Optimal. Leaf size=73 \[ \frac {a+b \sin ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {4677, 206} \[ \frac {a+b \sin ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 4677
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {a+b \sin ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \sin ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.70 \[ \frac {a-b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)+b \sin ^{-1}(c x)}{c^2 d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 279, normalized size = 3.82 \[ \left [\frac {{\left (b c^{2} x^{2} - b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{4 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}}, -\frac {{\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) + 2 \, \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{2 \, {\left (c^{4} d^{2} x^{2} - c^{2} d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 194, normalized size = 2.66 \[ \frac {a}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {1}{2} \, {\left (\sqrt {c x + 1} \sqrt {-c x + 1} c^{3} d^{2} {\left (\frac {2 \, x}{c^{2} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{3} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{3} d^{2}}\right )} + 2 \, \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} b}{\sqrt {c x + 1} \sqrt {-c x + 1} c^{2} d^{\frac {3}{2}}} + \frac {a}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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