\(\int (a+b \arccos (1+d x^2))^4 \, dx\) [73]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4899, 8} \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=\frac {192 b^3 \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )}{d x}-48 b^2 x \left (a+b \arccos \left (d x^2+1\right )\right )^2-\frac {8 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \arccos \left (d x^2+1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2+1\right )\right )^4+384 b^4 x \]

[In]

Int[(a + b*ArcCos[1 + d*x^2])^4,x]

[Out]

384*b^4*x + (192*b^3*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcCos[1 + d*x^2]))/(d*x) - 48*b^2*x*(a + b*ArcCos[1 + d*
x^2])^2 - (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

Rule 8

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

Rule 4899

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

Rubi steps \begin{align*} \text {integral}& = -\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-\left (48 b^2\right ) \int \left (a+b \arccos \left (1+d x^2\right )\right )^2 \, dx \\ & = \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+\left (384 b^4\right ) \int 1 \, 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 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.96 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=\frac {\left (a^4-48 a^2 b^2+384 b^4\right ) d x^2-8 a b \left (a^2-24 b^2\right ) \sqrt {-d x^2 \left (2+d x^2\right )}+4 b \left (a^3 d x^2-24 a b^2 d x^2-6 a^2 b \sqrt {-d x^2 \left (2+d x^2\right )}+48 b^3 \sqrt {-d x^2 \left (2+d x^2\right )}\right ) \arccos \left (1+d x^2\right )+6 b^2 \left (a^2 d x^2-8 b^2 d x^2-4 a b \sqrt {-d x^2 \left (2+d x^2\right )}\right ) \arccos \left (1+d x^2\right )^2+4 b^3 \left (a d x^2-2 b \sqrt {-d x^2 \left (2+d x^2\right )}\right ) \arccos \left (1+d x^2\right )^3+b^4 d x^2 \arccos \left (1+d x^2\right )^4}{d x} \]

[In]

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

[Out]

((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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.63 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=\frac {b^{4} d x^{2} \arccos \left (d x^{2} + 1\right )^{4} + 4 \, a b^{3} d x^{2} \arccos \left (d x^{2} + 1\right )^{3} + 6 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} d x^{2} \arccos \left (d x^{2} + 1\right )^{2} + 4 \, {\left (a^{3} b - 24 \, a b^{3}\right )} d x^{2} \arccos \left (d x^{2} + 1\right ) + {\left (a^{4} - 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} - 8 \, {\left (b^{4} \arccos \left (d x^{2} + 1\right )^{3} + 3 \, a b^{3} \arccos \left (d x^{2} + 1\right )^{2} + a^{3} b - 24 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} \arccos \left (d x^{2} + 1\right )\right )} \sqrt {-d^{2} x^{4} - 2 \, d x^{2}}}{d x} \]

[In]

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

[Out]

(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*arccos(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]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: sign: argument cannot be imaginary; 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. 577 vs. \(2 (123) = 246\).

Time = 1.04 (sec) , antiderivative size = 577, normalized size of antiderivative = 4.54 \[ \int \left (a+b \arccos \left (1+d x^2\right )\right )^4 \, dx=4 \, {\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^{3} b + 6 \, {\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^{2} b^{2} + 4 \, {\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 )} a b^{3} + {\left (x \arccos \left (d x^{2} + 1\right )^{4} - \frac {48 \, {\left (\sqrt {2} \pi ^{2} \sqrt {-d} - 8 \, \sqrt {2} \sqrt {-d}\right )} \mathrm {sgn}\left (x\right )}{{\left | d \right |}} - \frac {8 \, {\left (\sqrt {-d^{2} x^{2} - 2 \, d} \arccos \left (d x^{2} + 1\right )^{3} - \frac {6 \, {\left (\sqrt {2} \pi ^{2} \sqrt {-d} - \sqrt {d^{2} x^{2}} \arccos \left (\frac {d^{2} x^{2} + d}{d}\right )^{2} - 8 \, \sqrt {2} \sqrt {-d} + \frac {2 \, {\left (\pi \sqrt {-d^{2} x^{2} - 2 \, d} + 2 \, \sqrt {-d^{2} x^{2} - 2 \, d} \arcsin \left (-\frac {d^{2} x^{2} + d}{d}\right ) - \frac {4 \, {\left (\sqrt {2} \sqrt {-d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}} + \frac {4 \, \sqrt {2} \sqrt {-d} d}{{\left | d \right |}}\right )} d}{{\left | d \right |}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} b^{4} + a^{4} x \]

[In]

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

[Out]

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*sqr
t(2)*sqrt(-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)*a
rcsin(-(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

Mupad [F(-1)]

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

[In]

int((a + b*acos(d*x^2 + 1))^4,x)

[Out]

int((a + b*acos(d*x^2 + 1))^4, x)