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 - (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

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Rubi [A]  time = 0.158683, antiderivative size = 119, normalized size of antiderivative = 1., 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 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]

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 8

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

Mathematica [A]  time = 0.129566, 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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Maple [B]  time = 0.035, size = 255, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ( dx+c \right ) +{b}^{4} \left ( \left ( dx+c \right ) \left ( \arcsin \left ( dx+c \right ) \right ) ^{4}+4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}\sqrt{1- \left ( dx+c \right ) ^{2}}-12\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) +24\,dx+24\,c-24\,\sqrt{1- \left ( dx+c \right ) ^{2}}\arcsin \left ( dx+c \right ) \right ) +4\,a{b}^{3} \left ( \left ( \arcsin \left ( dx+c \right ) \right ) ^{3} \left ( dx+c \right ) +3\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\sqrt{1- \left ( dx+c \right ) ^{2}}-6\,\sqrt{1- \left ( dx+c \right ) ^{2}}-6\, \left ( dx+c \right ) \arcsin \left ( dx+c \right ) \right ) +6\,{a}^{2}{b}^{2} \left ( \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) -2\,dx-2\,c+2\,\sqrt{1- \left ( dx+c \right ) ^{2}}\arcsin \left ( dx+c \right ) \right ) +4\,{a}^{3}b \left ( \left ( dx+c \right ) \arcsin \left ( dx+c \right ) +\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) \right ) } \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.37543, size = 540, normalized size = 4.54 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 2.5461, size = 444, normalized size = 3.73 \begin{align*} \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}{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}

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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Giac [B]  time = 1.19167, size = 444, normalized size = 3.73 \begin{align*} \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} \end{align*}

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