3.2.86 \(\int \frac {x (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx\) [186]

3.2.86.1 Optimal result

Integrand size = 25, antiderivative size = 117 \[ \int \frac {x (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\frac {i (a+b \arcsin (c x))^3}{3 b c^2 d}-\frac {(a+b \arcsin (c x))^2 \log \left (1+e^{2 i \arcsin (c x)}\right )}{c^2 d}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{2 c^2 d} \]

output

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

3.2.86.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(117)=234\).

Time = 0.66 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.92 \[ \int \frac {x (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\frac {-12 i a b \pi \arcsin (c x)+6 i a b \arcsin (c x)^2+2 i b^2 \arcsin (c x)^3-24 a b \pi \log \left (1+e^{-i \arcsin (c x)}\right )-6 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-12 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+6 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )-12 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-6 b^2 \arcsin (c x)^2 \log \left (1+e^{2 i \arcsin (c x)}\right )-3 a^2 \log \left (1-c^2 x^2\right )+24 a b \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-6 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+6 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+12 i a b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+12 i a b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+6 i b^2 \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-3 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )}{6 c^2 d} \]

input

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

output

((-12*I)*a*b*Pi*ArcSin[c*x] + (6*I)*a*b*ArcSin[c*x]^2 + (2*I)*b^2*ArcSin[c 
*x]^3 - 24*a*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 6*a*b*Pi*Log[1 - I*E^(I* 
ArcSin[c*x])] - 12*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 6*a*b*Pi 
*Log[1 + I*E^(I*ArcSin[c*x])] - 12*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c 
*x])] - 6*b^2*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] - 3*a^2*Log[1 - 
 c^2*x^2] + 24*a*b*Pi*Log[Cos[ArcSin[c*x]/2]] - 6*a*b*Pi*Log[-Cos[(Pi + 2* 
ArcSin[c*x])/4]] + 6*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (12*I)*a*b* 
PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (12*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c 
*x])] + (6*I)*b^2*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - 3*b^2*P 
olyLog[3, -E^((2*I)*ArcSin[c*x])])/(6*c^2*d)
 

3.2.86.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5180, 3042, 4202, 2620, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^2 d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {i (a+b \arcsin (c x))^3}{3 b}-2 i \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 i \arcsin (c x)}\right )\right )-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{c^2 d}\)

input

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

output

(((I/3)*(a + b*ArcSin[c*x])^3)/b - (2*I)*((-1/2*I)*(a + b*ArcSin[c*x])^2*L 
og[1 + E^((2*I)*ArcSin[c*x])] + I*b*((I/2)*(a + b*ArcSin[c*x])*PolyLog[2, 
-E^((2*I)*ArcSin[c*x])] - (b*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/4)))/(c^2 
*d)
 

3.2.86.3.1 Defintions of rubi rules used

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4202

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

rule 5180

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.2.86.4 Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.76

method	result	size
parts	\(-\frac {a^{2} \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{2}}-\frac {b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d \,c^{2}}-\frac {2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d \,c^{2}}\)	\(206\)
derivativedivides	\(\frac {-\frac {a^{2} \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}-\frac {2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(208\)
default	\(\frac {-\frac {a^{2} \left (\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b^{2} \left (-\frac {i \arcsin \left (c x \right )^{3}}{3}+\arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}-\frac {2 a b \left (-\frac {i \arcsin \left (c x \right )^{2}}{2}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(208\)

input

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

output

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

3.2.86.5 Fricas [F]

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

input

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

output

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

3.2.86.6 Sympy [F]

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

input

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

output

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

3.2.86.7 Maxima [F]

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

input

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

output

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

3.2.86.8 Giac [F]

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

input

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

output

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

3.2.86.9 Mupad [F(-1)]

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

input

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

output

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