\(\int (a+b \arccos (1+d x^2))^3 \, dx\) [51]

Optimal result

Integrand size = 14, antiderivative size = 110 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^3 \, dx=-24 a b^2 x+\frac {48 b^3 \sqrt {-2 d x^2-d^2 x^4}}{d x}-24 b^3 x \arccos \left (1+d x^2\right )-\frac {6 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \arccos \left (1+d x^2\right )\right )^2}{d x}+x \left (a+b \arccos \left (1+d x^2\right )\right )^3 \] Output:

-24*a*b^2*x+48*b^3*(-d^2*x^4-2*d*x^2)^(1/2)/d/x-24*b^3*x*arccos(d*x^2+1)-6 
*b*(-d^2*x^4-2*d*x^2)^(1/2)*(a+b*arccos(d*x^2+1))^2/d/x+x*(a+b*arccos(d*x^ 
2+1))^3
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.47 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^3 \, dx=\frac {a \left (a^2-24 b^2\right ) d x^2-6 b \left (a^2-8 b^2\right ) \sqrt {-d x^2 \left (2+d x^2\right )}+3 b \left (a^2 d x^2-8 b^2 d x^2-4 a b \sqrt {-d x^2 \left (2+d x^2\right )}\right ) \arccos \left (1+d x^2\right )+3 b^2 \left (a d x^2-2 b \sqrt {-d x^2 \left (2+d x^2\right )}\right ) \arccos \left (1+d x^2\right )^2+b^3 d x^2 \arccos \left (1+d x^2\right )^3}{d x} \] Input:

Integrate[(a + b*ArcCos[1 + d*x^2])^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5314, 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.

\(\displaystyle \int \left (a+b \arccos \left (d x^2+1\right )\right )^3 \, dx\)

\(\Big \downarrow \) 5314

\(\displaystyle -24 b^2 \int \left (a+b \arccos \left (d x^2+1\right )\right )dx-\frac {6 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^2}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^3\)

\(\Big \downarrow \) 2009

\(\displaystyle -24 b^2 \left (a x+b x \arccos \left (d x^2+1\right )-\frac {2 b \sqrt {-d^2 x^4-2 d x^2}}{d x}\right )-\frac {6 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^2}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^3\)

Input:

Int[(a + b*ArcCos[1 + d*x^2])^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

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. \(437\) vs. \(2(106)=212\).

Time = 0.12 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.98

method	result	size
orering	\(x {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{3}+\frac {6 \left (d \,x^{2}+4\right ) {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{2} b x}{\sqrt {-\left (d \,x^{2}+1\right )^{2}+1}}-\frac {2 \left (d \,x^{2}+2\right ) x \left (\frac {24 \left (a +b \arccos \left (d \,x^{2}+1\right )\right ) b^{2} d^{2} x^{2}}{-\left (d \,x^{2}+1\right )^{2}+1}-\frac {6 {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{2} b d}{\sqrt {-\left (d \,x^{2}+1\right )^{2}+1}}-\frac {12 {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{2} b \,d^{2} x^{2} \left (d \,x^{2}+1\right )}{{\left (-\left (d \,x^{2}+1\right )^{2}+1\right )}^{\frac {3}{2}}}\right )}{d}-\frac {\left (d \,x^{2}+2\right )^{2} \left (-\frac {48 b^{3} d^{3} x^{3}}{{\left (-\left (d \,x^{2}+1\right )^{2}+1\right )}^{\frac {3}{2}}}+\frac {72 \left (a +b \arccos \left (d \,x^{2}+1\right )\right ) b^{2} d^{2} x}{-\left (d \,x^{2}+1\right )^{2}+1}+\frac {144 \left (a +b \arccos \left (d \,x^{2}+1\right )\right ) b^{2} d^{3} x^{3} \left (d \,x^{2}+1\right )}{{\left (-\left (d \,x^{2}+1\right )^{2}+1\right )}^{2}}-\frac {36 {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{2} b \,d^{2} \left (d \,x^{2}+1\right ) x}{{\left (-\left (d \,x^{2}+1\right )^{2}+1\right )}^{\frac {3}{2}}}-\frac {72 {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{2} b \,d^{3} x^{3} \left (d \,x^{2}+1\right )^{2}}{{\left (-\left (d \,x^{2}+1\right )^{2}+1\right )}^{\frac {5}{2}}}-\frac {24 {\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{2} b \,d^{3} x^{3}}{{\left (-\left (d \,x^{2}+1\right )^{2}+1\right )}^{\frac {3}{2}}}\right )}{d^{2}}\)	\(438\)

Input:

int((a+b*arccos(d*x^2+1))^3,x,method=_RETURNVERBOSE)
 

Output:

x*(a+b*arccos(d*x^2+1))^3+6*(d*x^2+4)*(a+b*arccos(d*x^2+1))^2*b*x/(-(d*x^2 
+1)^2+1)^(1/2)-2/d*(d*x^2+2)*x*(24*(a+b*arccos(d*x^2+1))*b^2*d^2*x^2/(-(d* 
x^2+1)^2+1)-6*(a+b*arccos(d*x^2+1))^2*b*d/(-(d*x^2+1)^2+1)^(1/2)-12*(a+b*a 
rccos(d*x^2+1))^2*b*d^2*x^2/(-(d*x^2+1)^2+1)^(3/2)*(d*x^2+1))-1/d^2*(d*x^2 
+2)^2*(-48*b^3*d^3*x^3/(-(d*x^2+1)^2+1)^(3/2)+72*(a+b*arccos(d*x^2+1))*b^2 
*d^2*x/(-(d*x^2+1)^2+1)+144*(a+b*arccos(d*x^2+1))*b^2*d^3*x^3/(-(d*x^2+1)^ 
2+1)^2*(d*x^2+1)-36*(a+b*arccos(d*x^2+1))^2*b*d^2/(-(d*x^2+1)^2+1)^(3/2)*( 
d*x^2+1)*x-72*(a+b*arccos(d*x^2+1))^2*b*d^3*x^3/(-(d*x^2+1)^2+1)^(5/2)*(d* 
x^2+1)^2-24*(a+b*arccos(d*x^2+1))^2*b*d^3*x^3/(-(d*x^2+1)^2+1)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.31 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^3 \, dx=\frac {b^{3} d x^{2} \arccos \left (d x^{2} + 1\right )^{3} + 3 \, a b^{2} d x^{2} \arccos \left (d x^{2} + 1\right )^{2} + 3 \, {\left (a^{2} b - 8 \, b^{3}\right )} d x^{2} \arccos \left (d x^{2} + 1\right ) + {\left (a^{3} - 24 \, a b^{2}\right )} d x^{2} - 6 \, \sqrt {-d^{2} x^{4} - 2 \, d x^{2}} {\left (b^{3} \arccos \left (d x^{2} + 1\right )^{2} + 2 \, a b^{2} \arccos \left (d x^{2} + 1\right ) + a^{2} b - 8 \, b^{3}\right )}}{d x} \] Input:

integrate((a+b*arccos(d*x^2+1))^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a+b*acos(d*x**2+1))**3,x)
 

Output:

Integral((a + b*acos(d*x**2 + 1))**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^3 \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccos(d*x^2+1))^3,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imagi 
nary; found sqrt((-_SAGE_VAR_d*_SAGE_VAR_x^2)-2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (106) = 212\).

Time = 0.53 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.05 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^3 \, dx=3 \, {\left (x \arccos \left (d x^{2} + 1\right ) + \frac {2 \, \sqrt {2} \sqrt {-d} \mathrm {sgn}\left (x\right )}{d} - \frac {2 \, \sqrt {-d^{2} x^{2} - 2 \, d}}{d \mathrm {sgn}\left (x\right )}\right )} a^{2} b + 3 \, {\left (x \arccos \left (d x^{2} + 1\right )^{2} - \frac {8 \, \sqrt {2} \sqrt {-d} \mathrm {sgn}\left (x\right )}{{\left | d \right |}} - \frac {4 \, {\left (\sqrt {-d^{2} x^{2} - 2 \, d} \arccos \left (d x^{2} + 1\right ) - \frac {2 \, {\left (\sqrt {2} \sqrt {-d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} a b^{2} + {\left (x \arccos \left (d x^{2} + 1\right )^{3} - \frac {24 \, {\left (\sqrt {2} \pi \sqrt {-d} {\left | d \right |} + 2 \, \sqrt {2} \sqrt {-d} d\right )} \mathrm {sgn}\left (x\right )}{d^{2}} - \frac {6 \, {\left (\sqrt {-d^{2} x^{2} - 2 \, d} \arccos \left (d x^{2} + 1\right )^{2} + \frac {4 \, {\left (\sqrt {d^{2} x^{2}} \arccos \left (\frac {d^{2} x^{2} + d}{d}\right ) + \frac {2 \, {\left (\sqrt {2} \sqrt {-d} - \sqrt {-d^{2} x^{2} - 2 \, d}\right )} d}{{\left | d \right |}} - \frac {\sqrt {2} \pi \sqrt {-d} {\left | d \right |} + 2 \, \sqrt {2} \sqrt {-d} d}{{\left | d \right |}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} b^{3} + a^{3} x \] Input:

integrate((a+b*arccos(d*x^2+1))^3,x, algorithm="giac")
 

Output:

3*(x*arccos(d*x^2 + 1) + 2*sqrt(2)*sqrt(-d)*sgn(x)/d - 2*sqrt(-d^2*x^2 - 2 
*d)/(d*sgn(x)))*a^2*b + 3*(x*arccos(d*x^2 + 1)^2 - 8*sqrt(2)*sqrt(-d)*sgn( 
x)/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*b^2 + (x*arccos(d*x^2 + 1)^3 - 
24*(sqrt(2)*pi*sqrt(-d)*abs(d) + 2*sqrt(2)*sqrt(-d)*d)*sgn(x)/d^2 - 6*(sqr 
t(-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) - (sqrt(2)*p 
i*sqrt(-d)*abs(d) + 2*sqrt(2)*sqrt(-d)*d)/abs(d))*d/abs(d))/(d*sgn(x)))*b^ 
3 + a^3*x
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^3 \,d x \] Input:

int((a + b*acos(d*x^2 + 1))^3,x)
 

Output:

int((a + b*acos(d*x^2 + 1))^3, x)
 

Reduce [F]

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

int((a+b*acos(d*x^2+1))^3,x)
 

Output:

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