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
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)
Time = 0.32 (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 |
\(\displaystyle -48 b^2 \int \left (a+b \arccos \left (d x^2+1\right )\right )^2dx-\frac {8 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^4\) |
\(\Big \downarrow \) 5314 |
\(\displaystyle -48 b^2 \left (-8 b^2 \int 1dx-\frac {4 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^2\right )-\frac {8 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^4\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -48 b^2 \left (-\frac {4 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^2-8 b^2 x\right )-\frac {8 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^4\) |
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)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(970\) vs. \(2(123)=246\).
Time = 0.20 (sec) , antiderivative size = 971, normalized size of antiderivative = 7.65
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...
Time = 0.14 (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)
\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=\int \left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{4}\, dx \] Input:
integrate((a+b*acos(d*x**2+1))**4,x)
Output:
Integral((a + b*acos(d*x**2 + 1))**4, x)
Exception generated. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccos(d*x^2+1))^4,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)
Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (123) = 246\).
Time = 0.91 (sec) , antiderivative size = 577, normalized size of antiderivative = 4.54 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx =\text {Too large to display} \] Input:
integrate((a+b*arccos(d*x^2+1))^4,x, algorithm="giac")
Output:
4*(x*arccos(d*x^2 + 1) + 2*sqrt(2)*sqrt(-d)*sgn(x)/d - 2*sqrt(-d^2*x^2 - 2 *d)/(d*sgn(x)))*a^3*b + 6*(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^2*b^2 + 4*(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* (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) - (sqrt( 2)*pi*sqrt(-d)*abs(d) + 2*sqrt(2)*sqrt(-d)*d)/abs(d))*d/abs(d))/(d*sgn(x)) )*a*b^3 + (x*arccos(d*x^2 + 1)^4 - 48*(sqrt(2)*pi^2*sqrt(-d) - 8*sqrt(2)*s qrt(-d))*sgn(x)/abs(d) - 8*(sqrt(-d^2*x^2 - 2*d)*arccos(d*x^2 + 1)^3 - 6*( sqrt(2)*pi^2*sqrt(-d) - sqrt(d^2*x^2)*arccos((d^2*x^2 + d)/d)^2 - 8*sqrt(2 )*sqrt(-d) + 2*(pi*sqrt(-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) + 4*sqrt(2 )*sqrt(-d)*d/abs(d))*d/abs(d))*d/abs(d))/(d*sgn(x)))*b^4 + a^4*x
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)
\[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=4 \left (\int \mathit {acos} \left (d \,x^{2}+1\right )d x \right ) a^{3} b +\left (\int \mathit {acos} \left (d \,x^{2}+1\right )^{4}d x \right ) b^{4}+4 \left (\int \mathit {acos} \left (d \,x^{2}+1\right )^{3}d x \right ) a \,b^{3}+6 \left (\int \mathit {acos} \left (d \,x^{2}+1\right )^{2}d x \right ) a^{2} b^{2}+a^{4} x \] 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