(2*(-1/b)^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2 )*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)-2*(-1/b)^(1/ 2)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^( 1/2)/b)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)+sin(-2*(a+b*arcsin(c*x))/b+2*a/b) )/(a+b*arcsin(c*x))^(1/2)
Exception generated. \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]