3.162 \(\int \frac{(d-c^2 d x^2) (a+b \sin ^{-1}(c x))^2}{x^2} \, dx\)

Optimal. Leaf size=149 \[ 2 i b^2 c d \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-2 b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-2 c^2 d x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+2 b^2 c^2 d x \]

[Out]

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

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Rubi [A]  time = 0.297993, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4695, 4619, 4677, 8, 4697, 4709, 4183, 2279, 2391} \[ 2 i b^2 c d \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 i b^2 c d \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )-2 b c d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x}-2 c^2 d x \left (a+b \sin ^{-1}(c x)\right )^2-4 b c d \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+2 b^2 c^2 d x \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^2,x]

[Out]

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

Rule 4695

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

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]

Rule 4697

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

Rule 4709

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

Rule 4183

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

Rule 2279

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.410173, size = 203, normalized size = 1.36 \[ -\frac{d \left (-i b^2 \left (2 c x \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-2 c x \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+i \sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 c x \left (\log \left (1+e^{i \sin ^{-1}(c x)}\right )-\log \left (1-e^{i \sin ^{-1}(c x)}\right )\right )\right )\right )+a^2 c^2 x^2+a^2+2 a b c x \left (\sqrt{1-c^2 x^2}+c x \sin ^{-1}(c x)\right )+2 a b \left (c x \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\sin ^{-1}(c x)\right )+b^2 c x \left (2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x \left (\sin ^{-1}(c x)^2-2\right )\right )\right )}{x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^2,x]

[Out]

-((d*(a^2 + a^2*c^2*x^2 + 2*a*b*c*x*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) + b^2*c*x*(2*Sqrt[1 - c^2*x^2]*ArcSi
n[c*x] + c*x*(-2 + ArcSin[c*x]^2)) + 2*a*b*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) - I*b^2*(I*ArcSin[c*
x]*(ArcSin[c*x] + 2*c*x*(-Log[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c*x])])) + 2*c*x*PolyLog[2, -E^(I*A
rcSin[c*x])] - 2*c*x*PolyLog[2, E^(I*ArcSin[c*x])])))/x)

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Maple [A]  time = 0.221, size = 269, normalized size = 1.8 \begin{align*} -d{a}^{2}{c}^{2}x-{\frac{d{a}^{2}}{x}}-2\,cd{b}^{2}\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1}-d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{2}x+2\,{b}^{2}{c}^{2}dx-{\frac{d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{x}}-2\,cd{b}^{2}\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,cd{b}^{2}\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,i{b}^{2}cd{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,i{b}^{2}cd{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,dab{c}^{2}x\arcsin \left ( cx \right ) -2\,{\frac{dab\arcsin \left ( cx \right ) }{x}}-2\,cdab\sqrt{-{c}^{2}{x}^{2}+1}-2\,cdab{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^2/x^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -b^{2} c^{2} d x \arcsin \left (c x\right )^{2} + 2 \, b^{2} c^{2} d{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} - a^{2} c^{2} d x - 2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b c d - 2 \,{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} a b d - \frac{{\left (2 \, c x \int \frac{\sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{\sqrt{c x + 1}{\left (c x - 1\right )} x}\,{d x} + \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2}\right )} b^{2} d}{x} - \frac{a^{2} d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcsin(c*
x))/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*(Integral(a**2*c**2, x) + Integral(-a**2/x**2, x) + Integral(b**2*c**2*asin(c*x)**2, x) + Integral(-b**2*as
in(c*x)**2/x**2, x) + Integral(2*a*b*c**2*asin(c*x), x) + Integral(-2*a*b*asin(c*x)/x**2, x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (c^{2} d x^{2} - d\right )}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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