3.2.94 \(\int (d+e x^2) \arcsin (a x)^2 \log (c x^n) \, dx\) [194]

3.2.94.1 Optimal result

Integrand size = 20, antiderivative size = 482 \[ \int \left (d+e x^2\right ) \arcsin (a x)^2 \log \left (c x^n\right ) \, dx=2 d n x+\frac {2 e n x}{27 a^2}+\frac {4}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{27} e n x^3-\frac {2 d n \sqrt {1-a^2 x^2} \arcsin (a x)}{a}-\frac {4 e n \sqrt {1-a^2 x^2} \arcsin (a x)}{27 a^3}-\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^3}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{27 a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)}{27 a^3}-d n x \arcsin (a x)^2-\frac {1}{9} e n x^3 \arcsin (a x)^2+\frac {4 \left (9 a^2 d+2 e\right ) n \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )+\frac {2 d \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a}+d x \arcsin (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arcsin (a x)^2 \log \left (c x^n\right )-\frac {2 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )}{9 a^3} \]

output

2*d*n*x+2/27*e*n*x/a^2+4/9*(9*d+2*e/a^2)*n*x+2/27*e*n*x^3+2/27*e*n*(-a^2*x 
^2+1)^(3/2)*arcsin(a*x)/a^3-d*n*x*arcsin(a*x)^2-1/9*e*n*x^3*arcsin(a*x)^2+ 
4/9*(9*a^2*d+2*e)*n*arcsin(a*x)*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))/a^3-2*d* 
x*ln(c*x^n)-4/9*e*x*ln(c*x^n)/a^2-2/27*e*x^3*ln(c*x^n)+d*x*arcsin(a*x)^2*l 
n(c*x^n)+1/3*e*x^3*arcsin(a*x)^2*ln(c*x^n)-2/9*I*(9*a^2*d+2*e)*n*polylog(2 
,-I*a*x-(-a^2*x^2+1)^(1/2))/a^3+2/9*I*(9*a^2*d+2*e)*n*polylog(2,I*a*x+(-a^ 
2*x^2+1)^(1/2))/a^3-2*d*n*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a-4/27*e*n*arcsin 
(a*x)*(-a^2*x^2+1)^(1/2)/a^3-2/9*(9*a^2*d+2*e)*n*arcsin(a*x)*(-a^2*x^2+1)^ 
(1/2)/a^3-2/27*e*n*x^2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a+2*d*arcsin(a*x)*ln 
(c*x^n)*(-a^2*x^2+1)^(1/2)/a+4/9*e*arcsin(a*x)*ln(c*x^n)*(-a^2*x^2+1)^(1/2 
)/a^3+2/9*e*x^2*arcsin(a*x)*ln(c*x^n)*(-a^2*x^2+1)^(1/2)/a
 

3.2.94.2 Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 456, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right ) \arcsin (a x)^2 \log \left (c x^n\right ) \, dx=\frac {162 a^3 d n x+26 a e n x+2 a^3 e n x^3-108 a^2 d n \sqrt {1-a^2 x^2} \arcsin (a x)-14 e n \sqrt {1-a^2 x^2} \arcsin (a x)-4 a^2 e n x^2 \sqrt {1-a^2 x^2} \arcsin (a x)-27 a^3 d n x \arcsin (a x)^2-3 a^3 e n x^3 \arcsin (a x)^2-54 a^2 d n \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )-12 e n \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )+54 a^2 d n \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )+12 e n \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )-54 a^3 d x \log \left (c x^n\right )-12 a e x \log \left (c x^n\right )-2 a^3 e x^3 \log \left (c x^n\right )+54 a^2 d \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )+12 e \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )+6 a^2 e x^2 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )+27 a^3 d x \arcsin (a x)^2 \log \left (c x^n\right )+9 a^3 e x^3 \arcsin (a x)^2 \log \left (c x^n\right )-6 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+6 i \left (9 a^2 d+2 e\right ) n \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )}{27 a^3} \]

input

Integrate[(d + e*x^2)*ArcSin[a*x]^2*Log[c*x^n],x]
 

output

(162*a^3*d*n*x + 26*a*e*n*x + 2*a^3*e*n*x^3 - 108*a^2*d*n*Sqrt[1 - a^2*x^2 
]*ArcSin[a*x] - 14*e*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - 4*a^2*e*n*x^2*Sqrt[ 
1 - a^2*x^2]*ArcSin[a*x] - 27*a^3*d*n*x*ArcSin[a*x]^2 - 3*a^3*e*n*x^3*ArcS 
in[a*x]^2 - 54*a^2*d*n*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] - 12*e*n*Arc 
Sin[a*x]*Log[1 - E^(I*ArcSin[a*x])] + 54*a^2*d*n*ArcSin[a*x]*Log[1 + E^(I* 
ArcSin[a*x])] + 12*e*n*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] - 54*a^3*d*x 
*Log[c*x^n] - 12*a*e*x*Log[c*x^n] - 2*a^3*e*x^3*Log[c*x^n] + 54*a^2*d*Sqrt 
[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n] + 12*e*Sqrt[1 - a^2*x^2]*ArcSin[a*x]* 
Log[c*x^n] + 6*a^2*e*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n] + 27*a^3 
*d*x*ArcSin[a*x]^2*Log[c*x^n] + 9*a^3*e*x^3*ArcSin[a*x]^2*Log[c*x^n] - (6* 
I)*(9*a^2*d + 2*e)*n*PolyLog[2, -E^(I*ArcSin[a*x])] + (6*I)*(9*a^2*d + 2*e 
)*n*PolyLog[2, E^(I*ArcSin[a*x])])/(27*a^3)
 

3.2.94.3 Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2834, 6, 2009}

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.

\(\displaystyle \int \arcsin (a x)^2 \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\)

\(\Big \downarrow \) 2834

\(\displaystyle -n \int \left (\frac {1}{3} e \arcsin (a x)^2 x^2-\frac {2 e x^2}{27}+\frac {2 e \sqrt {1-a^2 x^2} \arcsin (a x) x}{9 a}+d \arcsin (a x)^2-\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right )+\frac {2 d \sqrt {1-a^2 x^2} \arcsin (a x)}{a x}+\frac {4 e \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^3 x}\right )dx+\frac {2 d \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{a}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {4 e \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a^3}+d x \arcsin (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arcsin (a x)^2 \log \left (c x^n\right )-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )\)

\(\Big \downarrow \) 6

\(\displaystyle -n \int \left (\frac {1}{3} e \arcsin (a x)^2 x^2-\frac {2 e x^2}{27}+\frac {2 e \sqrt {1-a^2 x^2} \arcsin (a x) x}{9 a}+d \arcsin (a x)^2-\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right )+\frac {\left (\frac {2 d}{a}+\frac {4 e}{9 a^3}\right ) \sqrt {1-a^2 x^2} \arcsin (a x)}{x}\right )dx+\frac {2 d \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{a}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {4 e \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a^3}+d x \arcsin (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arcsin (a x)^2 \log \left (c x^n\right )-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{a}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-n \left (\frac {2 d \sqrt {1-a^2 x^2} \arcsin (a x)}{a}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{27 a}-\frac {4}{9} x \left (\frac {2 e}{a^2}+9 d\right )-\frac {2 e x}{27 a^2}-\frac {4 \arcsin (a x) \left (9 a^2 d+2 e\right ) \text {arctanh}\left (e^{i \arcsin (a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )}{9 a^3}-\frac {2 i \left (9 a^2 d+2 e\right ) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )}{9 a^3}+\frac {2 \sqrt {1-a^2 x^2} \arcsin (a x) \left (9 a^2 d+2 e\right )}{9 a^3}-\frac {2 e \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)}{27 a^3}+\frac {4 e \sqrt {1-a^2 x^2} \arcsin (a x)}{27 a^3}+d x \arcsin (a x)^2+\frac {1}{9} e x^3 \arcsin (a x)^2-2 d x-\frac {2 e x^3}{27}\right )+\frac {4 e \sqrt {1-a^2 x^2} \arcsin (a x) \log \left (c x^n\right )}{9 a^3}+d x \arcsin (a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \arcsin (a x)^2 \log \left (c x^n\right )-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )\)

input

Int[(d + e*x^2)*ArcSin[a*x]^2*Log[c*x^n],x]
 

output

-2*d*x*Log[c*x^n] - (4*e*x*Log[c*x^n])/(9*a^2) - (2*e*x^3*Log[c*x^n])/27 + 
 (2*d*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n])/a + (4*e*Sqrt[1 - a^2*x^2] 
*ArcSin[a*x]*Log[c*x^n])/(9*a^3) + (2*e*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]* 
Log[c*x^n])/(9*a) + d*x*ArcSin[a*x]^2*Log[c*x^n] + (e*x^3*ArcSin[a*x]^2*Lo 
g[c*x^n])/3 - n*(-2*d*x - (2*e*x)/(27*a^2) - (4*(9*d + (2*e)/a^2)*x)/9 - ( 
2*e*x^3)/27 + (2*d*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a + (4*e*Sqrt[1 - a^2*x^ 
2]*ArcSin[a*x])/(27*a^3) + (2*(9*a^2*d + 2*e)*Sqrt[1 - a^2*x^2]*ArcSin[a*x 
])/(9*a^3) + (2*e*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a) - (2*e*(1 - a^ 
2*x^2)^(3/2)*ArcSin[a*x])/(27*a^3) + d*x*ArcSin[a*x]^2 + (e*x^3*ArcSin[a*x 
]^2)/9 - (4*(9*a^2*d + 2*e)*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])])/(9*a^3 
) + (((2*I)/9)*(9*a^2*d + 2*e)*PolyLog[2, -E^(I*ArcSin[a*x])])/a^3 - (((2* 
I)/9)*(9*a^2*d + 2*e)*PolyLog[2, E^(I*ArcSin[a*x])])/a^3)
 

3.2.94.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2834

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)* 
(x_))]^(m_.), x_Symbol] :> With[{u = IntHide[Px*F[d*(e + f*x)]^m, x]}, Simp 
[(a + b*Log[c*x^n])   u, x] - Simp[b*n   Int[1/x   u, x], x]] /; FreeQ[{a, 
b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSi 
n, ArcCos, ArcSinh, ArcCosh}, F]
 

3.2.94.4 Maple [F]

\[\int \left (e \,x^{2}+d \right ) \arcsin \left (a x \right )^{2} \ln \left (c \,x^{n}\right )d x\]

input

int((e*x^2+d)*arcsin(a*x)^2*ln(c*x^n),x)
 

output

int((e*x^2+d)*arcsin(a*x)^2*ln(c*x^n),x)
 

3.2.94.5 Fricas [F]

\[ \int \left (d+e x^2\right ) \arcsin (a x)^2 \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \arcsin \left (a x\right )^{2} \log \left (c x^{n}\right ) \,d x } \]

input

integrate((e*x^2+d)*arcsin(a*x)^2*log(c*x^n),x, algorithm="fricas")
 

output

integral((e*x^2 + d)*arcsin(a*x)^2*log(c*x^n), x)
 

3.2.94.6 Sympy [F]

\[ \int \left (d+e x^2\right ) \arcsin (a x)^2 \log \left (c x^n\right ) \, dx=\int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {asin}^{2}{\left (a x \right )}\, dx \]

input

integrate((e*x**2+d)*asin(a*x)**2*ln(c*x**n),x)
 

output

Integral((d + e*x**2)*log(c*x**n)*asin(a*x)**2, x)
 

3.2.94.7 Maxima [F]

\[ \int \left (d+e x^2\right ) \arcsin (a x)^2 \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \arcsin \left (a x\right )^{2} \log \left (c x^{n}\right ) \,d x } \]

input

integrate((e*x^2+d)*arcsin(a*x)^2*log(c*x^n),x, algorithm="maxima")
 

output

1/3*(e*x^3 + 3*d*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2*log(x^n) 
- 1/9*((e*n - 3*e*log(c))*x^3 + 9*(d*n - d*log(c))*x)*arctan2(a*x, sqrt(a* 
x + 1)*sqrt(-a*x + 1))^2 + integrate(2/9*(3*(a*e*x^3 + 3*a*d*x)*arctan2(a* 
x, sqrt(a*x + 1)*sqrt(-a*x + 1))*log(x^n) - ((a*e*n - 3*a*e*log(c))*x^3 + 
9*(a*d*n - a*d*log(c))*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)))*sqrt 
(a*x + 1)*sqrt(-a*x + 1)/(a^2*x^2 - 1), x)
 

3.2.94.8 Giac [F]

\[ \int \left (d+e x^2\right ) \arcsin (a x)^2 \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \arcsin \left (a x\right )^{2} \log \left (c x^{n}\right ) \,d x } \]

input

integrate((e*x^2+d)*arcsin(a*x)^2*log(c*x^n),x, algorithm="giac")
 

output

integrate((e*x^2 + d)*arcsin(a*x)^2*log(c*x^n), x)
 

3.2.94.9 Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right ) \arcsin (a x)^2 \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,{\mathrm {asin}\left (a\,x\right )}^2\,\left (e\,x^2+d\right ) \,d x \]

input

int(log(c*x^n)*asin(a*x)^2*(d + e*x^2),x)
 

output

int(log(c*x^n)*asin(a*x)^2*(d + e*x^2), x)