3.3.91 \(\int (c-a^2 c x^2) \text {ArcSin}(a x)^3 \, dx\) [291]

Optimal. Leaf size=158 \[ -\frac {40 c \sqrt {1-a^2 x^2}}{9 a}-\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac {14}{3} c x \text {ArcSin}(a x)+\frac {2}{9} a^2 c x^3 \text {ArcSin}(a x)+\frac {2 c \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \text {ArcSin}(a x)^2}{3 a}+\frac {2}{3} c x \text {ArcSin}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {ArcSin}(a x)^3 \]

[Out]

-2/27*c*(-a^2*x^2+1)^(3/2)/a-14/3*c*x*arcsin(a*x)+2/9*a^2*c*x^3*arcsin(a*x)+1/3*c*(-a^2*x^2+1)^(3/2)*arcsin(a*
x)^2/a+2/3*c*x*arcsin(a*x)^3+1/3*c*x*(-a^2*x^2+1)*arcsin(a*x)^3-40/9*c*(-a^2*x^2+1)^(1/2)/a+2*c*arcsin(a*x)^2*
(-a^2*x^2+1)^(1/2)/a

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Rubi [A]
time = 0.15, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4743, 4715, 4767, 267, 4739, 455, 45} \begin {gather*} \frac {2}{9} a^2 c x^3 \text {ArcSin}(a x)+\frac {1}{3} c x \left (1-a^2 x^2\right ) \text {ArcSin}(a x)^3+\frac {c \left (1-a^2 x^2\right )^{3/2} \text {ArcSin}(a x)^2}{3 a}+\frac {2 c \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{a}-\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac {40 c \sqrt {1-a^2 x^2}}{9 a}+\frac {2}{3} c x \text {ArcSin}(a x)^3-\frac {14}{3} c x \text {ArcSin}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)*ArcSin[a*x]^3,x]

[Out]

(-40*c*Sqrt[1 - a^2*x^2])/(9*a) - (2*c*(1 - a^2*x^2)^(3/2))/(27*a) - (14*c*x*ArcSin[a*x])/3 + (2*a^2*c*x^3*Arc
Sin[a*x])/9 + (2*c*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + (c*(1 - a^2*x^2)^(3/2)*ArcSin[a*x]^2)/(3*a) + (2*c*x*A
rcSin[a*x]^3)/3 + (c*x*(1 - a^2*x^2)*ArcSin[a*x]^3)/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 4715

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 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4743

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

Rule 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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]

Rubi steps

\begin {align*} \int \left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^3 \, dx &=\frac {1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{3} (2 c) \int \sin ^{-1}(a x)^3 \, dx-(a c) \int x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \, dx\\ &=\frac {c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \sin ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3-\frac {1}{3} (2 c) \int \left (1-a^2 x^2\right ) \sin ^{-1}(a x) \, dx-(2 a c) \int \frac {x \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2}{3} c x \sin ^{-1}(a x)+\frac {2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac {2 c \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \sin ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3-(4 c) \int \sin ^{-1}(a x) \, dx+\frac {1}{3} (2 a c) \int \frac {x \left (1-\frac {a^2 x^2}{3}\right )}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {14}{3} c x \sin ^{-1}(a x)+\frac {2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac {2 c \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \sin ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{3} (a c) \text {Subst}\left (\int \frac {1-\frac {a^2 x}{3}}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )+(4 a c) \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {4 c \sqrt {1-a^2 x^2}}{a}-\frac {14}{3} c x \sin ^{-1}(a x)+\frac {2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac {2 c \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \sin ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{3} (a c) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-a^2 x}}+\frac {1}{3} \sqrt {1-a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {40 c \sqrt {1-a^2 x^2}}{9 a}-\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac {14}{3} c x \sin ^{-1}(a x)+\frac {2}{9} a^2 c x^3 \sin ^{-1}(a x)+\frac {2 c \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \sin ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 101, normalized size = 0.64 \begin {gather*} \frac {c \left (2 \sqrt {1-a^2 x^2} \left (-61+a^2 x^2\right )+6 a x \left (-21+a^2 x^2\right ) \text {ArcSin}(a x)-9 \sqrt {1-a^2 x^2} \left (-7+a^2 x^2\right ) \text {ArcSin}(a x)^2-9 a x \left (-3+a^2 x^2\right ) \text {ArcSin}(a x)^3\right )}{27 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)*ArcSin[a*x]^3,x]

[Out]

(c*(2*Sqrt[1 - a^2*x^2]*(-61 + a^2*x^2) + 6*a*x*(-21 + a^2*x^2)*ArcSin[a*x] - 9*Sqrt[1 - a^2*x^2]*(-7 + a^2*x^
2)*ArcSin[a*x]^2 - 9*a*x*(-3 + a^2*x^2)*ArcSin[a*x]^3))/(27*a)

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Maple [A]
time = 0.06, size = 132, normalized size = 0.84

method	result	size
derivativedivides	\(-\frac {c \left (9 a^{3} x^{3} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-27 a x \arcsin \left (a x \right )^{3}-6 a^{3} x^{3} \arcsin \left (a x \right )-63 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arcsin \left (a x \right )+122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\)	\(132\)
default	\(-\frac {c \left (9 a^{3} x^{3} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-27 a x \arcsin \left (a x \right )^{3}-6 a^{3} x^{3} \arcsin \left (a x \right )-63 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arcsin \left (a x \right )+122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)*arcsin(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/27/a*c*(9*a^3*x^3*arcsin(a*x)^3+9*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^2*x^2-27*a*x*arcsin(a*x)^3-6*a^3*x^3*a
rcsin(a*x)-63*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-2*a^2*x^2*(-a^2*x^2+1)^(1/2)+126*a*x*arcsin(a*x)+122*(-a^2*x^2+
1)^(1/2))

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Maxima [A]
time = 0.49, size = 128, normalized size = 0.81 \begin {gather*} -\frac {1}{3} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} - \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \arcsin \left (a x\right )^{3} + \frac {2}{27} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} + \frac {3 \, {\left (a^{2} c x^{3} - 21 \, c x\right )} \arcsin \left (a x\right )}{a} - \frac {61 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)*arcsin(a*x)^3,x, algorithm="maxima")

[Out]

-1/3*(sqrt(-a^2*x^2 + 1)*c*x^2 - 7*sqrt(-a^2*x^2 + 1)*c/a^2)*a*arcsin(a*x)^2 - 1/3*(a^2*c*x^3 - 3*c*x)*arcsin(
a*x)^3 + 2/27*(sqrt(-a^2*x^2 + 1)*c*x^2 + 3*(a^2*c*x^3 - 21*c*x)*arcsin(a*x)/a - 61*sqrt(-a^2*x^2 + 1)*c/a^2)*
a

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Fricas [A]
time = 1.74, size = 95, normalized size = 0.60 \begin {gather*} -\frac {9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \arcsin \left (a x\right )^{3} - 6 \, {\left (a^{3} c x^{3} - 21 \, a c x\right )} \arcsin \left (a x\right ) - {\left (2 \, a^{2} c x^{2} - 9 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \arcsin \left (a x\right )^{2} - 122 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)*arcsin(a*x)^3,x, algorithm="fricas")

[Out]

-1/27*(9*(a^3*c*x^3 - 3*a*c*x)*arcsin(a*x)^3 - 6*(a^3*c*x^3 - 21*a*c*x)*arcsin(a*x) - (2*a^2*c*x^2 - 9*(a^2*c*
x^2 - 7*c)*arcsin(a*x)^2 - 122*c)*sqrt(-a^2*x^2 + 1))/a

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Sympy [A]
time = 0.34, size = 150, normalized size = 0.95 \begin {gather*} \begin {cases} - \frac {a^{2} c x^{3} \operatorname {asin}^{3}{\left (a x \right )}}{3} + \frac {2 a^{2} c x^{3} \operatorname {asin}{\left (a x \right )}}{9} - \frac {a c x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3} + \frac {2 a c x^{2} \sqrt {- a^{2} x^{2} + 1}}{27} + c x \operatorname {asin}^{3}{\left (a x \right )} - \frac {14 c x \operatorname {asin}{\left (a x \right )}}{3} + \frac {7 c \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3 a} - \frac {122 c \sqrt {- a^{2} x^{2} + 1}}{27 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)*asin(a*x)**3,x)

[Out]

Piecewise((-a**2*c*x**3*asin(a*x)**3/3 + 2*a**2*c*x**3*asin(a*x)/9 - a*c*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**
2/3 + 2*a*c*x**2*sqrt(-a**2*x**2 + 1)/27 + c*x*asin(a*x)**3 - 14*c*x*asin(a*x)/3 + 7*c*sqrt(-a**2*x**2 + 1)*as
in(a*x)**2/(3*a) - 122*c*sqrt(-a**2*x**2 + 1)/(27*a), Ne(a, 0)), (0, True))

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Giac [A]
time = 0.44, size = 139, normalized size = 0.88 \begin {gather*} -\frac {1}{3} \, {\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right )^{3} + \frac {2}{3} \, c x \arcsin \left (a x\right )^{3} + \frac {2}{9} \, {\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right ) + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c \arcsin \left (a x\right )^{2}}{3 \, a} - \frac {40}{9} \, c x \arcsin \left (a x\right ) + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c \arcsin \left (a x\right )^{2}}{a} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{27 \, a} - \frac {40 \, \sqrt {-a^{2} x^{2} + 1} c}{9 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)*arcsin(a*x)^3,x, algorithm="giac")

[Out]

-1/3*(a^2*x^2 - 1)*c*x*arcsin(a*x)^3 + 2/3*c*x*arcsin(a*x)^3 + 2/9*(a^2*x^2 - 1)*c*x*arcsin(a*x) + 1/3*(-a^2*x
^2 + 1)^(3/2)*c*arcsin(a*x)^2/a - 40/9*c*x*arcsin(a*x) + 2*sqrt(-a^2*x^2 + 1)*c*arcsin(a*x)^2/a - 2/27*(-a^2*x
^2 + 1)^(3/2)*c/a - 40/9*sqrt(-a^2*x^2 + 1)*c/a

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {asin}\left (a\,x\right )}^3\,\left (c-a^2\,c\,x^2\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^3*(c - a^2*c*x^2),x)

[Out]

int(asin(a*x)^3*(c - a^2*c*x^2), x)

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