3.3.74 \(\int \frac {\text {ArcSin}(a x)^2}{(c-a^2 c x^2)^{5/2}} \, dx\) [274]

Optimal. Leaf size=283 \[ \frac {x}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\text {ArcSin}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \text {ArcSin}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {ArcSin}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}} \]

[Out]

1/3*x*arcsin(a*x)^2/c/(-a^2*c*x^2+c)^(3/2)+1/3*x/c^2/(-a^2*c*x^2+c)^(1/2)+2/3*x*arcsin(a*x)^2/c^2/(-a^2*c*x^2+
c)^(1/2)-1/3*arcsin(a*x)/a/c^2/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2)-2/3*I*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/
a/c^2/(-a^2*c*x^2+c)^(1/2)+4/3*arcsin(a*x)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^2*c
*x^2+c)^(1/2)-2/3*I*polylog(2,-(I*a*x+(-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {4747, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 197} \begin {gather*} -\frac {2 i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \text {ArcSin}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {\text {ArcSin}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \text {ArcSin}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{3 c^2 \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2),x]

[Out]

x/(3*c^2*Sqrt[c - a^2*c*x^2]) - ArcSin[a*x]/(3*a*c^2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x]^2
)/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*ArcSin[a*x]^2)/(3*c^2*Sqrt[c - a^2*c*x^2]) - (((2*I)/3)*Sqrt[1 - a^2*x^2]
*ArcSin[a*x]^2)/(a*c^2*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[1 + E^((2*I)*ArcSin[a*x])])
/(3*a*c^2*Sqrt[c - a^2*c*x^2]) - (((2*I)/3)*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[a*x])])/(a*c^2*Sqrt[
c - a^2*c*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4745

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSin[c
*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcSin
[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(p
+ 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rule 4765

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-e^(-1), Subst[In
t[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 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 \frac {\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {\left (2 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sin ^{-1}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (4 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (4 \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int x \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (8 i \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (4 \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (2 i \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{3 a c^2 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 149, normalized size = 0.53 \begin {gather*} \frac {a x+\left (-2 i \sqrt {1-a^2 x^2}+a x \left (2+\frac {1}{1-a^2 x^2}\right )\right ) \text {ArcSin}(a x)^2+\frac {\text {ArcSin}(a x) \left (-1+\left (4-4 a^2 x^2\right ) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}-2 i \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(a x)}\right )}{3 a c^2 \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2),x]

[Out]

(a*x + ((-2*I)*Sqrt[1 - a^2*x^2] + a*x*(2 + (1 - a^2*x^2)^(-1)))*ArcSin[a*x]^2 + (ArcSin[a*x]*(-1 + (4 - 4*a^2
*x^2)*Log[1 + E^((2*I)*ArcSin[a*x])]))/Sqrt[1 - a^2*x^2] - (2*I)*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin
[a*x])])/(3*a*c^2*Sqrt[c - a^2*c*x^2])

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Maple [A]
time = 0.22, size = 365, normalized size = 1.29

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-2 i \sqrt {-a^{2} x^{2}+1}-3 a x \right ) \left (-2 i \arcsin \left (a x \right ) a^{4} x^{4}-2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+i \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-a^{4} x^{4}+3 \arcsin \left (a x \right )^{2} a^{2} x^{2}+4 i \arcsin \left (a x \right ) a^{2} x^{2}+3 a x \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-i \sqrt {-a^{2} x^{2}+1}\, a x +3 a^{2} x^{2}-4 \arcsin \left (a x \right )^{2}-2 i \arcsin \left (a x \right )-2\right )}{3 c^{3} \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) a}+\frac {2 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (a x \right ) \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (a x \right )^{2}+\polylog \left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{3 a \,c^{3} \left (a^{2} x^{2}-1\right )}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(-c*(a^2*x^2-1))^(1/2)*(2*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+2*a^3*x^3-2*I*(-a^2*x^2+1)^(1/2)-3*a*x)*(-2*I*arcs
in(a*x)*a^4*x^4-2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^3*x^3+I*(-a^2*x^2+1)^(1/2)*a^3*x^3-a^4*x^4+3*arcsin(a*x)^2*
a^2*x^2+4*I*arcsin(a*x)*a^2*x^2+3*a*x*arcsin(a*x)*(-a^2*x^2+1)^(1/2)-I*(-a^2*x^2+1)^(1/2)*a*x+3*a^2*x^2-4*arcs
in(a*x)^2-2*I*arcsin(a*x)-2)/c^3/(3*a^6*x^6-10*a^4*x^4+11*a^2*x^2-4)/a+2/3*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1
))^(1/2)*(2*I*arcsin(a*x)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)+2*arcsin(a*x)^2+polylog(2,-(I*a*x+(-a^2*x^2+1)^(1
/2))^2))/a/c^3/(a^2*x^2-1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(5/2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^2/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**2/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(asin(a*x)**2/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^2/(c - a^2*c*x^2)^(5/2),x)

[Out]

int(asin(a*x)^2/(c - a^2*c*x^2)^(5/2), x)

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