\(\int (a+b \arccos (-1+d x^2))^4 \, dx\) [57]

Optimal result

Integrand size = 14, antiderivative size = 127 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=384 b^4 x+\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (-1+d x^2\right )\right )}{d x}-48 b^2 x \left (a+b \arccos \left (-1+d x^2\right )\right )^2-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (-1+d x^2\right )\right )^3}{d x}+x \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \] Output:

384*b^4*x+192*b^3*(-d^2*x^4+2*d*x^2)^(1/2)*(a+b*arccos(d*x^2-1))/d/x-48*b^ 
2*x*(a+b*arccos(d*x^2-1))^2-8*b*(-d^2*x^4+2*d*x^2)^(1/2)*(a+b*arccos(d*x^2 
-1))^3/d/x+x*(a+b*arccos(d*x^2-1))^4
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.96 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\frac {\left (a^4-48 a^2 b^2+384 b^4\right ) d x^2-8 a b \left (a^2-24 b^2\right ) \sqrt {d x^2 \left (2-d x^2\right )}+4 b \left (a^3 d x^2-24 a b^2 d x^2-6 a^2 b \sqrt {-d x^2 \left (-2+d x^2\right )}+48 b^3 \sqrt {-d x^2 \left (-2+d x^2\right )}\right ) \arccos \left (-1+d x^2\right )+6 b^2 \left (a^2 d x^2-8 b^2 d x^2-4 a b \sqrt {-d x^2 \left (-2+d x^2\right )}\right ) \arccos \left (-1+d x^2\right )^2+4 b^3 \left (a d x^2-2 b \sqrt {-d x^2 \left (-2+d x^2\right )}\right ) \arccos \left (-1+d x^2\right )^3+b^4 d x^2 \arccos \left (-1+d x^2\right )^4}{d x} \] Input:

Integrate[(a + b*ArcCos[-1 + d*x^2])^4,x]
 

Output:

((a^4 - 48*a^2*b^2 + 384*b^4)*d*x^2 - 8*a*b*(a^2 - 24*b^2)*Sqrt[d*x^2*(2 - 
 d*x^2)] + 4*b*(a^3*d*x^2 - 24*a*b^2*d*x^2 - 6*a^2*b*Sqrt[-(d*x^2*(-2 + d* 
x^2))] + 48*b^3*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x^2] + 6*b^2*(a 
^2*d*x^2 - 8*b^2*d*x^2 - 4*a*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d* 
x^2]^2 + 4*b^3*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x 
^2]^3 + b^4*d*x^2*ArcCos[-1 + d*x^2]^4)/(d*x)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5314, 5314, 24}

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.

\(\displaystyle \int \left (a+b \arccos \left (d x^2-1\right )\right )^4 \, dx\)

\(\Big \downarrow \) 5314

\(\displaystyle -48 b^2 \int \left (a+b \arccos \left (d x^2-1\right )\right )^2dx-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (d x^2-1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2-1\right )\right )^4\)

\(\Big \downarrow \) 5314

\(\displaystyle -48 b^2 \left (-8 b^2 \int 1dx-\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (d x^2-1\right )\right )}{d x}+x \left (a+b \arccos \left (d x^2-1\right )\right )^2\right )-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (d x^2-1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2-1\right )\right )^4\)

\(\Big \downarrow \) 24

\(\displaystyle -48 b^2 \left (-\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (d x^2-1\right )\right )}{d x}+x \left (a+b \arccos \left (d x^2-1\right )\right )^2-8 b^2 x\right )-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (d x^2-1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2-1\right )\right )^4\)

Input:

Int[(a + b*ArcCos[-1 + d*x^2])^4,x]
 

Output:

(-8*b*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2])^3)/(d*x) + x*(a + 
 b*ArcCos[-1 + d*x^2])^4 - 48*b^2*(-8*b^2*x - (4*b*Sqrt[2*d*x^2 - d^2*x^4] 
*(a + b*ArcCos[-1 + d*x^2]))/(d*x) + x*(a + b*ArcCos[-1 + d*x^2])^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 5314

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*( 
a + b*ArcCos[c + d*x^2])^n, x] + (-Simp[2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*(( 
a + b*ArcCos[c + d*x^2])^(n - 1)/(d*x)), x] - Simp[4*b^2*n*(n - 1)   Int[(a 
 + b*ArcCos[c + d*x^2])^(n - 2), x], x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c 
^2, 1] && GtQ[n, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(970\) vs. \(2(123)=246\).

Time = 0.19 (sec) , antiderivative size = 971, normalized size of antiderivative = 7.65

method	result	size
orering	\(\text {Expression too large to display}\)	\(971\)

Input:

int((a+b*arccos(d*x^2-1))^4,x,method=_RETURNVERBOSE)
 

Output:

x*(a+b*arccos(d*x^2-1))^4-32*(a+b*arccos(d*x^2-1))^3*b*x/(-(d*x^2-1)^2+1)^ 
(1/2)+1/d*x*(5*d*x^2-4)*(48*(a+b*arccos(d*x^2-1))^2*b^2*d^2*x^2/(-(d*x^2-1 
)^2+1)-8*(a+b*arccos(d*x^2-1))^3*b*d/(-(d*x^2-1)^2+1)^(1/2)-16*(a+b*arccos 
(d*x^2-1))^3*b*d^2*x^2/(-(d*x^2-1)^2+1)^(3/2)*(d*x^2-1))+(5*d^2*x^4-8*d*x^ 
2-4)/d^2*(-192*(a+b*arccos(d*x^2-1))*b^3*d^3*x^3/(-(d*x^2-1)^2+1)^(3/2)+14 
4*(a+b*arccos(d*x^2-1))^2*b^2*d^2*x/(-(d*x^2-1)^2+1)+288*(a+b*arccos(d*x^2 
-1))^2*b^2*d^3*x^3/(-(d*x^2-1)^2+1)^2*(d*x^2-1)-48*(a+b*arccos(d*x^2-1))^3 
*b*d^2/(-(d*x^2-1)^2+1)^(3/2)*(d*x^2-1)*x-96*(a+b*arccos(d*x^2-1))^3*b*d^3 
*x^3/(-(d*x^2-1)^2+1)^(5/2)*(d*x^2-1)^2-32*(a+b*arccos(d*x^2-1))^3*b*d^3*x 
^3/(-(d*x^2-1)^2+1)^(3/2))+1/d^2*x*(d*x^2-2)^2*(384*b^4*d^4*x^4/(-(d*x^2-1 
)^2+1)^2-1152*(a+b*arccos(d*x^2-1))*b^3*d^3*x^2/(-(d*x^2-1)^2+1)^(3/2)-230 
4*(a+b*arccos(d*x^2-1))*b^3*d^4*x^4/(-(d*x^2-1)^2+1)^(5/2)*(d*x^2-1)+144*( 
a+b*arccos(d*x^2-1))^2*b^2*d^2/(-(d*x^2-1)^2+1)+1728*(a+b*arccos(d*x^2-1)) 
^2*b^2*d^3*x^2/(-(d*x^2-1)^2+1)^2*(d*x^2-1)+2880*(a+b*arccos(d*x^2-1))^2*b 
^2*d^4*x^4/(-(d*x^2-1)^2+1)^3*(d*x^2-1)^2+768*(a+b*arccos(d*x^2-1))^2*b^2* 
d^4*x^4/(-(d*x^2-1)^2+1)^2-576*(a+b*arccos(d*x^2-1))^3*b*d^3/(-(d*x^2-1)^2 
+1)^(5/2)*(d*x^2-1)^2*x^2-192*(a+b*arccos(d*x^2-1))^3*b*d^3/(-(d*x^2-1)^2+ 
1)^(3/2)*x^2-48*(a+b*arccos(d*x^2-1))^3*b*d^2/(-(d*x^2-1)^2+1)^(3/2)*(d*x^ 
2-1)-960*(a+b*arccos(d*x^2-1))^3*b*d^4*x^4/(-(d*x^2-1)^2+1)^(7/2)*(d*x^2-1 
)^3-576*(a+b*arccos(d*x^2-1))^3*b*d^4*x^4/(-(d*x^2-1)^2+1)^(5/2)*(d*x^2...
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.63 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\frac {b^{4} d x^{2} \arccos \left (d x^{2} - 1\right )^{4} + 4 \, a b^{3} d x^{2} \arccos \left (d x^{2} - 1\right )^{3} + 6 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} d x^{2} \arccos \left (d x^{2} - 1\right )^{2} + 4 \, {\left (a^{3} b - 24 \, a b^{3}\right )} d x^{2} \arccos \left (d x^{2} - 1\right ) + {\left (a^{4} - 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} - 8 \, {\left (b^{4} \arccos \left (d x^{2} - 1\right )^{3} + 3 \, a b^{3} \arccos \left (d x^{2} - 1\right )^{2} + a^{3} b - 24 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} \arccos \left (d x^{2} - 1\right )\right )} \sqrt {-d^{2} x^{4} + 2 \, d x^{2}}}{d x} \] Input:

integrate((a+b*arccos(d*x^2-1))^4,x, algorithm="fricas")
 

Output:

(b^4*d*x^2*arccos(d*x^2 - 1)^4 + 4*a*b^3*d*x^2*arccos(d*x^2 - 1)^3 + 6*(a^ 
2*b^2 - 8*b^4)*d*x^2*arccos(d*x^2 - 1)^2 + 4*(a^3*b - 24*a*b^3)*d*x^2*arcc 
os(d*x^2 - 1) + (a^4 - 48*a^2*b^2 + 384*b^4)*d*x^2 - 8*(b^4*arccos(d*x^2 - 
 1)^3 + 3*a*b^3*arccos(d*x^2 - 1)^2 + a^3*b - 24*a*b^3 + 3*(a^2*b^2 - 8*b^ 
4)*arccos(d*x^2 - 1))*sqrt(-d^2*x^4 + 2*d*x^2))/(d*x)
 

Sympy [F]

\[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\int \left (a + b \operatorname {acos}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \] Input:

integrate((a+b*acos(d*x**2-1))**4,x)
 

Output:

Integral((a + b*acos(d*x**2 - 1))**4, x)
 

Maxima [F]

\[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\int { {\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{4} \,d x } \] Input:

integrate((a+b*arccos(d*x^2-1))^4,x, algorithm="maxima")
 

Output:

b^4*x*arctan2(sqrt(-d*x^2 + 2)*sqrt(d)*x, d*x^2 - 1)^4 + 4*(x*arccos(d*x^2 
 - 1) + 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(-d*x^2 + 2)*d))*a^3*b + a^4*x - 
integrate(2*(4*sqrt(-d*x^2 + 2)*b^4*sqrt(d)*x*arctan2(sqrt(-d*x^2 + 2)*sqr 
t(d)*x, d*x^2 - 1)^3 - 2*(a*b^3*d*x^2 - 2*a*b^3)*arctan2(sqrt(-d*x^2 + 2)* 
sqrt(d)*x, d*x^2 - 1)^3 - 3*(a^2*b^2*d*x^2 - 2*a^2*b^2)*arctan2(sqrt(-d*x^ 
2 + 2)*sqrt(d)*x, d*x^2 - 1)^2)/(d*x^2 - 2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (123) = 246\).

Time = 0.86 (sec) , antiderivative size = 586, normalized size of antiderivative = 4.61 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx =\text {Too large to display} \] Input:

integrate((a+b*arccos(d*x^2-1))^4,x, algorithm="giac")
 

Output:

4*(x*arccos(d*x^2 - 1) + 2*sqrt(2)*sgn(x)/sqrt(d) - 2*sqrt(-d^2*x^2 + 2*d) 
/(d*sgn(x)))*a^3*b + 6*(x*arccos(d*x^2 - 1)^2 + 4*(sqrt(2)*pi*sqrt(d)*abs( 
d) - 2*sqrt(2)*d^(3/2))*sgn(x)/(d*abs(d)) - 4*(sqrt(-d^2*x^2 + 2*d)*arccos 
(d*x^2 - 1) - 2*(sqrt(2)*sqrt(d) - sqrt(d^2*x^2))*d/abs(d))/(d*sgn(x)))*a^ 
2*b^2 + 4*(x*arccos(d*x^2 - 1)^3 + 6*(sqrt(2)*pi^2*sqrt(d) - 8*sqrt(2)*sqr 
t(d))*sgn(x)/d - 6*(sqrt(-d^2*x^2 + 2*d)*arccos(d*x^2 - 1)^2 + 4*(sqrt(d^2 
*x^2)*arccos((d^2*x^2 - d)/d) + 2*(sqrt(2)*sqrt(d) - sqrt(-d^2*x^2 + 2*d)) 
*d/abs(d) - 2*sqrt(2)*d^(3/2)/abs(d))*d/abs(d))/(d*sgn(x)))*a*b^3 + (x*arc 
cos(d*x^2 - 1)^4 + 8*(sqrt(2)*pi^3*sqrt(d)*abs(d) - 24*sqrt(2)*pi*sqrt(d)* 
abs(d) + 48*sqrt(2)*d^(3/2))*sgn(x)/(d*abs(d)) - 8*(sqrt(-d^2*x^2 + 2*d)*a 
rccos(d*x^2 - 1)^3 + 6*(sqrt(d^2*x^2)*arccos((d^2*x^2 - d)/d)^2 - 2*(pi*sq 
rt(-d^2*x^2 + 2*d) + 2*sqrt(-d^2*x^2 + 2*d)*arcsin(-(d^2*x^2 - d)/d) - 4*( 
sqrt(2)*sqrt(d) - sqrt(d^2*x^2))*d/abs(d) - 2*(sqrt(2)*pi*sqrt(d)*abs(d) - 
 2*sqrt(2)*d^(3/2))/abs(d))*d/abs(d) - 4*(sqrt(2)*pi*sqrt(d)*abs(d) - 2*sq 
rt(2)*d^(3/2))/d)*d/abs(d))/(d*sgn(x)))*b^4 + a^4*x
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\int {\left (a+b\,\mathrm {acos}\left (d\,x^2-1\right )\right )}^4 \,d x \] Input:

int((a + b*acos(d*x^2 - 1))^4,x)
 

Output:

int((a + b*acos(d*x^2 - 1))^4, x)
 

Reduce [F]

\[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=4 \left (\int \mathit {acos} \left (d \,x^{2}-1\right )d x \right ) a^{3} b +\left (\int \mathit {acos} \left (d \,x^{2}-1\right )^{4}d x \right ) b^{4}+4 \left (\int \mathit {acos} \left (d \,x^{2}-1\right )^{3}d x \right ) a \,b^{3}+6 \left (\int \mathit {acos} \left (d \,x^{2}-1\right )^{2}d x \right ) a^{2} b^{2}+a^{4} x \] Input:

int((a+b*acos(d*x^2-1))^4,x)
 

Output:

4*int(acos(d*x**2 - 1),x)*a**3*b + int(acos(d*x**2 - 1)**4,x)*b**4 + 4*int 
(acos(d*x**2 - 1)**3,x)*a*b**3 + 6*int(acos(d*x**2 - 1)**2,x)*a**2*b**2 + 
a**4*x