3.1.100 \(\int \sqrt {\pi -c^2 \pi x^2} (a+b \text {ArcSin}(c x)) \, dx\) [100]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {4741, 4737, 30} \begin {gather*} \frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \text {ArcSin}(c x))+\frac {\sqrt {\pi } (a+b \text {ArcSin}(c x))^2}{4 b c}-\frac {1}{4} \sqrt {\pi } b c x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(b*c*Sqrt[Pi]*x^2) + (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)

Rule 30

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

Rule 4737

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

Rule 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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]

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*}

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Mathematica [A]
time = 0.02, size = 87, normalized size = 1.28 \begin {gather*} \frac {\sqrt {\pi } \left (a^2-b^2 c^2 x^2+2 a b c x \sqrt {1-c^2 x^2}+2 b \left (a+b c x \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)+b^2 \text {ArcSin}(c x)^2\right )}{4 b c} \end {gather*}

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 = 0.08, size = 97, normalized size = 1.43

method	result	size
default	\(\frac {a x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \sqrt {\pi }\, \left (2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -c^{2} x^{2}+\arcsin \left (c x \right )^{2}\right )}{4 c}\)	\(97\)

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,method=_RETURNVERBOSE)

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 + sqrt(pi)*arcsin(c*x)/c)*a

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {\Pi -\Pi \,c^2\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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