Integrand size = 27, antiderivative size = 373 \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 b^{3/2} c^2}+\frac {5 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 b^{3/2} c^2}+\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{b^{3/2} c^2}+\frac {d^2 \sqrt {3 \pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {6 a}{b}\right )}{8 b^{3/2} c^2} \]
1/2*d^2*cos(4*a/b)*FresnelC(2*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+1/2*d^2*FresnelS(2*2^(1/2)/Pi^(1/2)*(a+ b*arcsin(c*x))^(1/2)/b^(1/2))*sin(4*a/b)*2^(1/2)*Pi^(1/2)/b^(3/2)/c^2+5/8* d^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+5/8*d^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+1/8*d^2*cos(6*a/b)*FresnelC(2*3^(1/2)/Pi ^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(3/2)/c^2+1/8*d ^2*FresnelS(2*3^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(6*a/b) *3^(1/2)*Pi^(1/2)/b^(3/2)/c^2-2*d^2*x*(-c^2*x^2+1)^(5/2)/b/c/(a+b*arcsin(c *x))^(1/2)
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.10 \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {i d^2 e^{-\frac {6 i a}{b}} \left (-5 \sqrt {2} e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+5 \sqrt {2} e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )-8 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {4 i (a+b \arcsin (c x))}{b}\right )+8 e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {4 i (a+b \arcsin (c x))}{b}\right )-\sqrt {6} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {6 i (a+b \arcsin (c x))}{b}\right )+\sqrt {6} e^{\frac {12 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {6 i (a+b \arcsin (c x))}{b}\right )+10 i e^{\frac {6 i a}{b}} \sin (2 \arcsin (c x))+8 i e^{\frac {6 i a}{b}} \sin (4 \arcsin (c x))+2 i e^{\frac {6 i a}{b}} \sin (6 \arcsin (c x))\right )}{32 b c^2 \sqrt {a+b \arcsin (c x)}} \]
((I/32)*d^2*(-5*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] + 5*Sqrt[2]*E^(((8*I)*a)/b)*Sq rt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] - 8*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c*x]))/b] + 8*E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b] *Gamma[1/2, ((4*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[6]*Sqrt[((-I)*(a + b*Arc Sin[c*x]))/b]*Gamma[1/2, ((-6*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[6]*E^(((12 *I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((6*I)*(a + b*ArcSin[ c*x]))/b] + (10*I)*E^(((6*I)*a)/b)*Sin[2*ArcSin[c*x]] + (8*I)*E^(((6*I)*a) /b)*Sin[4*ArcSin[c*x]] + (2*I)*E^(((6*I)*a)/b)*Sin[6*ArcSin[c*x]]))/(b*c^2 *E^(((6*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
Time = 2.04 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.62, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5214, 5168, 3042, 3793, 2009, 5224, 4906, 2009}
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.
| \(\displaystyle \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5214 |
| \(\displaystyle \frac {2 d^2 \int \frac {\left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 5168 |
| \(\displaystyle \frac {2 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 d^2 \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}+\frac {\pi }{2}\right )^4}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^2}-\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {2 d^2 \int \left (\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 \sqrt {a+b \arcsin (c x)}}+\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {a+b \arcsin (c x)}}+\frac {3}{8 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^2}-\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {12 c d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}+\frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 5224 |
| \(\displaystyle -\frac {12 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^2}+\frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle -\frac {12 d^2 \int \left (-\frac {\cos \left (\frac {6 a}{b}-\frac {6 (a+b \arcsin (c x))}{b}\right )}{32 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{16 \sqrt {a+b \arcsin (c x)}}+\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{32 \sqrt {a+b \arcsin (c x)}}+\frac {1}{16 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^2}+\frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d^2 \left (\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \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 )+\frac {1}{2} \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 )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {3}{4} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {12 d^2 \left (-\frac {1}{16} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \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 )+\frac {1}{32} \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 )-\frac {1}{16} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \sin \left (\frac {6 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}-\frac {2 d^2 x \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
(-2*d^2*x*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (2*d^2*((3* Sqrt[a + b*ArcSin[c*x]])/4 + (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2* Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi]*Cos[(2 *a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/2 + (Sqrt [b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[ (2*a)/b])/2 + (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin [c*x]])/Sqrt[b]]*Sin[(4*a)/b])/8))/(b^2*c^2) - (12*d^2*(Sqrt[a + b*ArcSin[ c*x]]/8 - (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/16 - (Sqrt[b]*Sqrt[Pi/3]*Cos[(6*a)/b]*FresnelC[ (2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/32 + (Sqrt[b]*Sqrt[Pi]*Co s[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/32 + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])] *Sin[(2*a)/b])/32 - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b* ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/16 - (Sqrt[b]*Sqrt[Pi/3]*FresnelS[(2* Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(6*a)/b])/32))/(b^2*c^2)
3.5.38.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Subst[Int[ x^n*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, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p* ((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 1)) )*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x] + Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2*x^2 )^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f }, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 , 0] && IGtQ[m, -3]
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 = 449, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(\frac {d^{2} \left (8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+\sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {-\frac {6}{b}}\, \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}-\sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {-\frac {6}{b}}\, \sin \left (\frac {6 a}{b}\right ) \operatorname {FresnelS}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \sqrt {\pi }\, \sqrt {2}+10 \sqrt {-\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }-10 \sqrt {-\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }+5 \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )+4 \sin \left (-\frac {4 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {4 a}{b}\right )+\sin \left (-\frac {6 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {6 a}{b}\right )\right )}{16 c^{2} b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(449\) |
1/16/c^2*d^2/b*(8*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*co s(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/ b)-8*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(4*a/b)*Fres nelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)+(a+b*arcsi n(c*x))^(1/2)*(-6/b)^(1/2)*cos(6*a/b)*FresnelC(6*2^(1/2)/Pi^(1/2)/(-6/b)^( 1/2)*(a+b*arcsin(c*x))^(1/2)/b)*Pi^(1/2)*2^(1/2)-(a+b*arcsin(c*x))^(1/2)*( -6/b)^(1/2)*sin(6*a/b)*FresnelS(6*2^(1/2)/Pi^(1/2)/(-6/b)^(1/2)*(a+b*arcsi n(c*x))^(1/2)/b)*Pi^(1/2)*2^(1/2)+10*(-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)-10*(-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)+5*sin( -2*(a+b*arcsin(c*x))/b+2*a/b)+4*sin(-4*(a+b*arcsin(c*x))/b+4*a/b)+sin(-6*( a+b*arcsin(c*x))/b+6*a/b))/(a+b*arcsin(c*x))^(1/2)
Exception generated. \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{3}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{5}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \]
d**2*(Integral(x/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c *x)), x) + Integral(-2*c**2*x**3/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*a sin(c*x))*asin(c*x)), x) + Integral(c**4*x**5/(a*sqrt(a + b*asin(c*x)) + b *sqrt(a + b*asin(c*x))*asin(c*x)), x))
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]