Optimal. Leaf size=49 \[ \frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {4737}
\begin {gather*} \frac {\sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4737
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 64, normalized size = 1.31 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \text {ArcSin}(c x) \left (3 a^2+3 a b \text {ArcSin}(c x)+b^2 \text {ArcSin}(c x)^2\right )}{3 c \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs.
\(2(43)=86\).
time = 0.08, size = 143, normalized size = 2.92
method | result | size |
default | \(\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{c d \left (c^{2} x^{2}-1\right )}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 47, normalized size = 0.96 \begin {gather*} \frac {b^{2} \arcsin \left (c x\right )^{3}}{3 \, c \sqrt {d}} + \frac {a b \arcsin \left (c x\right )^{2}}{c \sqrt {d}} + \frac {a^{2} \arcsin \left (c x\right )}{c \sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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