Optimal. Leaf size=178 \[ -i b d \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} b^2 d \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac{1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+d \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} b^2 c^2 d x^2 \]
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Rubi [A] time = 0.238255, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4699, 4625, 3717, 2190, 2531, 2282, 6589, 4647, 4641, 30} \[ -i b d \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} b^2 d \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac{1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+d \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} b^2 c^2 d x^2 \]
Antiderivative was successfully verified.
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Rule 4699
Rule 4625
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 4647
Rule 4641
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx &=\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-(b c d) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+d \operatorname{Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac{1}{2} (b c d) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{2} \left (b^2 c^2 d\right ) \int x \, dx\\ &=\frac{1}{4} b^2 c^2 d x^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-(2 i d) \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^2 d x^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-(2 b d) \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^2 d x^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (i b^2 d\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^2 d x^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} \left (b^2 d\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^2 d x^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{4} d \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{2} d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{i d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} b^2 d \text{Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.452689, size = 236, normalized size = 1.33 \[ \frac{1}{2} d \left (-2 i a b \left (\sin ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )\right )+\frac{1}{12} b^2 \left (24 i \sin ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(c x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(c x)}\right )+8 i \sin ^{-1}(c x)^3+24 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )-i \pi ^3\right )+a^2 \left (-c^2\right ) x^2+2 a^2 \log (x)-2 a b c^2 x^2 \sin ^{-1}(c x)+a b \left (\sin ^{-1}(c x)-c x \sqrt{1-c^2 x^2}\right )+4 a b \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{2} b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )+\frac{1}{4} b^2 \left (2 \sin ^{-1}(c x)^2-1\right ) \cos \left (2 \sin ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.229, size = 459, normalized size = 2.6 \begin{align*} -{\frac{d{a}^{2}{c}^{2}{x}^{2}}{2}}+d{a}^{2}\ln \left ( cx \right ) -2\,id{b}^{2}\arcsin \left ( cx \right ){\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{d{b}^{2}\arcsin \left ( cx \right ) cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2}}{2}}+{\frac{d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{4}}+{\frac{{b}^{2}{c}^{2}d{x}^{2}}{4}}-{\frac{d{b}^{2}}{8}}+d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{i}{3}}d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{3}+2\,d{b}^{2}{\it polylog} \left ( 3,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +d{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{2}\ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,idab{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,d{b}^{2}{\it polylog} \left ( 3,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,idab{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -{\frac{dabcx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}-dab\arcsin \left ( cx \right ){c}^{2}{x}^{2}+{\frac{dab\arcsin \left ( cx \right ) }{2}}+2\,dab\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,dab\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,id{b}^{2}\arcsin \left ( cx \right ){\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -idab \left ( \arcsin \left ( cx \right ) \right ) ^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a^{2} c^{2} d x^{2} + a^{2} d \log \left (x\right ) - \int \frac{{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - d \left (\int - \frac{a^{2}}{x}\, dx + \int a^{2} c^{2} x\, dx + \int - \frac{b^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x}\, dx + \int - \frac{2 a b \operatorname{asin}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{2} x \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x \operatorname{asin}{\left (c x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (c^{2} d x^{2} - d\right )}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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