Integrand size = 17, antiderivative size = 79 \[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=2 b d x+\frac {b \sqrt {1-e^2 x^2} (c+d \arccos (e x))}{e}-\frac {d \sqrt {1-e^2 x^2} (a+b \arcsin (e x))}{e}+x (c+d \arccos (e x)) (a+b \arcsin (e x)) \] Output:
2*b*d*x+b*(-e^2*x^2+1)^(1/2)*(c+d*arccos(e*x))/e-d*(-e^2*x^2+1)^(1/2)*(a+b *arcsin(e*x))/e+x*(c+d*arccos(e*x))*(a+b*arcsin(e*x))
Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.86 \[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=a c x+2 b d x+\frac {b c \sqrt {1-e^2 x^2}}{e}-\frac {a d \sqrt {1-e^2 x^2}}{e}+a d x \arccos (e x)-b d x \arccos (e x)^2+b c x \arcsin (e x)-\frac {b d \sqrt {1-e^2 x^2} (\arccos (e x)+\arcsin (e x))}{e}+\frac {b d \arccos (e x) \left (2 \sqrt {1-e^2 x^2}+e x (\arccos (e x)+\arcsin (e x))\right )}{e} \] Input:
Integrate[(c + d*ArcCos[e*x])*(a + b*ArcSin[e*x]),x]
Output:
a*c*x + 2*b*d*x + (b*c*Sqrt[1 - e^2*x^2])/e - (a*d*Sqrt[1 - e^2*x^2])/e + a*d*x*ArcCos[e*x] - b*d*x*ArcCos[e*x]^2 + b*c*x*ArcSin[e*x] - (b*d*Sqrt[1 - e^2*x^2]*(ArcCos[e*x] + ArcSin[e*x]))/e + (b*d*ArcCos[e*x]*(2*Sqrt[1 - e ^2*x^2] + e*x*(ArcCos[e*x] + ArcSin[e*x])))/e
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.
\(\displaystyle \int (a+b \arcsin (e x)) (d \arccos (e x)+c) \, dx\) |
\(\Big \downarrow \) 5300 |
\(\displaystyle \int (a+b \arcsin (e x)) (d \arccos (e x)+c)dx\) |
Input:
Int[(c + d*ArcCos[e*x])*(a + b*ArcSin[e*x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(159\) vs. \(2(75)=150\).
Time = 0.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.03
method | result | size |
orering | \(x \left (c +d \arccos \left (e x \right )\right ) \left (a +b \arcsin \left (e x \right )\right )+\frac {-\frac {d e \left (a +b \arcsin \left (e x \right )\right )}{\sqrt {-e^{2} x^{2}+1}}+\frac {\left (c +d \arccos \left (e x \right )\right ) b e}{\sqrt {-e^{2} x^{2}+1}}}{e^{2}}+\frac {x \left (e x -1\right ) \left (e x +1\right ) \left (-\frac {d \,e^{3} \left (a +b \arcsin \left (e x \right )\right ) x}{\left (-e^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {2 d \,e^{2} b}{-e^{2} x^{2}+1}+\frac {\left (c +d \arccos \left (e x \right )\right ) b \,e^{3} x}{\left (-e^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{e^{2}}\) | \(160\) |
Input:
int((c+d*arccos(e*x))*(a+b*arcsin(e*x)),x,method=_RETURNVERBOSE)
Output:
x*(c+d*arccos(e*x))*(a+b*arcsin(e*x))+1/e^2*(-d*e/(-e^2*x^2+1)^(1/2)*(a+b* arcsin(e*x))+(c+d*arccos(e*x))*b*e/(-e^2*x^2+1)^(1/2))+1/e^2*x*(e*x-1)*(e* x+1)*(-d*e^3/(-e^2*x^2+1)^(3/2)*(a+b*arcsin(e*x))*x-2*d*e^2/(-e^2*x^2+1)*b +(c+d*arccos(e*x))*b*e^3/(-e^2*x^2+1)^(3/2)*x)
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.22 \[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=-\frac {2 \, b d e x \arccos \left (e x\right )^{2} - {\left (\pi b d e - 2 \, {\left (b c - a d\right )} e\right )} x \arccos \left (e x\right ) - {\left (\pi b c e + 2 \, {\left (a c + 2 \, b d\right )} e\right )} x + \sqrt {-e^{2} x^{2} + 1} {\left (\pi b d - 4 \, b d \arccos \left (e x\right ) - 2 \, b c + 2 \, a d\right )}}{2 \, e} \] Input:
integrate((c+d*arccos(e*x))*(a+b*arcsin(e*x)),x, algorithm="fricas")
Output:
-1/2*(2*b*d*e*x*arccos(e*x)^2 - (pi*b*d*e - 2*(b*c - a*d)*e)*x*arccos(e*x) - (pi*b*c*e + 2*(a*c + 2*b*d)*e)*x + sqrt(-e^2*x^2 + 1)*(pi*b*d - 4*b*d*a rccos(e*x) - 2*b*c + 2*a*d))/e
Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.70 \[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\begin {cases} a c x + a d x \operatorname {acos}{\left (e x \right )} - \frac {a d \sqrt {- e^{2} x^{2} + 1}}{e} + b c x \operatorname {asin}{\left (e x \right )} + \frac {b c \sqrt {- e^{2} x^{2} + 1}}{e} + b d x \operatorname {acos}{\left (e x \right )} \operatorname {asin}{\left (e x \right )} + 2 b d x + \frac {b d \sqrt {- e^{2} x^{2} + 1} \operatorname {acos}{\left (e x \right )}}{e} - \frac {b d \sqrt {- e^{2} x^{2} + 1} \operatorname {asin}{\left (e x \right )}}{e} & \text {for}\: e \neq 0 \\a x \left (c + \frac {\pi d}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((c+d*acos(e*x))*(a+b*asin(e*x)),x)
Output:
Piecewise((a*c*x + a*d*x*acos(e*x) - a*d*sqrt(-e**2*x**2 + 1)/e + b*c*x*as in(e*x) + b*c*sqrt(-e**2*x**2 + 1)/e + b*d*x*acos(e*x)*asin(e*x) + 2*b*d*x + b*d*sqrt(-e**2*x**2 + 1)*acos(e*x)/e - b*d*sqrt(-e**2*x**2 + 1)*asin(e* x)/e, Ne(e, 0)), (a*x*(c + pi*d/2), True))
\[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\int { {\left (d \arccos \left (e x\right ) + c\right )} {\left (b \arcsin \left (e x\right ) + a\right )} \,d x } \] Input:
integrate((c+d*arccos(e*x))*(a+b*arcsin(e*x)),x, algorithm="maxima")
Output:
a*c*x + b*d*integrate(arctan2(e*x, sqrt(e*x + 1)*sqrt(-e*x + 1))*arctan2(s qrt(e*x + 1)*sqrt(-e*x + 1), e*x), x) + (e*x*arcsin(e*x) + sqrt(-e^2*x^2 + 1))*b*c/e + (e*x*arccos(e*x) - sqrt(-e^2*x^2 + 1))*a*d/e
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (75) = 150\).
Time = 0.16 (sec) , antiderivative size = 359, normalized size of antiderivative = 4.54 \[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=-\pi \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b d x \arccos \left (e x\right ) \left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor + \frac {1}{2} \, \pi \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b d x \arccos \left (e x\right ) - \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b d x \arccos \left (e x\right )^{2} - \pi \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b c x \left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor + \frac {1}{2} \, \pi \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b c x - \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b c x \arccos \left (e x\right ) + \frac {\pi \sqrt {-e^{2} x^{2} + 1} \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b d \left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor }{e} + 2 \, \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b d x + a d x \arccos \left (e x\right ) - \frac {\pi \sqrt {-e^{2} x^{2} + 1} \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b d}{2 \, e} + \frac {2 \, \sqrt {-e^{2} x^{2} + 1} \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b d \arccos \left (e x\right )}{e} + a c x + \frac {\sqrt {-e^{2} x^{2} + 1} \left (-1\right )^{\left \lfloor -\frac {\arccos \left (e x\right )}{\pi } + 1 \right \rfloor } b c}{e} - \frac {\sqrt {-e^{2} x^{2} + 1} a d}{e} \] Input:
integrate((c+d*arccos(e*x))*(a+b*arcsin(e*x)),x, algorithm="giac")
Output:
-pi*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*x*arccos(e*x)*floor(-arccos(e*x)/p i + 1) + 1/2*pi*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*x*arccos(e*x) - (-1)^f loor(-arccos(e*x)/pi + 1)*b*d*x*arccos(e*x)^2 - pi*(-1)^floor(-arccos(e*x) /pi + 1)*b*c*x*floor(-arccos(e*x)/pi + 1) + 1/2*pi*(-1)^floor(-arccos(e*x) /pi + 1)*b*c*x - (-1)^floor(-arccos(e*x)/pi + 1)*b*c*x*arccos(e*x) + pi*sq rt(-e^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*floor(-arccos(e*x)/pi + 1)/e + 2*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*x + a*d*x*arccos(e*x) - 1/ 2*pi*sqrt(-e^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 1)*b*d/e + 2*sqrt(-e^ 2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 1)*b*d*arccos(e*x)/e + a*c*x + sqr t(-e^2*x^2 + 1)*(-1)^floor(-arccos(e*x)/pi + 1)*b*c/e - sqrt(-e^2*x^2 + 1) *a*d/e
Timed out. \[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\int \left (a+b\,\mathrm {asin}\left (e\,x\right )\right )\,\left (c+d\,\mathrm {acos}\left (e\,x\right )\right ) \,d x \] Input:
int((a + b*asin(e*x))*(c + d*acos(e*x)),x)
Output:
int((a + b*asin(e*x))*(c + d*acos(e*x)), x)
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.43 \[ \int (c+d \arccos (e x)) (a+b \arcsin (e x)) \, dx=\frac {\mathit {acos} \left (e x \right ) \mathit {asin} \left (e x \right ) b d e x +\sqrt {-e^{2} x^{2}+1}\, \mathit {acos} \left (e x \right ) b d +\mathit {acos} \left (e x \right ) a d e x -\sqrt {-e^{2} x^{2}+1}\, \mathit {asin} \left (e x \right ) b d +\mathit {asin} \left (e x \right ) b c e x -\sqrt {-e^{2} x^{2}+1}\, a d +\sqrt {-e^{2} x^{2}+1}\, b c +a c e x +2 b d e x}{e} \] Input:
int((c+d*acos(e*x))*(a+b*asin(e*x)),x)
Output:
(acos(e*x)*asin(e*x)*b*d*e*x + sqrt( - e**2*x**2 + 1)*acos(e*x)*b*d + acos (e*x)*a*d*e*x - sqrt( - e**2*x**2 + 1)*asin(e*x)*b*d + asin(e*x)*b*c*e*x - sqrt( - e**2*x**2 + 1)*a*d + sqrt( - e**2*x**2 + 1)*b*c + a*c*e*x + 2*b*d *e*x)/e