3.100 \(\int \sqrt{\pi -c^2 \pi x^2} (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=68 \[ \frac{1}{2} x \sqrt{\pi -\pi c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi } \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c}-\frac{1}{4} \sqrt{\pi } b c x^2 \]

[Out]

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

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Rubi [A]  time = 0.0585542, antiderivative size = 116, normalized size of antiderivative = 1.71, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4647, 4641, 30} \[ \frac{1}{2} x \sqrt{\pi -\pi c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi -\pi c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{b c x^2 \sqrt{\pi -\pi c^2 x^2}}{4 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4647

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(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]

Rule 4641

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{2} x \sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi -c^2 \pi x^2} \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}-\frac{\left (b c \sqrt{\pi -c^2 \pi x^2}\right ) \int x \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c x^2 \sqrt{\pi -c^2 \pi x^2}}{4 \sqrt{1-c^2 x^2}}+\frac{1}{2} x \sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{\sqrt{\pi -c^2 \pi x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0508784, size = 87, normalized size = 1.28 \[ \frac{\sqrt{\pi } \left (a^2+2 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left (a+b c x \sqrt{1-c^2 x^2}\right )-b^2 c^2 x^2+b^2 \sin ^{-1}(c x)^2\right )}{4 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 101, normalized size = 1.5 \begin{align*}{\frac{ax}{2}\sqrt{-\pi \,{c}^{2}{x}^{2}+\pi }}+{\frac{a\pi }{2}\arctan \left ({x\sqrt{\pi \,{c}^{2}}{\frac{1}{\sqrt{-\pi \,{c}^{2}{x}^{2}+\pi }}}} \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{b\sqrt{\pi }\arcsin \left ( cx \right ) x}{2}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{bc{x}^{2}\sqrt{\pi }}{4}}+{\frac{b\sqrt{\pi } \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{4\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))*(-Pi*c^2*x^2+Pi)^(1/2),x)

[Out]

1/2*a*x*(-Pi*c^2*x^2+Pi)^(1/2)+1/2*a*Pi/(Pi*c^2)^(1/2)*arctan((Pi*c^2)^(1/2)*x/(-Pi*c^2*x^2+Pi)^(1/2))+1/2*b*P
i^(1/2)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x-1/4*b*c*x^2*Pi^(1/2)+1/4*b*Pi^(1/2)/c*arcsin(c*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))*(-pi*c^2*x^2+pi)^(1/2),x, algorithm="maxima")

[Out]

sqrt(pi)*b*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 1/2*(sqrt(p
i - pi*c^2*x^2)*x + pi*arcsin(c^2*x/sqrt(c^2))/sqrt(pi*c^2))*a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))*(-pi*c^2*x^2+pi)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(pi - pi*c^2*x^2)*(b*arcsin(c*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))*(-pi*c**2*x**2+pi)**(1/2),x)

[Out]

sqrt(pi)*(Integral(a*sqrt(-c**2*x**2 + 1), x) + Integral(b*sqrt(-c**2*x**2 + 1)*asin(c*x), x))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))*(-pi*c^2*x^2+pi)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError