Optimal. Leaf size=287 \[ 2 i b c^2 d^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-2 c^2 d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{4} b^2 c^4 d^2 x^2-b^2 c^2 d^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x) \]
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Rubi [A] time = 0.47, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4695, 4699, 4625, 3717, 2190, 2531, 2282, 6589, 4647, 4641, 30, 14} \[ 2 i b c^2 d^2 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-b^2 c^2 d^2 \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-2 c^2 d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{4} b^2 c^4 d^2 x^2+b^2 c^2 d^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 4641
Rule 4647
Rule 4695
Rule 4699
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (2 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b c d^2\right ) \int \frac {\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx\\ &=-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (2 c^2 d^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b^2 c^2 d^2\right ) \int \frac {1-c^2 x^2}{x} \, dx+\left (2 b c^3 d^2\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (3 b c^3 d^2\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (2 c^2 d^2\right ) \operatorname {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\left (b^2 c^2 d^2\right ) \int \left (\frac {1}{x}-c^2 x\right ) \, dx+\left (b c^3 d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} \left (3 b c^3 d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\left (b^2 c^4 d^2\right ) \int x \, dx+\frac {1}{2} \left (3 b^2 c^4 d^2\right ) \int x \, dx\\ &=-\frac {1}{4} b^2 c^4 d^2 x^2-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+b^2 c^2 d^2 \log (x)+\left (4 i c^2 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {1}{4} b^2 c^4 d^2 x^2-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+\left (4 b c^2 d^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {1}{4} b^2 c^4 d^2 x^2-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+2 i b c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\left (2 i b^2 c^2 d^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {1}{4} b^2 c^4 d^2 x^2-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+2 i b c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\left (b^2 c^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {1}{4} b^2 c^4 d^2 x^2-\frac {1}{2} b c^3 d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+2 i b c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-b^2 c^2 d^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.86, size = 343, normalized size = 1.20 \[ \frac {1}{2} d^2 \left (a^2 c^4 x^2-4 a^2 c^2 \log (x)-\frac {a^2}{x^2}+2 a b c^4 x^2 \sin ^{-1}(c x)+4 i a b c^2 \left (\sin ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\right )+a b c^2 \left (c x \sqrt {1-c^2 x^2}-\sin ^{-1}(c x)\right )-\frac {2 a b \left (c x \sqrt {1-c^2 x^2}+\sin ^{-1}(c x)\right )}{x^2}-8 a b c^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{6} i b^2 c^2 \left (-24 \sin ^{-1}(c x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )+12 i \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )-8 \sin ^{-1}(c x)^3+24 i \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+\pi ^3\right )-\frac {b^2 \left (-2 c^2 x^2 \log (c x)+2 c x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)+\sin ^{-1}(c x)^2\right )}{x^2}+\frac {1}{2} b^2 c^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )-\frac {1}{4} b^2 c^2 \left (2 \sin ^{-1}(c x)^2-1\right ) \cos \left (2 \sin ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.84, size = 767, normalized size = 2.67 \[ 4 i c^{2} d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+4 i c^{2} d^{2} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i c^{2} d^{2} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+4 i c^{2} d^{2} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i c^{2} d^{2} a b \arcsin \left (c x \right )^{2}-4 c^{2} d^{2} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 c^{2} d^{2} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {c^{3} d^{2} a b \sqrt {-c^{2} x^{2}+1}\, x}{2}+c^{4} d^{2} a b \arcsin \left (c x \right ) x^{2}-\frac {c \,d^{2} a b \sqrt {-c^{2} x^{2}+1}}{x}-\frac {c \,d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{x}+\frac {c^{3} d^{2} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x}{2}-\frac {c^{2} d^{2} a b \arcsin \left (c x \right )}{2}+\frac {c^{4} d^{2} b^{2} \arcsin \left (c x \right )^{2} x^{2}}{2}+i c^{2} d^{2} b^{2} \arcsin \left (c x \right )-2 c^{2} d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c^{2} d^{2} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {2 i c^{2} d^{2} b^{2} \arcsin \left (c x \right )^{3}}{3}+i c^{2} d^{2} a b -\frac {d^{2} a^{2}}{2 x^{2}}+\frac {d^{2} b^{2} c^{2}}{8}-\frac {b^{2} c^{4} d^{2} x^{2}}{4}-\frac {d^{2} a b \arcsin \left (c x \right )}{x^{2}}+\frac {c^{4} d^{2} a^{2} x^{2}}{2}-2 c^{2} d^{2} a^{2} \ln \left (c x \right )-\frac {c^{2} d^{2} b^{2} \arcsin \left (c x \right )^{2}}{4}-\frac {d^{2} b^{2} \arcsin \left (c x \right )^{2}}{2 x^{2}}+c^{2} d^{2} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+c^{2} d^{2} b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c^{2} d^{2} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 c^{2} d^{2} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 c^{2} d^{2} b^{2} \polylog \left (3, i c x +\sqrt {-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 \[ \frac {1}{2} \, a^{2} c^{4} d^{2} x^{2} - 2 \, a^{2} c^{2} d^{2} \log \relax (x) - a b d^{2} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c}{x} + \frac {\arcsin \left (c x\right )}{x^{2}}\right )} - \frac {a^{2} d^{2}}{2 \, x^{2}} + \int \frac {{\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2}{x^3} \,d x \]
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 \[ d^{2} \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \left (- \frac {2 a^{2} c^{2}}{x}\right )\, dx + \int a^{2} c^{4} x\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\, dx + \int \left (- \frac {2 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\right )\, dx + \int b^{2} c^{4} x \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- \frac {4 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int 2 a b c^{4} x \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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