\(\int x^3 (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx\) [186]

Optimal result

Integrand size = 20, antiderivative size = 215 \[ \int x^3 (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\frac {3 b d x^2}{32 e^2}+\frac {1}{32} b d x^4+\frac {3 b x \sqrt {1-e^2 x^2} (c+d \arccos (e x))}{32 e^3}+\frac {b x^3 \sqrt {1-e^2 x^2} (c+d \arccos (e x))}{16 e}+\frac {3 b (c+d \arccos (e x))^2}{64 d e^4}-\frac {3 d x \sqrt {1-e^2 x^2} (a+b \arcsin (e x))}{32 e^3}-\frac {d x^3 \sqrt {1-e^2 x^2} (a+b \arcsin (e x))}{16 e}+\frac {1}{4} x^4 (c+d \arccos (e x)) (a+b \arcsin (e x))+\frac {3 d (a+b \arcsin (e x))^2}{64 b e^4} \] Output:

3/32*b*d*x^2/e^2+1/32*b*d*x^4+3/32*b*x*(-e^2*x^2+1)^(1/2)*(c+d*arccos(e*x) 
)/e^3+1/16*b*x^3*(-e^2*x^2+1)^(1/2)*(c+d*arccos(e*x))/e+3/64*b*(c+d*arccos 
(e*x))^2/d/e^4-3/32*d*x*(-e^2*x^2+1)^(1/2)*(a+b*arcsin(e*x))/e^3-1/16*d*x^ 
3*(-e^2*x^2+1)^(1/2)*(a+b*arcsin(e*x))/e+1/4*x^4*(c+d*arccos(e*x))*(a+b*ar 
csin(e*x))+3/64*d*(a+b*arcsin(e*x))^2/b/e^4
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.04 \[ \int x^3 (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\frac {e x \left (8 a c e^3 x^3+b d e x \left (3+e^2 x^2\right )+b c \sqrt {1-e^2 x^2} \left (3+2 e^2 x^2\right )-a d \sqrt {1-e^2 x^2} \left (3+2 e^2 x^2\right )\right )+3 b d \arccos (e x)^2+\left (3 a d-b d e x \sqrt {1-e^2 x^2} \left (3+2 e^2 x^2\right )+b c \left (-3+8 e^4 x^4\right )\right ) \arcsin (e x)+3 b d \arcsin (e x)^2+d \arccos (e x) \left (8 a e^4 x^4+b e x \sqrt {1-e^2 x^2} \left (3+2 e^2 x^2\right )+b \left (3+8 e^4 x^4\right ) \arcsin (e x)\right )}{32 e^4} \] Input:

Integrate[x^3*(c + d*ArcCos[e*x])*(a + b*ArcSin[e*x]),x]
 

Output:

(e*x*(8*a*c*e^3*x^3 + b*d*e*x*(3 + e^2*x^2) + b*c*Sqrt[1 - e^2*x^2]*(3 + 2 
*e^2*x^2) - a*d*Sqrt[1 - e^2*x^2]*(3 + 2*e^2*x^2)) + 3*b*d*ArcCos[e*x]^2 + 
 (3*a*d - b*d*e*x*Sqrt[1 - e^2*x^2]*(3 + 2*e^2*x^2) + b*c*(-3 + 8*e^4*x^4) 
)*ArcSin[e*x] + 3*b*d*ArcSin[e*x]^2 + d*ArcCos[e*x]*(8*a*e^4*x^4 + b*e*x*S 
qrt[1 - e^2*x^2]*(3 + 2*e^2*x^2) + b*(3 + 8*e^4*x^4)*ArcSin[e*x]))/(32*e^4 
)
 

Rubi [F]

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.

\(\displaystyle \int x^3 (a+b \arcsin (e x)) (d \arccos (e x)+c) \, dx\)

\(\Big \downarrow \) 5300

\(\displaystyle \int x^3 (a+b \arcsin (e x)) (d \arccos (e x)+c)dx\)

Input:

Int[x^3*(c + d*ArcCos[e*x])*(a + b*ArcSin[e*x]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.15 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.50

method	result	size
orering	\(\frac {\left (37 e^{4} x^{4}+21 e^{2} x^{2}-60\right ) \left (c +d \arccos \left (e x \right )\right ) \left (a +b \arcsin \left (e x \right )\right )}{64 e^{4}}-\frac {\left (9 e^{4} x^{4}+11 e^{2} x^{2}-24\right ) \left (3 x^{2} \left (c +d \arccos \left (e x \right )\right ) \left (a +b \arcsin \left (e x \right )\right )-\frac {x^{3} d e \left (a +b \arcsin \left (e x \right )\right )}{\sqrt {-e^{2} x^{2}+1}}+\frac {x^{3} \left (c +d \arccos \left (e x \right )\right ) b e}{\sqrt {-e^{2} x^{2}+1}}\right )}{64 e^{4} x^{2}}+\frac {\left (e^{2} x^{2}+3\right ) \left (e x -1\right ) \left (e x +1\right ) \left (6 x \left (c +d \arccos \left (e x \right )\right ) \left (a +b \arcsin \left (e x \right )\right )-\frac {6 x^{2} d e \left (a +b \arcsin \left (e x \right )\right )}{\sqrt {-e^{2} x^{2}+1}}+\frac {6 x^{2} \left (c +d \arccos \left (e x \right )\right ) b e}{\sqrt {-e^{2} x^{2}+1}}-\frac {x^{4} d \,e^{3} \left (a +b \arcsin \left (e x \right )\right )}{\left (-e^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {2 x^{3} d \,e^{2} b}{-e^{2} x^{2}+1}+\frac {x^{4} \left (c +d \arccos \left (e x \right )\right ) b \,e^{3}}{\left (-e^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{64 x \,e^{4}}\)	\(322\)

Input:

int(x^3*(c+d*arccos(e*x))*(a+b*arcsin(e*x)),x,method=_RETURNVERBOSE)
 

Output:

1/64*(37*e^4*x^4+21*e^2*x^2-60)/e^4*(c+d*arccos(e*x))*(a+b*arcsin(e*x))-1/ 
64*(9*e^4*x^4+11*e^2*x^2-24)/e^4/x^2*(3*x^2*(c+d*arccos(e*x))*(a+b*arcsin( 
e*x))-x^3*d*e/(-e^2*x^2+1)^(1/2)*(a+b*arcsin(e*x))+x^3*(c+d*arccos(e*x))*b 
*e/(-e^2*x^2+1)^(1/2))+1/64/x*(e^2*x^2+3)/e^4*(e*x-1)*(e*x+1)*(6*x*(c+d*ar 
ccos(e*x))*(a+b*arcsin(e*x))-6*x^2*d*e/(-e^2*x^2+1)^(1/2)*(a+b*arcsin(e*x) 
)+6*x^2*(c+d*arccos(e*x))*b*e/(-e^2*x^2+1)^(1/2)-x^4*d*e^3/(-e^2*x^2+1)^(3 
/2)*(a+b*arcsin(e*x))-2*x^3*d*e^2/(-e^2*x^2+1)*b+x^4*(c+d*arccos(e*x))*b*e 
^3/(-e^2*x^2+1)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.90 \[ \int x^3 (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\frac {6 \, b d e^{2} x^{2} + 2 \, {\left (4 \, \pi b c e^{4} + {\left (8 \, a c + b d\right )} e^{4}\right )} x^{4} - 2 \, {\left (8 \, b d e^{4} x^{4} - 3 \, b d\right )} \arccos \left (e x\right )^{2} + {\left (8 \, {\left (\pi b d e^{4} - 2 \, {\left (b c - a d\right )} e^{4}\right )} x^{4} - 3 \, \pi b d + 6 \, b c - 6 \, a d\right )} \arccos \left (e x\right ) - \sqrt {-e^{2} x^{2} + 1} {\left (2 \, {\left (\pi b d e^{3} - 2 \, {\left (b c - a d\right )} e^{3}\right )} x^{3} + 3 \, {\left (\pi b d e - 2 \, {\left (b c - a d\right )} e\right )} x - 4 \, {\left (2 \, b d e^{3} x^{3} + 3 \, b d e x\right )} \arccos \left (e x\right )\right )}}{64 \, e^{4}} \] Input:

integrate(x^3*(c+d*arccos(e*x))*(a+b*arcsin(e*x)),x, algorithm="fricas")
 

Output:

1/64*(6*b*d*e^2*x^2 + 2*(4*pi*b*c*e^4 + (8*a*c + b*d)*e^4)*x^4 - 2*(8*b*d* 
e^4*x^4 - 3*b*d)*arccos(e*x)^2 + (8*(pi*b*d*e^4 - 2*(b*c - a*d)*e^4)*x^4 - 
 3*pi*b*d + 6*b*c - 6*a*d)*arccos(e*x) - sqrt(-e^2*x^2 + 1)*(2*(pi*b*d*e^3 
 - 2*(b*c - a*d)*e^3)*x^3 + 3*(pi*b*d*e - 2*(b*c - a*d)*e)*x - 4*(2*b*d*e^ 
3*x^3 + 3*b*d*e*x)*arccos(e*x)))/e^4
 

Sympy [A] (verification not implemented)

Time = 1.31 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.60 \[ \int x^3 (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\begin {cases} \frac {a c x^{4}}{4} + \frac {a d x^{4} \operatorname {acos}{\left (e x \right )}}{4} - \frac {a d x^{3} \sqrt {- e^{2} x^{2} + 1}}{16 e} - \frac {3 a d x \sqrt {- e^{2} x^{2} + 1}}{32 e^{3}} + \frac {3 a d \operatorname {asin}{\left (e x \right )}}{32 e^{4}} + \frac {b c x^{4} \operatorname {asin}{\left (e x \right )}}{4} + \frac {b c x^{3} \sqrt {- e^{2} x^{2} + 1}}{16 e} + \frac {3 b c x \sqrt {- e^{2} x^{2} + 1}}{32 e^{3}} - \frac {3 b c \operatorname {asin}{\left (e x \right )}}{32 e^{4}} + \frac {b d x^{4} \operatorname {acos}{\left (e x \right )} \operatorname {asin}{\left (e x \right )}}{4} + \frac {b d x^{4}}{32} + \frac {b d x^{3} \sqrt {- e^{2} x^{2} + 1} \operatorname {acos}{\left (e x \right )}}{16 e} - \frac {b d x^{3} \sqrt {- e^{2} x^{2} + 1} \operatorname {asin}{\left (e x \right )}}{16 e} + \frac {3 b d x^{2}}{32 e^{2}} + \frac {3 b d x \sqrt {- e^{2} x^{2} + 1} \operatorname {acos}{\left (e x \right )}}{32 e^{3}} - \frac {3 b d x \sqrt {- e^{2} x^{2} + 1} \operatorname {asin}{\left (e x \right )}}{32 e^{3}} - \frac {3 b d \operatorname {acos}{\left (e x \right )} \operatorname {asin}{\left (e x \right )}}{32 e^{4}} & \text {for}\: e \neq 0 \\\frac {a x^{4} \left (c + \frac {\pi d}{2}\right )}{4} & \text {otherwise} \end {cases} \] Input:

integrate(x**3*(c+d*acos(e*x))*(a+b*asin(e*x)),x)
 

Output:

Piecewise((a*c*x**4/4 + a*d*x**4*acos(e*x)/4 - a*d*x**3*sqrt(-e**2*x**2 + 
1)/(16*e) - 3*a*d*x*sqrt(-e**2*x**2 + 1)/(32*e**3) + 3*a*d*asin(e*x)/(32*e 
**4) + b*c*x**4*asin(e*x)/4 + b*c*x**3*sqrt(-e**2*x**2 + 1)/(16*e) + 3*b*c 
*x*sqrt(-e**2*x**2 + 1)/(32*e**3) - 3*b*c*asin(e*x)/(32*e**4) + b*d*x**4*a 
cos(e*x)*asin(e*x)/4 + b*d*x**4/32 + b*d*x**3*sqrt(-e**2*x**2 + 1)*acos(e* 
x)/(16*e) - b*d*x**3*sqrt(-e**2*x**2 + 1)*asin(e*x)/(16*e) + 3*b*d*x**2/(3 
2*e**2) + 3*b*d*x*sqrt(-e**2*x**2 + 1)*acos(e*x)/(32*e**3) - 3*b*d*x*sqrt( 
-e**2*x**2 + 1)*asin(e*x)/(32*e**3) - 3*b*d*acos(e*x)*asin(e*x)/(32*e**4), 
 Ne(e, 0)), (a*x**4*(c + pi*d/2)/4, True))
 

Maxima [F]

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

integrate(x^3*(c+d*arccos(e*x))*(a+b*arcsin(e*x)),x, algorithm="maxima")
 

Output:

1/4*a*c*x^4 + 1/32*(8*x^4*arcsin(e*x) + (2*sqrt(-e^2*x^2 + 1)*x^3/e^2 + 3* 
sqrt(-e^2*x^2 + 1)*x/e^4 - 3*arcsin(e^2*x/sqrt(e^2))/(sqrt(e^2)*e^4))*e)*b 
*c + 1/32*(8*x^4*arccos(e*x) - (2*sqrt(-e^2*x^2 + 1)*x^3/e^2 + 3*sqrt(-e^2 
*x^2 + 1)*x/e^4 - 3*arcsin(e^2*x/sqrt(e^2))/(sqrt(e^2)*e^4))*e)*a*d + b*d* 
integrate(x^3*arctan2(e*x, sqrt(e*x + 1)*sqrt(-e*x + 1))*arctan2(sqrt(e*x 
+ 1)*sqrt(-e*x + 1), e*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (189) = 378\).

Time = 0.18 (sec) , antiderivative size = 742, normalized size of antiderivative = 3.45 \[ \int x^3 (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\text {Too large to display} \] Input:

integrate(x^3*(c+d*arccos(e*x))*(a+b*arcsin(e*x)),x, algorithm="giac")
 

Output:

-1/4*pi*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*x^4*arccos(e*x)*floor(-arccos( 
e*x)/pi + 1) + 1/8*pi*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*x^4*arccos(e*x) 
- 1/4*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*x^4*arccos(e*x)^2 - 1/4*pi*(-1)^ 
floor(-arccos(e*x)/pi + 1)*b*c*x^4*floor(-arccos(e*x)/pi + 1) + 1/8*pi*(-1 
)^floor(-arccos(e*x)/pi + 1)*b*c*x^4 - 1/4*(-1)^floor(-arccos(e*x)/pi + 1) 
*b*c*x^4*arccos(e*x) + 1/16*pi*sqrt(-e^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/ 
pi + 1)*b*d*x^3*floor(-arccos(e*x)/pi + 1)/e + 1/32*(-1)^floor(-arccos(e*x 
)/pi + 1)*b*d*x^4 + 1/4*a*d*x^4*arccos(e*x) - 1/32*pi*sqrt(-e^2*x^2 + 1)*( 
-1)^floor(-arccos(e*x)/pi + 1)*b*d*x^3/e + 1/8*sqrt(-e^2*x^2 + 1)*(-1)^flo 
or(-arccos(e*x)/pi + 1)*b*d*x^3*arccos(e*x)/e + 1/4*a*c*x^4 + 1/16*sqrt(-e 
^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 1)*b*c*x^3/e - 1/16*sqrt(-e^2*x^2 
 + 1)*a*d*x^3/e + 3/32*pi*sqrt(-e^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 
1)*b*d*x*floor(-arccos(e*x)/pi + 1)/e^3 + 3/32*(-1)^floor(-arccos(e*x)/pi 
+ 1)*b*d*x^2/e^2 - 3/64*pi*sqrt(-e^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 
 1)*b*d*x/e^3 + 3/16*sqrt(-e^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 1)*b* 
d*x*arccos(e*x)/e^3 + 3/32*pi*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*arccos(e 
*x)*floor(-arccos(e*x)/pi + 1)/e^4 + 3/32*sqrt(-e^2*x^2 + 1)*(-1)^floor(-a 
rccos(e*x)/pi + 1)*b*c*x/e^3 - 3/64*pi*(-1)^floor(-arccos(e*x)/pi + 1)*b*d 
*arccos(e*x)/e^4 + 3/32*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*arccos(e*x)^2/ 
e^4 - 3/32*sqrt(-e^2*x^2 + 1)*a*d*x/e^3 + 3/32*(-1)^floor(-arccos(e*x)/...
 

Mupad [F(-1)]

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

int(x^3*(a + b*asin(e*x))*(c + d*acos(e*x)),x)
 

Output:

int(x^3*(a + b*asin(e*x))*(c + d*acos(e*x)), x)
 

Reduce [F]

\[ \int x^3 (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\frac {8 \mathit {acos} \left (e x \right ) a d \,e^{4} x^{4}+3 \mathit {asin} \left (e x \right ) a d +8 \mathit {asin} \left (e x \right ) b c \,e^{4} x^{4}-3 \mathit {asin} \left (e x \right ) b c -2 \sqrt {-e^{2} x^{2}+1}\, a d \,e^{3} x^{3}-3 \sqrt {-e^{2} x^{2}+1}\, a d e x +2 \sqrt {-e^{2} x^{2}+1}\, b c \,e^{3} x^{3}+3 \sqrt {-e^{2} x^{2}+1}\, b c e x +32 \left (\int \mathit {acos} \left (e x \right ) \mathit {asin} \left (e x \right ) x^{3}d x \right ) b d \,e^{4}+8 a c \,e^{4} x^{4}}{32 e^{4}} \] Input:

int(x^3*(c+d*acos(e*x))*(a+b*asin(e*x)),x)
 

Output:

(8*acos(e*x)*a*d*e**4*x**4 + 3*asin(e*x)*a*d + 8*asin(e*x)*b*c*e**4*x**4 - 
 3*asin(e*x)*b*c - 2*sqrt( - e**2*x**2 + 1)*a*d*e**3*x**3 - 3*sqrt( - e**2 
*x**2 + 1)*a*d*e*x + 2*sqrt( - e**2*x**2 + 1)*b*c*e**3*x**3 + 3*sqrt( - e* 
*2*x**2 + 1)*b*c*e*x + 32*int(acos(e*x)*asin(e*x)*x**3,x)*b*d*e**4 + 8*a*c 
*e**4*x**4)/(32*e**4)