Optimal. Leaf size=146 \[ \frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{c^2 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4767, 4715,
267} \begin {gather*} -\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{c^2 d}+\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4715
Rule 4767
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c \sqrt {d-c^2 d x^2}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c \sqrt {d-c^2 d x^2}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {2 a b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 86, normalized size = 0.59 \begin {gather*} \frac {\left (-1+c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+2 b \sqrt {1-c^2 x^2} \left (a c x+b \sqrt {1-c^2 x^2}+b c x \text {ArcSin}(c x)\right )}{c^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 316, normalized size = 2.16
method | result | size |
default | \(-\frac {a^{2} \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(316\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 130, normalized size = 0.89 \begin {gather*} 2 \, b^{2} {\left (\frac {x \arcsin \left (c x\right )}{c \sqrt {d}} + \frac {\sqrt {-c^{2} x^{2} + 1}}{c^{2} \sqrt {d}}\right )} + \frac {2 \, a b x}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} \arcsin \left (c x\right )^{2}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b \arcsin \left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2}}{c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.41, size = 147, normalized size = 1.01 \begin {gather*} -\frac {2 \, \sqrt {-c^{2} d x^{2} + d} {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1} + {\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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.
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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 = -0.01 \begin {gather*} \int \frac {x\,{\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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