3.204 \(\int \frac{x (a+b \sin ^{-1}(c x))^2}{(d-c^2 d x^2)^3} \, dx\)

Optimal. Leaf size=150 \[ -\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^3 \sqrt{1-c^2 x^2}}-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac{b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3} \]

[Out]

b^2/(12*c^2*d^3*(1 - c^2*x^2)) - (b*x*(a + b*ArcSin[c*x]))/(6*c*d^3*(1 - c^2*x^2)^(3/2)) - (b*x*(a + b*ArcSin[
c*x]))/(3*c*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])^2/(4*c^2*d^3*(1 - c^2*x^2)^2) - (b^2*Log[1 - c^2*x^2]
)/(6*c^2*d^3)

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Rubi [A]  time = 0.137028, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4677, 4655, 4651, 260, 261} \[ -\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^3 \sqrt{1-c^2 x^2}}-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac{b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]

[Out]

b^2/(12*c^2*d^3*(1 - c^2*x^2)) - (b*x*(a + b*ArcSin[c*x]))/(6*c*d^3*(1 - c^2*x^2)^(3/2)) - (b*x*(a + b*ArcSin[
c*x]))/(3*c*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])^2/(4*c^2*d^3*(1 - c^2*x^2)^2) - (b^2*Log[1 - c^2*x^2]
)/(6*c^2*d^3)

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 4655

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 - c^2*x^2)^FracPart[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 4651

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)/Sqrt[d], Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(d + e*x^2), x], x
] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]

Rule 260

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

Rule 261

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]

Rubi steps

\begin{align*} \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}\\ &=-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac{b^2 \int \frac{x}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^3}-\frac{b \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 c d^3}\\ &=\frac{b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac{b^2 \int \frac{x}{1-c^2 x^2} \, dx}{3 d^3}\\ &=\frac{b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{b x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac{b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.197349, size = 162, normalized size = 1.08 \[ \frac{3 a^2+4 a b c^3 x^3 \sqrt{1-c^2 x^2}-6 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left (3 a+b c x \sqrt{1-c^2 x^2} \left (2 c^2 x^2-3\right )\right )-b^2 c^2 x^2-2 b^2 \left (c^2 x^2-1\right )^2 \log \left (1-c^2 x^2\right )+3 b^2 \sin ^{-1}(c x)^2+b^2}{12 c^2 d^3 \left (c^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2 + b^2 - b^2*c^2*x^2 - 6*a*b*c*x*Sqrt[1 - c^2*x^2] + 4*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*(3*a + b*c*x*
Sqrt[1 - c^2*x^2]*(-3 + 2*c^2*x^2))*ArcSin[c*x] + 3*b^2*ArcSin[c*x]^2 - 2*b^2*(-1 + c^2*x^2)^2*Log[1 - c^2*x^2
])/(12*c^2*d^3*(-1 + c^2*x^2)^2)

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Maple [B]  time = 0.036, size = 335, normalized size = 2.2 \begin{align*}{\frac{{a}^{2}}{4\,{c}^{2}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}}+{\frac{{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{4\,{c}^{2}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}}-{\frac{{b}^{2}\arcsin \left ( cx \right ) x}{6\,c{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{b}^{2}}{12\,{c}^{2}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2}\arcsin \left ( cx \right ) x}{3\,c{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{b}^{2}\ln \left ( -{c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{2}{d}^{3}}}+{\frac{ab\arcsin \left ( cx \right ) }{2\,{c}^{2}{d}^{3} \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}}+{\frac{ab}{6\,{c}^{2}{d}^{3} \left ( cx-1 \right ) }\sqrt{- \left ( cx-1 \right ) ^{2}-2\,cx+2}}+{\frac{ab}{6\,{c}^{2}{d}^{3} \left ( cx+1 \right ) }\sqrt{- \left ( cx+1 \right ) ^{2}+2\,cx+2}}-{\frac{ab}{24\,{c}^{2}{d}^{3} \left ( cx-1 \right ) ^{2}}\sqrt{- \left ( cx-1 \right ) ^{2}-2\,cx+2}}+{\frac{ab}{24\,{c}^{2}{d}^{3} \left ( cx+1 \right ) ^{2}}\sqrt{- \left ( cx+1 \right ) ^{2}+2\,cx+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^3,x)

[Out]

1/4/c^2*a^2/d^3/(c^2*x^2-1)^2+1/4/c^2*b^2/d^3*arcsin(c*x)^2/(c^2*x^2-1)^2-1/6/c*b^2/d^3*arcsin(c*x)*(-c^2*x^2+
1)^(1/2)/(c^2*x^2-1)^2*x-1/12/c^2*b^2/d^3/(c^2*x^2-1)+1/3/c*b^2/d^3*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)
*x-1/6*b^2*ln(-c^2*x^2+1)/c^2/d^3+1/2/c^2*a*b/d^3/(c^2*x^2-1)^2*arcsin(c*x)+1/6/c^2*a*b/d^3/(c*x-1)*(-(c*x-1)^
2-2*c*x+2)^(1/2)+1/6/c^2*a*b/d^3/(c*x+1)*(-(c*x+1)^2+2*c*x+2)^(1/2)-1/24/c^2*a*b/d^3/(c*x-1)^2*(-(c*x-1)^2-2*c
*x+2)^(1/2)+1/24/c^2*a*b/d^3/(c*x+1)^2*(-(c*x+1)^2+2*c*x+2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2}}{4 \,{\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} + \frac{b^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} - 2 \,{\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )} \int \frac{4 \, a b c x \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) - \sqrt{c x + 1} \sqrt{-c x + 1} b^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{c^{7} d^{3} x^{6} - 3 \, c^{5} d^{3} x^{4} + 3 \, c^{3} d^{3} x^{2} - c d^{3}}\,{d x}}{4 \,{\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) + 1/4*(b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 4*(c
^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)*integrate(-1/2*(4*a*b*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - s
qrt(c*x + 1)*sqrt(-c*x + 1)*b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^7*d^3*x^6 - 3*c^5*d^3*x^4 + 3*c
^3*d^3*x^2 - c*d^3), x))/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)

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Fricas [A]  time = 2.71602, size = 360, normalized size = 2.4 \begin{align*} -\frac{b^{2} c^{2} x^{2} - 3 \, b^{2} \arcsin \left (c x\right )^{2} - 6 \, a b \arcsin \left (c x\right ) - 3 \, a^{2} - b^{2} + 2 \,{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) - 2 \,{\left (2 \, a b c^{3} x^{3} - 3 \, a b c x +{\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{12 \,{\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(b^2*c^2*x^2 - 3*b^2*arcsin(c*x)^2 - 6*a*b*arcsin(c*x) - 3*a^2 - b^2 + 2*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 +
b^2)*log(c^2*x^2 - 1) - 2*(2*a*b*c^3*x^3 - 3*a*b*c*x + (2*b^2*c^3*x^3 - 3*b^2*c*x)*arcsin(c*x))*sqrt(-c^2*x^2
+ 1))/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a^{2} x}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac{b^{2} x \operatorname{asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac{2 a b x \operatorname{asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(a**2*x/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(b**2*x*asin(c*x)**2/(c**6*x**6 -
3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(2*a*b*x*asin(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x
))/d**3

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Giac [B]  time = 1.46452, size = 533, normalized size = 3.55 \begin{align*} \frac{b^{2} c^{2} x^{4} \arcsin \left (c x\right )^{2}}{4 \,{\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac{a b c^{2} x^{4} \arcsin \left (c x\right )}{2 \,{\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac{a^{2} c^{2} x^{4}}{4 \,{\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac{b^{2} c x^{3} \arcsin \left (c x\right )}{6 \,{\left (c^{2} x^{2} - 1\right )} \sqrt{-c^{2} x^{2} + 1} d^{3}} - \frac{b^{2} x^{2} \arcsin \left (c x\right )^{2}}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{3}} + \frac{a b c x^{3}}{6 \,{\left (c^{2} x^{2} - 1\right )} \sqrt{-c^{2} x^{2} + 1} d^{3}} - \frac{a b x^{2} \arcsin \left (c x\right )}{{\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac{a^{2} x^{2}}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac{b^{2} x^{2}}{12 \,{\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac{b^{2} x \arcsin \left (c x\right )}{2 \, \sqrt{-c^{2} x^{2} + 1} c d^{3}} + \frac{b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{2} d^{3}} - \frac{a b x}{2 \, \sqrt{-c^{2} x^{2} + 1} c d^{3}} + \frac{a b \arcsin \left (c x\right )}{2 \, c^{2} d^{3}} - \frac{b^{2} \log \left (2\right )}{3 \, c^{2} d^{3}} - \frac{b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{6 \, c^{2} d^{3}} + \frac{a^{2}}{4 \, c^{2} d^{3}} + \frac{b^{2}}{12 \, c^{2} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*c^2*x^4*arcsin(c*x)^2/((c^2*x^2 - 1)^2*d^3) + 1/2*a*b*c^2*x^4*arcsin(c*x)/((c^2*x^2 - 1)^2*d^3) + 1/4*
a^2*c^2*x^4/((c^2*x^2 - 1)^2*d^3) + 1/6*b^2*c*x^3*arcsin(c*x)/((c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*d^3) - 1/2*b^2
*x^2*arcsin(c*x)^2/((c^2*x^2 - 1)*d^3) + 1/6*a*b*c*x^3/((c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*d^3) - a*b*x^2*arcsin
(c*x)/((c^2*x^2 - 1)*d^3) - 1/2*a^2*x^2/((c^2*x^2 - 1)*d^3) - 1/12*b^2*x^2/((c^2*x^2 - 1)*d^3) - 1/2*b^2*x*arc
sin(c*x)/(sqrt(-c^2*x^2 + 1)*c*d^3) + 1/4*b^2*arcsin(c*x)^2/(c^2*d^3) - 1/2*a*b*x/(sqrt(-c^2*x^2 + 1)*c*d^3) +
1/2*a*b*arcsin(c*x)/(c^2*d^3) - 1/3*b^2*log(2)/(c^2*d^3) - 1/6*b^2*log(abs(-c^2*x^2 + 1))/(c^2*d^3) + 1/4*a^2
/(c^2*d^3) + 1/12*b^2/(c^2*d^3)