\(\int (d-c^2 d x^2) \sqrt {a+b \arcsin (c x)} \, dx\) [81]

Optimal result

Integrand size = 24, antiderivative size = 381 \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\frac {2}{3} d x \sqrt {a+b \arcsin (c x)}+\frac {1}{3} d x \left (1-c^2 x^2\right ) \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c}-\frac {\sqrt {b} d \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{3 c}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{12 c}+\frac {\sqrt {b} d \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{3 c}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 c} \] Output:

2/3*d*x*(a+b*arcsin(c*x))^(1/2)+1/3*d*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^(1/ 
2)-3/8*b^(1/2)*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))/c-1/72*b^(1/2)*d*6^(1/2)*Pi^(1/2)*cos(3*a/b)*F 
resnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/c+3/8*b^(1/2)*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)/c+1/72*b^(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)/c
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.60 \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\frac {b d e^{-\frac {3 i a}{b}} \left (27 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+27 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )+\sqrt {3} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{72 c \sqrt {a+b \arcsin (c x)}} \] Input:

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

Output:

(b*d*(27*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((- 
I)*(a + b*ArcSin[c*x]))/b] + 27*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x] 
))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*(Sqrt[((-I)*(a + b*A 
rcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/ 
b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/ 
b])))/(72*c*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
 

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5158, 5130, 5224, 25, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833, 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.

\(\displaystyle \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx\)

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5224

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*x*(1 - c^2*x^2)*Sqrt[a + b*ArcSin[c*x]])/3 + (2*d*(x*Sqrt[a + b*ArcSin[ 
c*x]] - (Sqrt[b]*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSi 
n[c*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*Arc 
Sin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c)))/3 - (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*ArcSin[c*x]])/Sqrt[b]] 
)/2 - (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sq 
rt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*A 
rcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/2))/(6*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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

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]
 

rule 3786

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]
 

rule 3787

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]
 

rule 3832

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]
 

rule 3833

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]
 

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 5130

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar 
cSin[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - 
 c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 5158

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], 
x] - Simp[b*c*(n/(2*p + 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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

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 = 0.10 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {d \left (-27 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -27 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b -\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b +54 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b +54 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a +6 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b +6 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a \right )}{72 c \sqrt {a +b \arcsin \left (c x \right )}}\)	\(364\)

Input:

int((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/72/c*d/(a+b*arcsin(c*x))^(1/2)*(-27*2^(1/2)*Pi^(1/2)*(-1/b)^(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)*b-27*2^(1/2)*Pi^(1/2)*(-1/b)^(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)*b-2^(1/2)*Pi^(1/2)*(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(3*a/b)*F 
resnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b-2^(1/ 
2)*Pi^(1/2)*(-3/b)^(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)*b+54*arcsin(c*x)*sin 
(-(a+b*arcsin(c*x))/b+a/b)*b+54*sin(-(a+b*arcsin(c*x))/b+a/b)*a+6*arcsin(c 
*x)*sin(-3*(a+b*arcsin(c*x))/b+3*a/b)*b+6*sin(-3*(a+b*arcsin(c*x))/b+3*a/b 
)*a)
 

Fricas [F(-2)]

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

integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((-c**2*d*x**2+d)*(a+b*asin(c*x))**(1/2),x)
 

Output:

-d*(Integral(c**2*x**2*sqrt(a + b*asin(c*x)), x) + Integral(-sqrt(a + b*as 
in(c*x)), x))
 

Maxima [F]

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

integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 1103, normalized size of antiderivative = 2.90 \[ \int \left (d-c^2 d x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\text {Too large to display} \] Input:

integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^(1/2),x, algorithm="giac")
 

Output:

3/8*sqrt(2)*sqrt(pi)*a*b^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sq 
rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b) 
/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 3/16*I*sqrt(2)*sqrt(pi)*b^3 
*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*s 
qrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^ 
2*sqrt(abs(b)))*c) + 3/8*sqrt(2)*sqrt(pi)*a*b^2*d*erf(1/2*I*sqrt(2)*sqrt(b 
*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt( 
abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - 3/16* 
I*sqrt(2)*sqrt(pi)*b^3*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(ab 
s(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((- 
I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/4*sqrt(pi)*a*b^(3/2)*d*erf(- 
1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin( 
c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b) 
)*c) + 1/24*I*sqrt(pi)*b^(5/2)*d*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/ 
sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(3*I*a/b 
)/((sqrt(6)*b^2 + I*sqrt(6)*b^3/abs(b))*c) + 1/4*sqrt(pi)*a*b^(3/2)*d*erf( 
-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin 
(c*x) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/((sqrt(6)*b^2 - I*sqrt(6)*b^3/abs( 
b))*c) - 1/24*I*sqrt(pi)*b^(5/2)*d*erf(-1/2*sqrt(6)*sqrt(b*arcsin(c*x) + a 
)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(c*x) + a)*sqrt(b)/abs(b))*e^(-3...
 

Mupad [F(-1)]

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

int((a + b*asin(c*x))^(1/2)*(d - c^2*d*x^2),x)
 

Output:

int((a + b*asin(c*x))^(1/2)*(d - c^2*d*x^2), x)
 

Reduce [F]

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

int((-c^2*d*x^2+d)*(a+b*asin(c*x))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

d*(int(sqrt(asin(c*x)*b + a),x) - int(sqrt(asin(c*x)*b + a)*x**2,x)*c**2)