\(\int \frac {a+b \arcsin (c x)}{x (d-c^2 d x^2)^{3/2}} \, dx\) [123]

Optimal result

Integrand size = 27, antiderivative size = 224 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\frac {a}{d}+\frac {b \arcsin (c x)}{d}}{\sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \] Output:

(a/d+b*arcsin(c*x)/d)/(-c^2*d*x^2+d)^(1/2)-2*(-c^2*x^2+1)^(1/2)*(a+b*arcsi 
n(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)-b*(-c^2*x 
^2+1)^(1/2)*arctanh(c*x)/d/(-c^2*d*x^2+d)^(1/2)+I*b*(-c^2*x^2+1)^(1/2)*pol 
ylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)-I*b*(-c^2*x^2+1)^ 
(1/2)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.34 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-\frac {a \sqrt {d-c^2 d x^2}}{-1+c^2 x^2}+a \sqrt {d} \log (x)-a \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b d \left (\arcsin (c x)+\sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-\sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+\sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-\sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}}{d^2} \] Input:

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

Output:

(-((a*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2)) + a*Sqrt[d]*Log[x] - a*Sqrt[d]* 
Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*(ArcSin[c*x] + Sqrt[1 - c^2*x^ 
2]*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]* 
Log[1 + E^(I*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Si 
n[ArcSin[c*x]/2]] - Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[ 
c*x]/2]] + I*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])] - I*Sqrt[1 - 
 c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[d - c^2*d*x^2])/d^2
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.71, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5208, 219, 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.

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

\(\Big \downarrow \) 5208

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 5208

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c 
*(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 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]
 

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(224\)
parts	\(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(224\)

Input:

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

Output:

a/d/(-c^2*d*x^2+d)^(1/2)-a/d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2)) 
/x)+b*(-(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)+(-c^2*x^2+1)^(1 
/2)*(-d*(c^2*x^2-1))^(1/2)*(arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I 
*arctan(I*c*x+(-c^2*x^2+1)^(1/2))-I*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-I*di 
log(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2/(c^2*x^2-1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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