Optimal. Leaf size=132 \[ \frac{1}{2} i b d \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+d \log (x) \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2-b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b d \log (x) \sin ^{-1}(c x) \]
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Rubi [A] time = 0.239003, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632, Rules used = {14, 4732, 12, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac{1}{2} i b d \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+d \log (x) \left (a+b \cos ^{-1}(c x)\right )+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2-b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b d \log (x) \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 4732
Rule 12
Rule 6742
Rule 321
Rule 216
Rule 2326
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{x} \, dx &=\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+(b c) \int \frac{e x^2+2 d \log (x)}{2 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+\frac{1}{2} (b c) \int \frac{e x^2+2 d \log (x)}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+\frac{1}{2} (b c) \int \left (\frac{e x^2}{\sqrt{1-c^2 x^2}}+\frac{2 d \log (x)}{\sqrt{1-c^2 x^2}}\right ) \, dx\\ &=\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+(b c d) \int \frac{\log (x)}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{2} (b c e) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+b d \sin ^{-1}(c x) \log (x)-(b d) \int \frac{\sin ^{-1}(c x)}{x} \, dx+\frac{(b e) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 c}\\ &=-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+\frac{b e \sin ^{-1}(c x)}{4 c^2}+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+b d \sin ^{-1}(c x) \log (x)-(b d) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+b d \sin ^{-1}(c x) \log (x)+(2 i b d) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2-b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+b d \sin ^{-1}(c x) \log (x)+(b d) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2-b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+b d \sin ^{-1}(c x) \log (x)-\frac{1}{2} (i b d) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{1}{2} e x^2 \left (a+b \cos ^{-1}(c x)\right )+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2-b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+d \left (a+b \cos ^{-1}(c x)\right ) \log (x)+b d \sin ^{-1}(c x) \log (x)+\frac{1}{2} i b d \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.193091, size = 111, normalized size = 0.84 \[ \frac{1}{4} \left (-2 i b d \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )+4 a d \log (x)+2 a e x^2-\frac{b e x \sqrt{1-c^2 x^2}}{c}+\frac{b e \sin ^{-1}(c x)}{c^2}+2 b \cos ^{-1}(c x) \left (e x^2+2 d \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )\right )-2 i b d \cos ^{-1}(c x)^2\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.144, size = 130, normalized size = 1. \begin{align*}{\frac{a{x}^{2}e}{2}}+ad\ln \left ( cx \right ) -{\frac{i}{2}}b \left ( \arccos \left ( cx \right ) \right ) ^{2}d-{\frac{bex}{4\,c}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arccos \left ( cx \right ){x}^{2}e}{2}}-{\frac{b\arccos \left ( cx \right ) e}{4\,{c}^{2}}}+bd\arccos \left ( cx \right ) \ln \left ( 1+ \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) -{\frac{i}{2}}bd{\it polylog} \left ( 2,- \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) \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 e x^{2} + a d \log \left (x\right ) + \int \frac{{\left (b e x^{2} + b d\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\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 e x^{2} + a d +{\left (b e x^{2} + b d\right )} \arccos \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*} \int \frac{\left (a + b \operatorname{acos}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \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 (e x^{2} + d\right )}{\left (b \arccos \left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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