3.3.39 \(\int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx\) [239]

3.3.39.1 Optimal result

Integrand size = 12, antiderivative size = 191 \[ \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx=-\frac {\sqrt {1-(c+d x)^2}}{4 b d (a+b \arcsin (c+d x))^4}+\frac {c+d x}{12 b^2 d (a+b \arcsin (c+d x))^3}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d (a+b \arcsin (c+d x))^2}-\frac {c+d x}{24 b^4 d (a+b \arcsin (c+d x))}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{24 b^5 d}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{24 b^5 d} \]

output

1/12*(d*x+c)/b^2/d/(a+b*arcsin(d*x+c))^3+1/24*(-d*x-c)/b^4/d/(a+b*arcsin(d 
*x+c))+1/24*Ci((a+b*arcsin(d*x+c))/b)*cos(a/b)/b^5/d+1/24*Si((a+b*arcsin(d 
*x+c))/b)*sin(a/b)/b^5/d-1/4*(1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^4 
+1/24*(1-(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsin(d*x+c))^2
 

3.3.39.2 Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx=\frac {-\frac {6 b^4 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^4}+\frac {2 b^3 (c+d x)}{(a+b \arcsin (c+d x))^3}+\frac {b^2 \sqrt {1-(c+d x)^2}}{(a+b \arcsin (c+d x))^2}-\frac {b (c+d x)}{a+b \arcsin (c+d x)}+\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c+d x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c+d x)\right )}{24 b^5 d} \]

input

Integrate[(a + b*ArcSin[c + d*x])^(-5),x]
 

output

((-6*b^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^4 + (2*b^3*(c + d* 
x))/(a + b*ArcSin[c + d*x])^3 + (b^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[ 
c + d*x])^2 - (b*(c + d*x))/(a + b*ArcSin[c + d*x]) + Cos[a/b]*CosIntegral 
[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(24 
*b^5*d)
 

3.3.39.3 Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5302, 5132, 5222, 5132, 5222, 5134, 3042, 3784, 25, 3042, 3780, 3783}

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.

\(\displaystyle \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx\)

\(\Big \downarrow \) 5302

\(\displaystyle \frac {\int \frac {1}{(a+b \arcsin (c+d x))^5}d(c+d x)}{d}\)

\(\Big \downarrow \) 5132

\(\displaystyle \frac {-\frac {\int \frac {c+d x}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^4}d(c+d x)}{4 b}-\frac {\sqrt {1-(c+d x)^2}}{4 b (a+b \arcsin (c+d x))^4}}{d}\)

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5132

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

\(\Big \downarrow \) 5222

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

\(\Big \downarrow \) 5134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

\(\displaystyle \frac {-\frac {\frac {-\frac {\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c+d x)}{b}\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c+d x)}{b}\right )}{b^2}-\frac {c+d x}{b (a+b \arcsin (c+d x))}}{2 b}-\frac {\sqrt {1-(c+d x)^2}}{2 b (a+b \arcsin (c+d x))^2}}{3 b}-\frac {c+d x}{3 b (a+b \arcsin (c+d x))^3}}{4 b}-\frac {\sqrt {1-(c+d x)^2}}{4 b (a+b \arcsin (c+d x))^4}}{d}\)

input

Int[(a + b*ArcSin[c + d*x])^(-5),x]
 

output

(-1/4*Sqrt[1 - (c + d*x)^2]/(b*(a + b*ArcSin[c + d*x])^4) - (-1/3*(c + d*x 
)/(b*(a + b*ArcSin[c + d*x])^3) + (-1/2*Sqrt[1 - (c + d*x)^2]/(b*(a + b*Ar 
cSin[c + d*x])^2) - (-((c + d*x)/(b*(a + b*ArcSin[c + d*x]))) + (Cos[a/b]* 
CosIntegral[(a + b*ArcSin[c + d*x])/b] + Sin[a/b]*SinIntegral[(a + b*ArcSi 
n[c + d*x])/b])/b^2)/(2*b))/(3*b))/(4*b))/d
 

3.3.39.3.1 Defintions of rubi rules used

rule 25

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

rule 3042

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

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 5132

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

rule 5134

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   Su 
bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, 
c, n}, x]
 

rule 5222

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) 
+ (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^ 
2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Simp[f*(m/(b*c*(n 
 + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[(f*x)^(m - 1)*(a + b* 
ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2* 
d + e, 0] && LtQ[n, -1]
 

rule 5302

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d 
  Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, 
n}, x]
 

3.3.39.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(386\) vs. \(2(178)=356\).

Time = 0.52 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.03

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{4 \left (a +b \arcsin \left (d x +c \right )\right )^{4} b}+\frac {\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} b^{3}+\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} b^{3}+3 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a \,b^{2}+3 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{2} b +3 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{2} b -\arcsin \left (d x +c \right )^{2} b^{3} \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\, \arcsin \left (d x +c \right ) b^{3}+\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{3}+\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{3}-2 \arcsin \left (d x +c \right ) a \,b^{2} \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\, a \,b^{2}-a^{2} b \left (d x +c \right )+2 \left (d x +c \right ) b^{3}}{24 \left (a +b \arcsin \left (d x +c \right )\right )^{3} b^{5}}}{d}\)	\(387\)
default	\(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{4 \left (a +b \arcsin \left (d x +c \right )\right )^{4} b}+\frac {\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} b^{3}+\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} b^{3}+3 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a \,b^{2}+3 \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{2} b +3 \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{2} b -\arcsin \left (d x +c \right )^{2} b^{3} \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\, \arcsin \left (d x +c \right ) b^{3}+\sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{3}+\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) a^{3}-2 \arcsin \left (d x +c \right ) a \,b^{2} \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\, a \,b^{2}-a^{2} b \left (d x +c \right )+2 \left (d x +c \right ) b^{3}}{24 \left (a +b \arcsin \left (d x +c \right )\right )^{3} b^{5}}}{d}\)	\(387\)

input

int(1/(a+b*arcsin(d*x+c))^5,x,method=_RETURNVERBOSE)
 

output

1/d*(-1/4*(1-(d*x+c)^2)^(1/2)/(a+b*arcsin(d*x+c))^4/b+1/24*(sin(a/b)*Si(ar 
csin(d*x+c)+a/b)*arcsin(d*x+c)^3*b^3+cos(a/b)*Ci(arcsin(d*x+c)+a/b)*arcsin 
(d*x+c)^3*b^3+3*sin(a/b)*Si(arcsin(d*x+c)+a/b)*arcsin(d*x+c)^2*a*b^2+3*cos 
(a/b)*Ci(arcsin(d*x+c)+a/b)*arcsin(d*x+c)^2*a*b^2+3*sin(a/b)*Si(arcsin(d*x 
+c)+a/b)*arcsin(d*x+c)*a^2*b+3*cos(a/b)*Ci(arcsin(d*x+c)+a/b)*arcsin(d*x+c 
)*a^2*b-arcsin(d*x+c)^2*b^3*(d*x+c)+(1-(d*x+c)^2)^(1/2)*arcsin(d*x+c)*b^3+ 
sin(a/b)*Si(arcsin(d*x+c)+a/b)*a^3+cos(a/b)*Ci(arcsin(d*x+c)+a/b)*a^3-2*ar 
csin(d*x+c)*a*b^2*(d*x+c)+(1-(d*x+c)^2)^(1/2)*a*b^2-a^2*b*(d*x+c)+2*(d*x+c 
)*b^3)/(a+b*arcsin(d*x+c))^3/b^5)
 

3.3.39.5 Fricas [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{5}} \,d x } \]

input

integrate(1/(a+b*arcsin(d*x+c))^5,x, algorithm="fricas")
 

output

integral(1/(b^5*arcsin(d*x + c)^5 + 5*a*b^4*arcsin(d*x + c)^4 + 10*a^2*b^3 
*arcsin(d*x + c)^3 + 10*a^3*b^2*arcsin(d*x + c)^2 + 5*a^4*b*arcsin(d*x + c 
) + a^5), x)
 

3.3.39.6 Sympy [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{5}}\, dx \]

input

integrate(1/(a+b*asin(d*x+c))**5,x)
 

output

Integral((a + b*asin(c + d*x))**(-5), x)
 

3.3.39.7 Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx=\text {Timed out} \]

input

integrate(1/(a+b*arcsin(d*x+c))^5,x, algorithm="maxima")
 

output

Timed out
 

3.3.39.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1915 vs. \(2 (175) = 350\).

Time = 0.31 (sec) , antiderivative size = 1915, normalized size of antiderivative = 10.03 \[ \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx=\text {Too large to display} \]

input

integrate(1/(a+b*arcsin(d*x+c))^5,x, algorithm="giac")
 

output

1/24*b^4*arcsin(d*x + c)^4*cos(a/b)*cos_integral(a/b + arcsin(d*x + c))/(b 
^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin( 
d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/24*b^4*arcsin(d* 
x + c)^4*sin(a/b)*sin_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + 
c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3 
*b^6*d*arcsin(d*x + c) + a^4*b^5*d) + 1/6*a*b^3*arcsin(d*x + c)^3*cos(a/b) 
*cos_integral(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d* 
arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x 
 + c) + a^4*b^5*d) + 1/6*a*b^3*arcsin(d*x + c)^3*sin(a/b)*sin_integral(a/b 
 + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 
 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d 
) - 1/24*(d*x + c)*b^4*arcsin(d*x + c)^3/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^ 
8*d*arcsin(d*x + c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin 
(d*x + c) + a^4*b^5*d) + 1/4*a^2*b^2*arcsin(d*x + c)^2*cos(a/b)*cos_integr 
al(a/b + arcsin(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x 
+ c)^3 + 6*a^2*b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4 
*b^5*d) + 1/4*a^2*b^2*arcsin(d*x + c)^2*sin(a/b)*sin_integral(a/b + arcsin 
(d*x + c))/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*arcsin(d*x + c)^3 + 6*a^2* 
b^7*d*arcsin(d*x + c)^2 + 4*a^3*b^6*d*arcsin(d*x + c) + a^4*b^5*d) - 1/8*( 
d*x + c)*a*b^3*arcsin(d*x + c)^2/(b^9*d*arcsin(d*x + c)^4 + 4*a*b^8*d*a...
 

3.3.39.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^5} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^5} \,d x \]

input

int(1/(a + b*asin(c + d*x))^5,x)
 

output

int(1/(a + b*asin(c + d*x))^5, x)