\(\int (d+e x^2) \arcsin (a x) \log (c x^n) \, dx\) [192]

Optimal result

Integrand size = 18, antiderivative size = 246 \[ \int \left (d+e x^2\right ) \arcsin (a x) \log \left (c x^n\right ) \, dx=-\frac {d n \sqrt {1-a^2 x^2}}{a}-\frac {\left (3 a^2 d+e\right ) n \sqrt {1-a^2 x^2}}{3 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-d n x \arcsin (a x)-\frac {1}{9} e n x^3 \arcsin (a x)-\frac {e n \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{9 a^3}+\frac {\left (3 a^2 d+e\right ) n \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{3 a^3}+\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \arcsin (a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \arcsin (a x) \log \left (c x^n\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^2\right ) \arcsin (a x) \log \left (c x^n\right ) \, dx=\frac {-54 a^2 d n \sqrt {1-a^2 x^2}-7 e n \sqrt {1-a^2 x^2}-2 a^2 e n x^2 \sqrt {1-a^2 x^2}-3 \left (9 a^2 d+2 e\right ) n \log (x)+27 a^2 d \sqrt {1-a^2 x^2} \log \left (c x^n\right )+6 e \sqrt {1-a^2 x^2} \log \left (c x^n\right )+3 a^2 e x^2 \sqrt {1-a^2 x^2} \log \left (c x^n\right )-3 a^3 x \arcsin (a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+27 a^2 d n \log \left (1+\sqrt {1-a^2 x^2}\right )+6 e n \log \left (1+\sqrt {1-a^2 x^2}\right )}{27 a^3} \] Input:

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

Output:

(-54*a^2*d*n*Sqrt[1 - a^2*x^2] - 7*e*n*Sqrt[1 - a^2*x^2] - 2*a^2*e*n*x^2*S 
qrt[1 - a^2*x^2] - 3*(9*a^2*d + 2*e)*n*Log[x] + 27*a^2*d*Sqrt[1 - a^2*x^2] 
*Log[c*x^n] + 6*e*Sqrt[1 - a^2*x^2]*Log[c*x^n] + 3*a^2*e*x^2*Sqrt[1 - a^2* 
x^2]*Log[c*x^n] - 3*a^3*x*ArcSin[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*L 
og[c*x^n]) + 27*a^2*d*n*Log[1 + Sqrt[1 - a^2*x^2]] + 6*e*n*Log[1 + Sqrt[1 
- a^2*x^2]])/(27*a^3)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2834, 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.

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

\(\Big \downarrow \) 2834

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(n*((d*Sqrt[1 - a^2*x^2])/a + ((3*a^2*d + e)*Sqrt[1 - a^2*x^2])/(3*a^3) - 
 (2*e*(1 - a^2*x^2)^(3/2))/(27*a^3) + d*x*ArcSin[a*x] + (e*x^3*ArcSin[a*x] 
)/9 + (e*ArcTanh[Sqrt[1 - a^2*x^2]])/(9*a^3) - ((3*a^2*d + e)*ArcTanh[Sqrt 
[1 - a^2*x^2]])/(3*a^3))) + ((3*a^2*d + e)*Sqrt[1 - a^2*x^2]*Log[c*x^n])/( 
3*a^3) - (e*(1 - a^2*x^2)^(3/2)*Log[c*x^n])/(9*a^3) + d*x*ArcSin[a*x]*Log[ 
c*x^n] + (e*x^3*ArcSin[a*x]*Log[c*x^n])/3
 

Defintions of rubi rules used

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 7.07 (sec) , antiderivative size = 6894, normalized size of antiderivative = 28.02

method	result	size
default	\(\text {Expression too large to display}\)	\(6894\)

Input:

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

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right ) \arcsin (a x) \log \left (c x^n\right ) \, dx=\frac {18 \, {\left (a^{3} e x^{3} + 3 \, a^{3} d x\right )} \arcsin \left (a x\right ) \log \left (c\right ) + 18 \, {\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \arcsin \left (a x\right ) \log \left (x\right ) + 3 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 3 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) - 6 \, {\left (a^{3} e n x^{3} + 9 \, a^{3} d n x\right )} \arcsin \left (a x\right ) - 2 \, {\left (2 \, a^{2} e n x^{2} + {\left (54 \, a^{2} d + 7 \, e\right )} n - 3 \, {\left (a^{2} e x^{2} + 9 \, a^{2} d + 2 \, e\right )} \log \left (c\right ) - 3 \, {\left (a^{2} e n x^{2} + {\left (9 \, a^{2} d + 2 \, e\right )} n\right )} \log \left (x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{54 \, a^{3}} \] Input:

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

Output:

1/54*(18*(a^3*e*x^3 + 3*a^3*d*x)*arcsin(a*x)*log(c) + 18*(a^3*e*n*x^3 + 3* 
a^3*d*n*x)*arcsin(a*x)*log(x) + 3*(9*a^2*d + 2*e)*n*log(sqrt(-a^2*x^2 + 1) 
 + 1) - 3*(9*a^2*d + 2*e)*n*log(sqrt(-a^2*x^2 + 1) - 1) - 6*(a^3*e*n*x^3 + 
 9*a^3*d*n*x)*arcsin(a*x) - 2*(2*a^2*e*n*x^2 + (54*a^2*d + 7*e)*n - 3*(a^2 
*e*x^2 + 9*a^2*d + 2*e)*log(c) - 3*(a^2*e*n*x^2 + (9*a^2*d + 2*e)*n)*log(x 
))*sqrt(-a^2*x^2 + 1))/a^3
 

Sympy [A] (verification not implemented)

Time = 52.72 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.77 \[ \int \left (d+e x^2\right ) \arcsin (a x) \log \left (c x^n\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

a*e*n*Piecewise((-Piecewise((x**2*sqrt(-a**2*x**2 + 1)/3 - sqrt(-a**2*x**2 
 + 1)/(3*a**2), Ne(a, 0)), (x**2/2, True))/(3*a**2) - 2*Piecewise((I*sqrt( 
a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x 
**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 
+ 1) + 1), True))/(3*a**4), (a > -oo) & (a < oo) & Ne(a, 0)), (x**4/16, Tr 
ue))/3 + a*e*n*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a* 
*2*x**2 + 1)/(3*a**4), Ne(a**2, 0)), (x**4/4, True))/9 - a*e*Piecewise((-x 
**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a* 
*2, 0)), (x**4/4, True))*log(c*x**n)/3 - d*n*Piecewise((0, Eq(a, 0)), (Pie 
cewise((x*asin(a*x) + sqrt(-a**2*x**2 + 1)/a, Ne(a, 0)), (0, True)) + Piec 
ewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x 
)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sq 
rt(-a**2*x**2 + 1) + 1), True))/a, True)) + d*Piecewise((0, Eq(a, 0)), (x* 
asin(a*x) + sqrt(-a**2*x**2 + 1)/a, True))*log(c*x**n) - e*n*x**3*asin(a*x 
)/9 + e*x**3*log(c*x**n)*asin(a*x)/3
 

Maxima [F]

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

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

Output:

-1/54*(-I*(27*a^2*d*n*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/a^3) + a^ 
2*e*n*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5) - 
162*a^2*e*n*integrate(1/9*x^4*log(x)/(a^2*x^2 - 1), x) - 486*a^2*d*n*integ 
rate(1/9*x^2*log(x)/(a^2*x^2 - 1), x) - 27*a^2*d*(2*x/a^2 - log(a*x + 1)/a 
^3 + log(a*x - 1)/a^3)*log(c) - 3*a^2*e*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x 
 + 1)/a^5 + 3*log(a*x - 1)/a^5)*log(c))*a^3 - 2*(-2*I*a^3*e*n + 3*I*a^3*e* 
log(c))*x^3 - 54*a^3*integrate(-1/9*((a*e*n - 3*a*e*log(c))*x^3 + 9*(a*d*n 
 - a*d*log(c))*x - 3*(a*e*x^3 + 3*a*d*x)*log(x^n))*sqrt(a*x + 1)*sqrt(-a*x 
 + 1)/(a^2*x^2 - 1), x) - 9*(3*I*a^2*d + I*e)*n*dilog(a*x) - 9*(-3*I*a^2*d 
 - I*e)*n*dilog(-a*x) - 6*(9*I*a^3*d*log(c) + 3*I*a*e*log(c) + 2*(-9*I*a^3 
*d - 2*I*a*e)*n)*x + 6*((a^3*e*n - 3*a^3*e*log(c))*x^3 + 9*(a^3*d*n - a^3* 
d*log(c))*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)) - 3*(-9*I*a^2*d*lo 
g(c) + (9*I*a^2*d + I*e)*n - 3*I*e*log(c))*log(a*x + 1) - 3*(9*I*a^2*d*log 
(c) + (-9*I*a^2*d - I*e)*n + 3*I*e*log(c))*log(a*x - 1) - 3*(2*I*a^3*e*x^3 
 + 6*(3*I*a^3*d + I*a*e)*x + 6*(a^3*e*x^3 + 3*a^3*d*x)*arctan2(a*x, sqrt(a 
*x + 1)*sqrt(-a*x + 1)) + 3*(-3*I*a^2*d - I*e)*log(a*x + 1) + 3*(3*I*a^2*d 
 + I*e)*log(-a*x + 1))*log(x^n))/a^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5306 vs. \(2 (216) = 432\).

Time = 0.26 (sec) , antiderivative size = 5306, normalized size of antiderivative = 21.57 \[ \int \left (d+e x^2\right ) \arcsin (a x) \log \left (c x^n\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/54*(54*a^3*d*n*x*arcsin(a*x)*log(a*x) - 54*a^3*d*n*x*arcsin(a*x)*log(a) 
+ 54*a^3*d*x*arcsin(a*x)*log(c) - 108*a^3*d*n*x*arcsin(a*x)/(sqrt(-a^2*x^2 
 + 1)*a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1 
)^2 + sqrt(-a^2*x^2 + 1) + 1) - 54*a^4*d*n*x^2*log(abs(a)*abs(x))/((a^2*x^ 
2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1)*(sqrt(-a^2*x^2 + 1) + 1)^2) + 18*(a^2*x^ 
2 - 1)*a*e*x*arcsin(a*x)*log(c) + 54*a^4*d*n*x^2*log(sqrt(-a^2*x^2 + 1) + 
1)/((a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1)*(sqrt(-a^2*x^2 + 1) + 1)^2) + 
 54*sqrt(-a^2*x^2 + 1)*a^2*d*n*log(a*x) - 54*sqrt(-a^2*x^2 + 1)*a^2*d*n*lo 
g(a) + 108*a^4*d*n*x^2/((a^2*x^2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1)*(sqrt(-a^ 
2*x^2 + 1) + 1)^2) + 18*a*e*x*arcsin(a*x)*log(c) + 54*sqrt(-a^2*x^2 + 1)*a 
^2*d*log(c) - 54*a^2*d*n*log(abs(a)*abs(x))/(a^2*x^2/(sqrt(-a^2*x^2 + 1) + 
 1)^2 + 1) + 54*a^2*d*n*log(sqrt(-a^2*x^2 + 1) + 1)/(a^2*x^2/(sqrt(-a^2*x^ 
2 + 1) + 1)^2 + 1) - 6*(-a^2*x^2 + 1)^(3/2)*e*log(c) - 108*a^2*d*n/(a^2*x^ 
2/(sqrt(-a^2*x^2 + 1) + 1)^2 + 1) + (18*(a^2*x^2 - 1)*a*x*arcsin(a*x)*log( 
a*x) - 18*(a^2*x^2 - 1)*a*x*arcsin(a*x)*log(a) + 18*a*x*arcsin(a*x)*log(a* 
x) - 18*a*x*arcsin(a*x)*log(a) - 6*(-a^2*x^2 + 1)^(3/2)*log(a*x) + 6*(-a^2 
*x^2 + 1)^(3/2)*log(a) + 18*sqrt(-a^2*x^2 + 1)*log(a*x) - 18*sqrt(-a^2*x^2 
 + 1)*log(a) - (192*(a^2*x^2 - 1)^2*a^8*x^8*log(abs(a)*abs(x))/((4*(-a^2*x 
^2 + 1)^(3/2) - 3*sqrt(-a^2*x^2 + 1) + 1)^2*(sqrt(-a^2*x^2 + 1) + 1)^6) - 
192*(a^2*x^2 - 1)^2*a^8*x^8*log(sqrt(-a^2*x^2 + 1) + 1)/((4*(-a^2*x^2 +...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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