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

Optimal. Leaf size=57 \[ \frac{a+b \sin ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}} \]

[Out]

-(b*x)/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(2*c^2*d^2*(1 - c^2*x^2))

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Rubi [A]  time = 0.047586, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4677, 191} \[ \frac{a+b \sin ^{-1}(c x)}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*x)/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(2*c^2*d^2*(1 - c^2*x^2))

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 191

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]

Rubi steps

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

Mathematica [A]  time = 0.047155, size = 50, normalized size = 0.88 \[ \frac{a-b c x \sqrt{1-c^2 x^2}+b \sin ^{-1}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 98, normalized size = 1.7 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{\frac{a}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b}{{d}^{2}} \left ( -{\frac{\arcsin \left ( cx \right ) }{2\,{c}^{2}{x}^{2}-2}}+{\frac{1}{4\,cx-4}\sqrt{- \left ( cx-1 \right ) ^{2}-2\,cx+2}}+{\frac{1}{4\,cx+4}\sqrt{- \left ( cx+1 \right ) ^{2}+2\,cx+2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(-1/2*a/d^2/(c^2*x^2-1)+b/d^2*(-1/2/(c^2*x^2-1)*arcsin(c*x)+1/4/(c*x-1)*(-(c*x-1)^2-2*c*x+2)^(1/2)+1/4/(
c*x+1)*(-(c*x+1)^2+2*c*x+2)^(1/2)))

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Maxima [B]  time = 1.74219, size = 225, normalized size = 3.95 \begin{align*} \frac{1}{4} \,{\left (\frac{{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} + \sqrt{c^{6} d^{4}} c^{4} d^{2} x} - \frac{\sqrt{-c^{2} x^{2} + 1} c^{2} d^{2}}{c^{6} d^{4} - \sqrt{c^{6} d^{4}} c^{4} d^{2} x}\right )} c^{5} d^{2}}{\sqrt{c^{6} d^{4}}} - \frac{2 \, \arcsin \left (c x\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} b - \frac{a}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((sqrt(-c^2*x^2 + 1)*c^2*d^2/(c^6*d^4 + sqrt(c^6*d^4)*c^4*d^2*x) - sqrt(-c^2*x^2 + 1)*c^2*d^2/(c^6*d^4 - s
qrt(c^6*d^4)*c^4*d^2*x))*c^5*d^2/sqrt(c^6*d^4) - 2*arcsin(c*x)/(c^4*d^2*x^2 - c^2*d^2))*b - 1/2*a/(c^4*d^2*x^2
- c^2*d^2)

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Fricas [A]  time = 2.04174, size = 115, normalized size = 2.02 \begin{align*} -\frac{a c^{2} x^{2} - \sqrt{-c^{2} x^{2} + 1} b c x + b \arcsin \left (c x\right )}{2 \,{\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.40863, size = 120, normalized size = 2.11 \begin{align*} -\frac{b x^{2} \arcsin \left (c x\right )}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac{a x^{2}}{2 \,{\left (c^{2} x^{2} - 1\right )} d^{2}} - \frac{b x}{2 \, \sqrt{-c^{2} x^{2} + 1} c d^{2}} + \frac{b \arcsin \left (c x\right )}{2 \, c^{2} d^{2}} + \frac{a}{2 \, c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^2*arcsin(c*x)/((c^2*x^2 - 1)*d^2) - 1/2*a*x^2/((c^2*x^2 - 1)*d^2) - 1/2*b*x/(sqrt(-c^2*x^2 + 1)*c*d^2
) + 1/2*b*arcsin(c*x)/(c^2*d^2) + 1/2*a/(c^2*d^2)