3.3.73 \(\int \frac {\text {ArcSin}(a x)^2}{(c-a^2 c x^2)^{3/2}} \, dx\) [273]

Optimal. Leaf size=179 \[ \frac {x \text {ArcSin}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4745, 4765, 3800, 2221, 2317, 2438} \begin {gather*} -\frac {i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {x \text {ArcSin}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )}{a c \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)^(3/2),x]

[Out]

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

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 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]

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {\left (2 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int x \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (4 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 )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {\left (2 \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 )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (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 )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sin ^{-1}(a x)^2}{c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 108, normalized size = 0.60 \begin {gather*} \frac {\text {ArcSin}(a x) \left (a x \text {ArcSin}(a x)+\sqrt {1-a^2 x^2} \left (-i \text {ArcSin}(a x)+2 \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )\right )\right )-i \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(a x)}\right )}{a c \sqrt {c \left (1-a^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.17, size = 169, normalized size = 0.94

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}+a x \right ) \arcsin \left (a x \right )^{2}}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (a x \right ) \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (a x \right )^{2}+\polylog \left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}\)	\(169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-c*(a^2*x^2-1))^(1/2)*(I*(-a^2*x^2+1)^(1/2)+a*x)*arcsin(a*x)^2/c^2/a/(a^2*x^2-1)+I*(-a^2*x^2+1)^(1/2)*(-c*(a
^2*x^2-1))^(1/2)/c^2/a/(a^2*x^2-1)*(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))

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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)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Integral(asin(a*x)**2/(-c*(a*x - 1)*(a*x + 1))**(3/2), 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)^(3/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^2/(-a^2*c*x^2 + c)^(3/2), x)

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

[Out]

int(asin(a*x)^2/(c - a^2*c*x^2)^(3/2), x)

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