3.1.43 \(\int \frac {a+b \arcsin (c x)}{x^2 (d-c^2 d x^2)^2} \, dx\) [43]

3.1.43.1 Optimal result

Integrand size = 25, antiderivative size = 186 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c}{2 d^2 \sqrt {1-c^2 x^2}}-\frac {a+b \arcsin (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{2 d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 d^2} \]

output

(-a-b*arcsin(c*x))/d^2/x/(-c^2*x^2+1)+3/2*c^2*x*(a+b*arcsin(c*x))/d^2/(-c^ 
2*x^2+1)-3*I*c*(a+b*arcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/d^2-b*c* 
arctanh((-c^2*x^2+1)^(1/2))/d^2+3/2*I*b*c*polylog(2,-I*(I*c*x+(-c^2*x^2+1) 
^(1/2)))/d^2-3/2*I*b*c*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/d^2-1/2*b*c 
/d^2/(-c^2*x^2+1)^(1/2)
 

3.1.43.2 Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.87 \[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {\frac {4 a}{x}+\frac {b c \sqrt {1-c^2 x^2}}{1-c x}+\frac {b c \sqrt {1-c^2 x^2}}{1+c x}+\frac {2 a c^2 x}{-1+c^2 x^2}+3 i b c \pi \arcsin (c x)+\frac {4 b \arcsin (c x)}{x}+\frac {b c \arcsin (c x)}{-1+c x}+\frac {b c \arcsin (c x)}{1+c x}+4 b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-3 b c \pi \log \left (1-i e^{i \arcsin (c x)}\right )-6 b c \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-3 b c \pi \log \left (1+i e^{i \arcsin (c x)}\right )+6 b c \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+3 a c \log (1-c x)-3 a c \log (1+c x)+3 b c \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+3 b c \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-6 i b c \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+6 i b c \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{4 d^2} \]

input

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

output

-1/4*((4*a)/x + (b*c*Sqrt[1 - c^2*x^2])/(1 - c*x) + (b*c*Sqrt[1 - c^2*x^2] 
)/(1 + c*x) + (2*a*c^2*x)/(-1 + c^2*x^2) + (3*I)*b*c*Pi*ArcSin[c*x] + (4*b 
*ArcSin[c*x])/x + (b*c*ArcSin[c*x])/(-1 + c*x) + (b*c*ArcSin[c*x])/(1 + c* 
x) + 4*b*c*ArcTanh[Sqrt[1 - c^2*x^2]] - 3*b*c*Pi*Log[1 - I*E^(I*ArcSin[c*x 
])] - 6*b*c*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*b*c*Pi*Log[1 + I* 
E^(I*ArcSin[c*x])] + 6*b*c*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a* 
c*Log[1 - c*x] - 3*a*c*Log[1 + c*x] + 3*b*c*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x 
])/4]] + 3*b*c*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*b*c*PolyLog[2, 
(-I)*E^(I*ArcSin[c*x])] + (6*I)*b*c*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^2
 

3.1.43.3 Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5204, 27, 243, 61, 73, 221, 5162, 241, 5164, 3042, 4669, 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.

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

\(\Big \downarrow \) 5204

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5162

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

\(\displaystyle \frac {3 c^2 \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {a+b \arcsin (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{2 d^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {3 c^2 \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {a+b \arcsin (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{2 d^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {3 c^2 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{d^2}-\frac {a+b \arcsin (c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {b c \left (\frac {2}{\sqrt {1-c^2 x^2}}-2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{2 d^2}\)

input

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

output

-((a + b*ArcSin[c*x])/(d^2*x*(1 - c^2*x^2))) + (b*c*(2/Sqrt[1 - c^2*x^2] - 
 2*ArcTanh[Sqrt[1 - c^2*x^2]]))/(2*d^2) + (3*c^2*(-1/2*b/(c*Sqrt[1 - c^2*x 
^2]) + (x*(a + b*ArcSin[c*x]))/(2*(1 - c^2*x^2)) + ((-2*I)*(a + b*ArcSin[c 
*x])*ArcTan[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - 
I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*c)))/d^2
 

3.1.43.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 4669

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

rule 5162

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cSin[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] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 5164

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

3.1.43.4 Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.44

method	result	size
derivativedivides	\(c \left (\frac {a \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(268\)
default	\(c \left (\frac {a \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\)	\(268\)
parts	\(\frac {a \left (-\frac {1}{x}-\frac {c}{4 \left (c x -1\right )}-\frac {3 c \ln \left (c x -1\right )}{4}-\frac {c}{4 \left (c x +1\right )}+\frac {3 c \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b c \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\)	\(268\)

input

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

output

c*(a/d^2*(-1/c/x-1/4/(c*x-1)-3/4*ln(c*x-1)-1/4/(c*x+1)+3/4*ln(c*x+1))+b/d^ 
2*(-1/2*(3*c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-2*arcsin(c*x))/c/x/( 
c^2*x^2-1)+ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)-ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-3 
/2*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/2*arcsin(c*x)*ln(1-I*( 
I*c*x+(-c^2*x^2+1)^(1/2)))+3/2*I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/2 
*I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))))
 

3.1.43.5 Fricas [F]

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

input

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

output

integral((b*arcsin(c*x) + a)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x)
 

3.1.43.6 Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]

input

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

output

(Integral(a/(c**4*x**6 - 2*c**2*x**4 + x**2), x) + Integral(b*asin(c*x)/(c 
**4*x**6 - 2*c**2*x**4 + x**2), x))/d**2
 

3.1.43.7 Maxima [F]

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

input

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

output

-1/4*a*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 3 
*c*log(c*x - 1)/d^2) + 1/4*(3*(c^3*x^3 - c*x)*arctan2(c*x, sqrt(c*x + 1)*s 
qrt(-c*x + 1))*log(c*x + 1) - 3*(c^3*x^3 - c*x)*arctan2(c*x, sqrt(c*x + 1) 
*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(3*c^2*x^2 - 2)*arctan2(c*x, sqrt(c*x + 
 1)*sqrt(-c*x + 1)) + 4*(c^2*d^2*x^3 - d^2*x)*integrate(-1/4*(6*c^3*x^2 - 
3*(c^4*x^3 - c^2*x)*log(c*x + 1) + 3*(c^4*x^3 - c^2*x)*log(-c*x + 1) - 4*c 
)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x))* 
b/(c^2*d^2*x^3 - d^2*x)
 

3.1.43.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.1.43.9 Mupad [F(-1)]

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

input

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

output

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