Optimal. Leaf size=66 \[ -\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{d x}+\frac {b c \sqrt {1-c^2 x^2} \log (x)}{\sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {4771, 29}
\begin {gather*} \frac {b c \sqrt {1-c^2 x^2} \log (x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 4771
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{x^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 )}{d x}+\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}+\frac {b c \sqrt {1-c^2 x^2} \log (x)}{\sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 69, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{d x}+\frac {b c \sqrt {d-c^2 d x^2} \log (x)}{d \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.21, size = 216, normalized size = 3.27
method | result | size |
default | \(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{d x}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c}{d \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x \,c^{2}}{d \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d x \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d \left (c^{2} x^{2}-1\right )}\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 104, normalized size = 1.58 \begin {gather*} -\frac {{\left (\left (-1\right )^{-2 \, c^{2} d x^{2} + 2 \, d} \sqrt {d} \log \left (-2 \, c^{2} d + \frac {2 \, d}{x^{2}}\right ) + \sqrt {d} \log \left (x^{2} - \frac {1}{c^{2}}\right )\right )} b c}{2 \, d} - \frac {\sqrt {-c^{2} d x^{2} + d} b \arcsin \left (c x\right )}{d x} - \frac {\sqrt {-c^{2} d x^{2} + d} a}{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.13, size = 218, normalized size = 3.30 \begin {gather*} \left [\frac {b c \sqrt {d} x \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) - 2 \, \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{2 \, d x}, \frac {b c \sqrt {-d} x \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{d x}\right ] \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 {a + b \operatorname {asin}{\left (c x \right )}}{x^{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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {a+b\,\mathrm {asin}\left (c\,x\right )}{x^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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