Optimal. Leaf size=210 \[ \frac {i b c^2 \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac {i b c^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {2 c^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d} \]
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Rubi [A] time = 0.38, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4701, 4679, 4419, 4183, 2531, 2282, 6589, 4681, 29} \[ \frac {i b c^2 \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac {i b c^2 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac {b^2 c^2 \text {PolyLog}\left (3,-e^{2 i \sin ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )}{2 d}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {2 c^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {b^2 c^2 \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2282
Rule 2531
Rule 4183
Rule 4419
Rule 4679
Rule 4681
Rule 4701
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+c^2 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {a+b \sin ^{-1}(c x)}{x^2 \sqrt {1-c^2 x^2}} \, dx}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {c^2 \operatorname {Subst}\left (\int (a+b x)^2 \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x} \, dx}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}+\frac {b^2 c^2 \log (x)}{d}+\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int (a+b x)^2 \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}-\frac {\left (2 b c^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (2 b c^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {\left (i b^2 c^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac {\left (i b^2 c^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d x^2}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.19, size = 353, normalized size = 1.68 \[ -\frac {a^2 c^2 \log \left (1-c^2 x^2\right )-2 a^2 c^2 \log (x)+\frac {a^2}{x^2}+2 a b c^2 \left (\frac {\sqrt {1-c^2 x^2}}{c x}+\frac {\sin ^{-1}(c x)}{c^2 x^2}-i \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )+i \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+2 \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )\right )+2 b^2 c^2 \left (\frac {\sin ^{-1}(c x)^2}{2 c^2 x^2}+\frac {\sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c x}-i \sin ^{-1}(c x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )-i \sin ^{-1}(c x) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )-\log (c x)-\frac {2}{3} i \sin ^{-1}(c x)^3-\sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x)^2 \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )+\frac {i \pi ^3}{24}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}{c^{2} d x^{5} - d x^{3}}, x\right ) \]
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 \[ \int -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 793, normalized size = 3.78 \[ \frac {i c^{2} a b}{d}+\frac {c^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {c^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i c^{2} b^{2} \arcsin \left (c x \right )}{d}-\frac {c^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {2 c^{2} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 c^{2} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i c^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i c^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i c^{2} a b \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {i c^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {2 i c^{2} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 i c^{2} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 c^{2} a b \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {a b \arcsin \left (c x \right )}{d \,x^{2}}-\frac {a^{2}}{2 d \,x^{2}}+\frac {c^{2} a^{2} \ln \left (c x \right )}{d}-\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 d \,x^{2}}-\frac {c^{2} a^{2} \ln \left (c x +1\right )}{2 d}-\frac {c^{2} a^{2} \ln \left (c x -1\right )}{2 d}+\frac {c^{2} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )}{d}+\frac {c^{2} b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {2 c^{2} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} c^{2} \polylog \left (3, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}+\frac {2 c^{2} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {2 c^{2} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {c \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{d x}-\frac {c a b \sqrt {-c^{2} x^{2}+1}}{d x} \]
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 \[ -\frac {1}{2} \, {\left (\frac {c^{2} \log \left (c x + 1\right )}{d} + \frac {c^{2} \log \left (c x - 1\right )}{d} - \frac {2 \, c^{2} \log \relax (x)}{d} + \frac {1}{d x^{2}}\right )} a^{2} - \int \frac {b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{2} d x^{5} - d x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3\,\left (d-c^2\,d\,x^2\right )} \,d x \]
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 \[ - \frac {\int \frac {a^{2}}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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