Optimal. Leaf size=220 \[ \frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}} \]
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Rubi [A] time = 0.31, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {4705, 4713, 4709, 4183, 2279, 2391, 206} \[ \frac {i b \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2279
Rule 2391
Rule 4183
Rule 4705
Rule 4709
Rule 4713
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx}{d}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \sin ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 1.16, size = 300, normalized size = 1.36 \[ \frac {-\frac {a \sqrt {d-c^2 d x^2}}{c^2 x^2-1}-a \sqrt {d} \log \left (\sqrt {d} \sqrt {d-c^2 d x^2}+d\right )+a \sqrt {d} \log (x)+\frac {b d \left (i \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-i \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )+\sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-\sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )+\sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )-\sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+\sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}}{d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 344, normalized size = 1.56 \[ \frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \dilog \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{2} \left (c^{2} x^{2}-1\right )} \]
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 \[ -a {\left (\frac {\log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {1}{\sqrt {-c^{2} d x^{2} + d} d}\right )} - \frac {\frac {b \int \frac {\arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} \sqrt {-c x + 1} x}\,{d x}}{d}}{\sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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