3.209 \(\int (a+b \sin ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=119 \[ -\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+24 b^4 x \]

[Out]

24*b^4*x-12*b^2*(d*x+c)*(a+b*arcsin(d*x+c))^2/d+(d*x+c)*(a+b*arcsin(d*x+c))^4/d-24*b^3*(a+b*arcsin(d*x+c))*(1-
(d*x+c)^2)^(1/2)/d+4*b*(a+b*arcsin(d*x+c))^3*(1-(d*x+c)^2)^(1/2)/d

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Rubi [A]  time = 0.16, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4803, 4619, 4677, 8} \[ -\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+24 b^4 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])^4,x]

[Out]

24*b^4*x - (24*b^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/d - (12*b^2*(c + d*x)*(a + b*ArcSin[c + d*x]
)^2)/d + (4*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcSin[c + d*x])^4)/d

Rule 8

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

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4803

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps

\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {\left (24 b^3\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}+\frac {\left (24 b^4\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=24 b^4 x-\frac {24 b^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}-\frac {12 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}+\frac {4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4}{d}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 115, normalized size = 0.97 \[ \frac {-12 b^2 \left (2 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2-2 b^2 (c+d x)\right )+(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^4+4 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x])^4,x]

[Out]

(4*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 + (c + d*x)*(a + b*ArcSin[c + d*x])^4 - 12*b^2*(-2*b^2*(c
+ d*x) + 2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (c + d*x)*(a + b*ArcSin[c + d*x])^2))/d

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fricas [B]  time = 0.50, size = 233, normalized size = 1.96 \[ \frac {{\left (b^{4} d x + b^{4} c\right )} \arcsin \left (d x + c\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{4} - 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x + 6 \, {\left ({\left (a^{2} b^{2} - 2 \, b^{4}\right )} d x + {\left (a^{2} b^{2} - 2 \, b^{4}\right )} c\right )} \arcsin \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{3} b - 6 \, a b^{3}\right )} d x + {\left (a^{3} b - 6 \, a b^{3}\right )} c\right )} \arcsin \left (d x + c\right ) + 4 \, {\left (b^{4} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{3} \arcsin \left (d x + c\right )^{2} + a^{3} b - 6 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^4,x, algorithm="fricas")

[Out]

((b^4*d*x + b^4*c)*arcsin(d*x + c)^4 + 4*(a*b^3*d*x + a*b^3*c)*arcsin(d*x + c)^3 + (a^4 - 12*a^2*b^2 + 24*b^4)
*d*x + 6*((a^2*b^2 - 2*b^4)*d*x + (a^2*b^2 - 2*b^4)*c)*arcsin(d*x + c)^2 + 4*((a^3*b - 6*a*b^3)*d*x + (a^3*b -
6*a*b^3)*c)*arcsin(d*x + c) + 4*(b^4*arcsin(d*x + c)^3 + 3*a*b^3*arcsin(d*x + c)^2 + a^3*b - 6*a*b^3 + 3*(a^2
*b^2 - 2*b^4)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d

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giac [B]  time = 0.25, size = 329, normalized size = 2.76 \[ \frac {{\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{4}}{d} + \frac {4 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )^{3}}{d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right )^{3}}{d} + \frac {6 \, {\left (d x + c\right )} a^{2} b^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {12 \, {\left (d x + c\right )} b^{4} \arcsin \left (d x + c\right )^{2}}{d} + \frac {12 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3} \arcsin \left (d x + c\right )^{2}}{d} + \frac {4 \, {\left (d x + c\right )} a^{3} b \arcsin \left (d x + c\right )}{d} - \frac {24 \, {\left (d x + c\right )} a b^{3} \arcsin \left (d x + c\right )}{d} + \frac {12 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{2} \arcsin \left (d x + c\right )}{d} - \frac {24 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{4} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{4}}{d} - \frac {12 \, {\left (d x + c\right )} a^{2} b^{2}}{d} + \frac {24 \, {\left (d x + c\right )} b^{4}}{d} + \frac {4 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{3} b}{d} - \frac {24 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{3}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^4,x, algorithm="giac")

[Out]

(d*x + c)*b^4*arcsin(d*x + c)^4/d + 4*(d*x + c)*a*b^3*arcsin(d*x + c)^3/d + 4*sqrt(-(d*x + c)^2 + 1)*b^4*arcsi
n(d*x + c)^3/d + 6*(d*x + c)*a^2*b^2*arcsin(d*x + c)^2/d - 12*(d*x + c)*b^4*arcsin(d*x + c)^2/d + 12*sqrt(-(d*
x + c)^2 + 1)*a*b^3*arcsin(d*x + c)^2/d + 4*(d*x + c)*a^3*b*arcsin(d*x + c)/d - 24*(d*x + c)*a*b^3*arcsin(d*x
+ c)/d + 12*sqrt(-(d*x + c)^2 + 1)*a^2*b^2*arcsin(d*x + c)/d - 24*sqrt(-(d*x + c)^2 + 1)*b^4*arcsin(d*x + c)/d
+ (d*x + c)*a^4/d - 12*(d*x + c)*a^2*b^2/d + 24*(d*x + c)*b^4/d + 4*sqrt(-(d*x + c)^2 + 1)*a^3*b/d - 24*sqrt(
-(d*x + c)^2 + 1)*a*b^3/d

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maple [B]  time = 0.07, size = 255, normalized size = 2.14 \[ \frac {a^{4} \left (d x +c \right )+b^{4} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{4}+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a \,b^{3} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{3}+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+6 a^{2} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+4 a^{3} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))^4,x)

[Out]

1/d*(a^4*(d*x+c)+b^4*((d*x+c)*arcsin(d*x+c)^4+4*arcsin(d*x+c)^3*(1-(d*x+c)^2)^(1/2)-12*arcsin(d*x+c)^2*(d*x+c)
+24*d*x+24*c-24*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2))+4*a*b^3*((d*x+c)*arcsin(d*x+c)^3+3*arcsin(d*x+c)^2*(1-(d*x+
c)^2)^(1/2)-6*(1-(d*x+c)^2)^(1/2)-6*(d*x+c)*arcsin(d*x+c))+6*a^2*b^2*(arcsin(d*x+c)^2*(d*x+c)-2*d*x-2*c+2*arcs
in(d*x+c)*(1-(d*x+c)^2)^(1/2))+4*a^3*b*((d*x+c)*arcsin(d*x+c)+(1-(d*x+c)^2)^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b^{4} x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{4} + a^{4} x + \frac {4 \, {\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} a^{3} b}{d} + \int \frac {2 \, {\left (2 \, \sqrt {d x + c + 1} \sqrt {-d x - c + 1} b^{4} d x \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 2 \, {\left (a b^{3} d^{2} x^{2} + 2 \, a b^{3} c d x + a b^{3} c^{2} - a b^{3}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{3} + 3 \, {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{2} b^{2} c d x + a^{2} b^{2} c^{2} - a^{2} b^{2}\right )} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^4,x, algorithm="maxima")

[Out]

b^4*x*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + a^4*x + 4*((d*x + c)*arcsin(d*x + c) + sqrt(-
(d*x + c)^2 + 1))*a^3*b/d + integrate(2*(2*sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*b^4*d*x*arctan2(d*x + c, sqrt(
d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 2*(a*b^3*d^2*x^2 + 2*a*b^3*c*d*x + a*b^3*c^2 - a*b^3)*arctan2(d*x + c, sq
rt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 3*(a^2*b^2*d^2*x^2 + 2*a^2*b^2*c*d*x + a^2*b^2*c^2 - a^2*b^2)*arctan2(
d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^2)/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)

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mupad [B]  time = 0.58, size = 229, normalized size = 1.92 \[ a^4\,x+\frac {b^4\,\left (c+d\,x\right )\,\left ({\mathrm {asin}\left (c+d\,x\right )}^4-12\,{\mathrm {asin}\left (c+d\,x\right )}^2+24\right )}{d}-\frac {b^4\,\left (24\,\mathrm {asin}\left (c+d\,x\right )-4\,{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}+\frac {6\,a^2\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {4\,a^3\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {4\,a\,b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}-\frac {4\,a\,b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))^4,x)

[Out]

a^4*x + (b^4*(c + d*x)*(asin(c + d*x)^4 - 12*asin(c + d*x)^2 + 24))/d - (b^4*(24*asin(c + d*x) - 4*asin(c + d*
x)^3)*(1 - (c + d*x)^2)^(1/2))/d + (6*a^2*b^2*(2*asin(c + d*x)*(1 - (c + d*x)^2)^(1/2) + (asin(c + d*x)^2 - 2)
*(c + d*x)))/d + (4*a^3*b*((1 - (c + d*x)^2)^(1/2) + asin(c + d*x)*(c + d*x)))/d + (4*a*b^3*(3*asin(c + d*x)^2
- 6)*(1 - (c + d*x)^2)^(1/2))/d - (4*a*b^3*(6*asin(c + d*x) - asin(c + d*x)^3)*(c + d*x))/d

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sympy [A]  time = 1.54, size = 444, normalized size = 3.73 \[ \begin {cases} a^{4} x + \frac {4 a^{3} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a^{3} b x \operatorname {asin}{\left (c + d x \right )} + \frac {4 a^{3} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {6 a^{2} b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 12 a^{2} b^{2} x + \frac {12 a^{2} b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {4 a b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 24 a b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {12 a b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {b^{4} c \operatorname {asin}^{4}{\left (c + d x \right )}}{d} - \frac {12 b^{4} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + b^{4} x \operatorname {asin}^{4}{\left (c + d x \right )} - 12 b^{4} x \operatorname {asin}^{2}{\left (c + d x \right )} + 24 b^{4} x + \frac {4 b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {24 b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\relax (c )}\right )^{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x+c))**4,x)

[Out]

Piecewise((a**4*x + 4*a**3*b*c*asin(c + d*x)/d + 4*a**3*b*x*asin(c + d*x) + 4*a**3*b*sqrt(-c**2 - 2*c*d*x - d*
*2*x**2 + 1)/d + 6*a**2*b**2*c*asin(c + d*x)**2/d + 6*a**2*b**2*x*asin(c + d*x)**2 - 12*a**2*b**2*x + 12*a**2*
b**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d + 4*a*b**3*c*asin(c + d*x)**3/d - 24*a*b**3*c*asin(
c + d*x)/d + 4*a*b**3*x*asin(c + d*x)**3 - 24*a*b**3*x*asin(c + d*x) + 12*a*b**3*sqrt(-c**2 - 2*c*d*x - d**2*x
**2 + 1)*asin(c + d*x)**2/d - 24*a*b**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d + b**4*c*asin(c + d*x)**4/d -
12*b**4*c*asin(c + d*x)**2/d + b**4*x*asin(c + d*x)**4 - 12*b**4*x*asin(c + d*x)**2 + 24*b**4*x + 4*b**4*sqrt(
-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/d - 24*b**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d
*x)/d, Ne(d, 0)), (x*(a + b*asin(c))**4, True))

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