\(\int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx\) [390]

Optimal result

Integrand size = 24, antiderivative size = 253 \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}-\frac {3 d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {d \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {3 d \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c}+\frac {d \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{b^{3/2} c} \] Output:

-2*d*(-c^2*x^2+1)^(3/2)/b/c/(a+b*arcsin(c*x))^(1/2)-3/2*d*2^(1/2)*Pi^(1/2) 
*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(3/ 
2)/c-1/2*d*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcs 
in(c*x))^(1/2)/b^(1/2))/b^(3/2)/c+3/2*d*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/ 
Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)/b^(3/2)/c+1/2*d*6^(1/2) 
*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(3 
*a/b)/b^(3/2)/c
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.38 \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {d e^{-\frac {3 i (a+b \arcsin (c x))}{b}} \left (-e^{\frac {3 i a}{b}}-3 e^{\frac {3 i a}{b}+2 i \arcsin (c x)}-3 e^{\frac {3 i a}{b}+4 i \arcsin (c x)}-e^{\frac {3 i (a+2 b \arcsin (c x))}{b}}+3 e^{\frac {2 i a}{b}+3 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+3 e^{\frac {4 i a}{b}+3 i \arcsin (c x)} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c x))}{b}\right )+\sqrt {3} e^{3 i \arcsin (c x)} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+\sqrt {3} e^{3 i \left (\frac {2 a}{b}+\arcsin (c x)\right )} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )}{4 b c \sqrt {a+b \arcsin (c x)}} \] Input:

Integrate[(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^(3/2),x]
 

Output:

(d*(-E^(((3*I)*a)/b) - 3*E^(((3*I)*a)/b + (2*I)*ArcSin[c*x]) - 3*E^(((3*I) 
*a)/b + (4*I)*ArcSin[c*x]) - E^(((3*I)*(a + 2*b*ArcSin[c*x]))/b) + 3*E^((( 
2*I)*a)/b + (3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/ 
2, ((-I)*(a + b*ArcSin[c*x]))/b] + 3*E^(((4*I)*a)/b + (3*I)*ArcSin[c*x])*S 
qrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] + Sqr 
t[3]*E^((3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ( 
(-3*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*E^((3*I)*((2*a)/b + ArcSin[c*x])) 
*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b] 
))/(4*b*c*E^(((3*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, 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.

\(\displaystyle \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx\)

\(\Big \downarrow \) 5166

\(\displaystyle -\frac {6 c d \int \frac {x \sqrt {1-c^2 x^2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 5224

\(\displaystyle -\frac {6 d \int -\frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {6 d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {6 d \int \left (\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c}-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {6 d \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )\right )}{b^2 c}-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \arcsin (c x)}}\)

Input:

Int[(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^(3/2),x]
 

Output:

(-2*d*(1 - c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (6*d*((Sqrt[b]* 
Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]] 
)/2 + (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*Arc 
Sin[c*x]])/Sqrt[b]])/2 - (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + 
 b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt 
[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2))/(b^2*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4906

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]
 

rule 5166

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]
 

rule 5224

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]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.20

method	result	size
default	\(-\frac {d \left (-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-3 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-\sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}-\sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}+3 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )+\cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )\right )}{2 c b \sqrt {a +b \arcsin \left (c x \right )}}\)	\(304\)

Input:

int((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2/c*d/b/(a+b*arcsin(c*x))^(1/2)*(-3*(-1/b)^(1/2)*Pi^(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*ar 
csin(c*x))^(1/2)/b)-3*(-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)-Pi^(1/2)*2^(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*arcsin(c*x))^(1/2)/b)*(-3/b)^(1/2)-Pi^(1/2)*2^( 
1/2)*(a+b*arcsin(c*x))^(1/2)*sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b) 
^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(-3/b)^(1/2)+3*cos(-(a+b*arcsin(c*x))/b+ 
a/b)+cos(-3*(a+b*arcsin(c*x))/b+3*a/b))
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=- d \left (\int \frac {c^{2} x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx\right ) \] Input:

integrate((-c**2*d*x**2+d)/(a+b*asin(c*x))**(3/2),x)
 

Output:

-d*(Integral(c**2*x**2/(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))
 

Maxima [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")
 

Output:

-integrate((c^2*d*x^2 - d)/(b*arcsin(c*x) + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")
 

Output:

integrate(-(c^2*d*x^2 - d)/(b*arcsin(c*x) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {d-c^2\,d\,x^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:

int((d - c^2*d*x^2)/(a + b*asin(c*x))^(3/2),x)
 

Output:

int((d - c^2*d*x^2)/(a + b*asin(c*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^{3/2}} \, dx =\text {Too large to display} \] Input:

int((-c^2*d*x^2+d)/(a+b*asin(c*x))^(3/2),x)
 

Output:

(d*(12*sqrt(asin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*asin(c*x)*b + 12*asin( 
c*x)*int(sqrt(asin(c*x)*b + a)/(asin(c*x)**2*b**2*c**2*x**2 - asin(c*x)**2 
*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a**2*c**2*x**2 - a** 
2),x)*a*b**2*c - 12*asin(c*x)*int((sqrt(asin(c*x)*b + a)*x**4)/(asin(c*x)* 
*2*b**2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin 
(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a*b**2*c**5 + 24*asin(c*x)*int((sqrt 
(asin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*asin(c*x)*x**3)/(asin(c*x)**2*b** 
2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)* 
a*b + a**2*c**2*x**2 - a**2),x)*a*b**2*c**4 - 12*asin(c*x)*int((sqrt(asin( 
c*x)*b + a)*sqrt( - c**2*x**2 + 1)*asin(c*x)**2*x)/(asin(c*x)**2*b**2*c**2 
*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + 
a**2*c**2*x**2 - a**2),x)*b**3*c**2 + 24*asin(c*x)*int((sqrt(asin(c*x)*b + 
 a)*sqrt( - c**2*x**2 + 1)*x**3)/(asin(c*x)**2*b**2*c**2*x**2 - asin(c*x)* 
*2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a**2*c**2*x**2 - a 
**2),x)*a**2*b*c**4 + 12*asin(c*x)*int((sqrt(asin(c*x)*b + a)*sqrt( - c**2 
*x**2 + 1)*x)/(asin(c*x)**2*b**2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c* 
x)*a*b*c**2*x**2 - 2*asin(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*a**2*b*c**2 
 + 3*asin(c*x)*int((sqrt(asin(c*x)*b + a)*sqrt( - c**2*x**2 + 1)*x)/(asin( 
c*x)**2*b**2*c**2*x**2 - asin(c*x)**2*b**2 + 2*asin(c*x)*a*b*c**2*x**2 - 2 
*asin(c*x)*a*b + a**2*c**2*x**2 - a**2),x)*b**3*c**2 - 8*sqrt(asin(c*x)...