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

Optimal. Leaf size=80 \[ \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0363379, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4653, 260} \[ \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 c d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4653

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[1 - c^2*x^2])/(d*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 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]

Rubi steps

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

Mathematica [A]  time = 0.202672, size = 77, normalized size = 0.96 \[ -\frac{\sqrt{d-c^2 d x^2} \left (2 a c x+b \sqrt{1-c^2 x^2} \log \left (c^2 x^2-1\right )+2 b c x \sin ^{-1}(c x)\right )}{2 c d^2 \left (c^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.086, size = 177, normalized size = 2.2 \begin{align*}{\frac{ax}{d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}}+{\frac{ib\arcsin \left ( cx \right ) }{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) x}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{c{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\ln \left ( 1+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.7018, size = 93, normalized size = 1.16 \begin{align*} -\frac{b c \sqrt{\frac{1}{c^{4} d}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{2 \, d} + \frac{b x \arcsin \left (c x\right )}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a x}{\sqrt{-c^{2} d x^{2} + d} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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