Integrand size = 26, antiderivative size = 390 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {5 d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c}-\frac {5 d^2 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d^2 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {5 d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 b^{3/2} c}+\frac {5 d^2 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^{3/2} c}+\frac {d^2 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{4 b^{3/2} c} \]
-5/4*d^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+5/4*d^2*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsi n(c*x))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/b^(3/2)/c-5/8*d^2*cos(3*a /b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^ (1/2)/b^(3/2)/c+5/8*d^2*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/ b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/b^(3/2)/c-1/8*d^2*cos(5*a/b)*FresnelS (10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*10^(1/2)*Pi^(1/2)/b^(3 /2)/c+1/8*d^2*FresnelC(10^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))* sin(5*a/b)*10^(1/2)*Pi^(1/2)/b^(3/2)/c-2*d^2*(-c^2*x^2+1)^(5/2)/b/c/(a+b*a rcsin(c*x))^(1/2)
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {d^2 e^{-\frac {5 i (a+b \arcsin (c x))}{b}} \left (-e^{\frac {5 i a}{b}}-5 e^{\frac {5 i a}{b}+2 i \arcsin (c x)}-10 e^{\frac {5 i a}{b}+4 i \arcsin (c x)}-10 e^{\frac {5 i a}{b}+6 i \arcsin (c x)}-5 e^{\frac {5 i a}{b}+8 i \arcsin (c x)}-e^{\frac {5 i (a+2 b \arcsin (c x))}{b}}+10 e^{\frac {4 i a}{b}+5 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+10 e^{\frac {6 i a}{b}+5 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )+5 \sqrt {3} e^{\frac {2 i a}{b}+5 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+5 \sqrt {3} e^{\frac {8 i a}{b}+5 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )+\sqrt {5} e^{5 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {5 i (a+b \arcsin (c x))}{b}\right )+\sqrt {5} e^{\frac {5 i (2 a+b \arcsin (c x))}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {5 i (a+b \arcsin (c x))}{b}\right )\right )}{16 b c \sqrt {a+b \arcsin (c x)}} \]
(d^2*(-E^(((5*I)*a)/b) - 5*E^(((5*I)*a)/b + (2*I)*ArcSin[c*x]) - 10*E^(((5 *I)*a)/b + (4*I)*ArcSin[c*x]) - 10*E^(((5*I)*a)/b + (6*I)*ArcSin[c*x]) - 5 *E^(((5*I)*a)/b + (8*I)*ArcSin[c*x]) - E^(((5*I)*(a + 2*b*ArcSin[c*x]))/b) + 10*E^(((4*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/ b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 10*E^(((6*I)*a)/b + (5*I)*Ar cSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x ]))/b] + 5*Sqrt[3]*E^(((2*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*A rcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 5*Sqrt[3]*E^( ((8*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2 , ((3*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[5]*E^((5*I)*ArcSin[c*x])*Sqrt[((-I )*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] + Sqr t[5]*E^(((5*I)*(2*a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*G amma[1/2, ((5*I)*(a + b*ArcSin[c*x]))/b]))/(16*b*c*E^(((5*I)*(a + b*ArcSin [c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])
Time = 0.83 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5166, 5224, 25, 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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 5166 |
\(\displaystyle -\frac {10 c d^2 \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {10 d^2 \int -\frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {10 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {10 d^2 \int \left (\frac {\sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c x))}{b}\right )}{16 \sqrt {a+b \arcsin (c x)}}+\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{16 \sqrt {a+b \arcsin (c x)}}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{8 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {10 d^2 \left (-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{10}} \sqrt {b} \sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {3 \pi }{2}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{10}} \sqrt {b} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
(-2*d^2*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (10*d^2*((Sqr t[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqr t[b]])/4 + (Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[ a + b*ArcSin[c*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi/10]*Cos[(5*a)/b]*Fresne lS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi/2] *FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/4 - (Sqr t[b]*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] *Sin[(3*a)/b])/8 - (Sqrt[b]*Sqrt[Pi/10]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*A rcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/8))/(b^2*c)
3.5.39.3.1 Defintions of rubi rules used
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[Sqrt[1 - c^2*x^2]*(d + e*x^2)^p*((a + b*ArcSin[c*x])^(n + 1 )/(b*c*(n + 1))), x] + Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, 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 = 0.14 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.16
method | result | size |
default | \(-\frac {d^{2} \left (-5 \sqrt {2}\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }-5 \sqrt {2}\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {\pi }-\cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {-\frac {5}{b}}-\sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {-\frac {5}{b}}-10 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }-10 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }+10 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )+5 \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )+\cos \left (-\frac {5 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {5 a}{b}\right )\right )}{8 c b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(451\) |
-1/8/c*d^2/b/(a+b*arcsin(c*x))^(1/2)*(-5*2^(1/2)*(-3/b)^(1/2)*(a+b*arcsin( c*x))^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsi n(c*x))^(1/2)/b)*Pi^(1/2)-5*2^(1/2)*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*s in(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2) /b)*Pi^(1/2)-cos(5*a/b)*FresnelS(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcs in(c*x))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*(-5/b)^(1/2)-si n(5*a/b)*FresnelC(5*2^(1/2)/Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/ b)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*(-5/b)^(1/2)-10*2^(1/2)*(a+b*a rcsin(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arc sin(c*x))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)-10*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)+10*cos(-(a+b*arcsin(c*x))/b+a/b)+5*cos(-3*(a+b*ar csin(c*x))/b+3*a/b)+cos(-5*(a+b*arcsin(c*x))/b+5*a/b))
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\int \left (- \frac {2 c^{2} x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \frac {1}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \]
d**2*(Integral(-2*c**2*x**2/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c *x))*asin(c*x)), x) + Integral(c**4*x**4/(a*sqrt(a + b*asin(c*x)) + b*sqrt (a + b*asin(c*x))*asin(c*x)), x) + Integral(1/(a*sqrt(a + b*asin(c*x)) + b *sqrt(a + b*asin(c*x))*asin(c*x)), x))
\[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]