3.2.17 \(\int \frac {a+b \arcsin (c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx\) [117]

3.2.17.1 Optimal result

Integrand size = 27, antiderivative size = 229 \[ \int \frac {a+b \arcsin (c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 d x^2}-\frac {c^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 \sqrt {d-c^2 d x^2}}-\frac {i b c^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 \sqrt {d-c^2 d x^2}} \]

output

-1/2*b*c*(-c^2*x^2+1)^(1/2)/x/(-c^2*d*x^2+d)^(1/2)-c^2*(a+b*arcsin(c*x))*a 
rctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1 
/2*I*b*c^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/(-c^2*d 
*x^2+d)^(1/2)-1/2*I*b*c^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1) 
^(1/2)/(-c^2*d*x^2+d)^(1/2)-1/2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/d/x 
^2
 

3.2.17.2 Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \arcsin (c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\frac {-\frac {4 a \sqrt {d-c^2 d x^2}}{x^2}+4 a c^2 \sqrt {d} \log (x)-4 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b c^2 d^2 \left (1-c^2 x^2\right )^{3/2} \left (-2 \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )+4 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-4 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+4 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-4 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+\arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{\left (d-c^2 d x^2\right )^{3/2}}}{8 d} \]

input

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

output

((-4*a*Sqrt[d - c^2*d*x^2])/x^2 + 4*a*c^2*Sqrt[d]*Log[x] - 4*a*c^2*Sqrt[d] 
*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*c^2*d^2*(1 - c^2*x^2)^(3/2)*(-2 
*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 4*ArcSin[c*x]*Log 
[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (4*I) 
*PolyLog[2, -E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + Ar 
cSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(d - c^2*d*x^2)^(3 
/2))/(8*d)
 

3.2.17.3 Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5204, 15, 5218, 3042, 4671, 2715, 2838}

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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

\(\displaystyle \frac {c^2 \sqrt {1-c^2 x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {c^2 \sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {c^2 \sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 d x^2}-\frac {b c \sqrt {1-c^2 x^2}}{2 x \sqrt {d-c^2 d x^2}}\)

input

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

output

-1/2*(b*c*Sqrt[1 - c^2*x^2])/(x*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2 
]*(a + b*ArcSin[c*x]))/(2*d*x^2) + (c^2*Sqrt[1 - c^2*x^2]*(-2*(a + b*ArcSi 
n[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - 
I*b*PolyLog[2, E^(I*ArcSin[c*x])]))/(2*Sqrt[d - c^2*d*x^2])
 

3.2.17.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5204

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* 
c*(n/(f*(m + 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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

rule 5218

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

3.2.17.4 Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.21

method	result	size
default	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{2 d \,x^{2}}-\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}\right )\)	\(277\)
parts	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{2 d \,x^{2}}-\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}\right )\)	\(277\)

input

int((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

output

-1/2*a/d/x^2*(-c^2*d*x^2+d)^(1/2)-1/2*a*c^2/d^(1/2)*ln((2*d+2*d^(1/2)*(-c^ 
2*d*x^2+d)^(1/2))/x)+b*(-1/2*(c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-a 
rcsin(c*x))*(-d*(c^2*x^2-1))^(1/2)/x^2/d/(c^2*x^2-1)-1/2*I*(-c^2*x^2+1)^(1 
/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)*(I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x 
^2+1)^(1/2))-I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-polylog(2,I*c*x+ 
(-c^2*x^2+1)^(1/2))+polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2)))*c^2)
 

3.2.17.5 Fricas [F]

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

input

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas" 
)
 

output

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^2*d*x^5 - d*x^3), x)
 

3.2.17.6 Sympy [F]

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

input

integrate((a+b*asin(c*x))/x**3/(-c**2*d*x**2+d)**(1/2),x)
 

output

Integral((a + b*asin(c*x))/(x**3*sqrt(-d*(c*x - 1)*(c*x + 1))), x)
 

3.2.17.7 Maxima [F]

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

input

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima" 
)
 

output

-1/2*(c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d) 
+ sqrt(-c^2*d*x^2 + d)/(d*x^2))*a + b*integrate(arctan2(c*x, sqrt(c*x + 1) 
*sqrt(-c*x + 1))/(sqrt(c*x + 1)*sqrt(-c*x + 1)*x^3), x)/sqrt(d)
 

3.2.17.8 Giac [F(-2)]

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

input

integrate((a+b*arcsin(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
 

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

3.2.17.9 Mupad [F(-1)]

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

input

int((a + b*asin(c*x))/(x^3*(d - c^2*d*x^2)^(1/2)),x)
 

output

int((a + b*asin(c*x))/(x^3*(d - c^2*d*x^2)^(1/2)), x)