Optimal. Leaf size=132 \[ d \log (x) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {b e x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e \sin ^{-1}(c x)}{4 c^2}-\frac {1}{2} i b d \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} i b d \sin ^{-1}(c x)^2+b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d \log (x) \sin ^{-1}(c x) \]
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Rubi [A] time = 0.24, antiderivative size = 132, normalized size of antiderivative = 1.00, 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, 4731, 12, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac {1}{2} i b d \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+d \log (x) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {b e x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e \sin ^{-1}(c x)}{4 c^2}-\frac {1}{2} i b d \sin ^{-1}(c x)^2+b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d \log (x) \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 216
Rule 321
Rule 2190
Rule 2279
Rule 2326
Rule 2391
Rule 3717
Rule 4625
Rule 4731
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )+d \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )+d \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )+d \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )+d \left (a+b \sin ^{-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 \sin ^{-1}(c x)\right )-b d \sin ^{-1}(c x) \log (x)+d \left (a+b \sin ^{-1}(c x)\right ) \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 {b e \sin ^{-1}(c x)}{4 c^2}+\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )-b d \sin ^{-1}(c x) \log (x)+d \left (a+b \sin ^{-1}(c x)\right ) \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 {b e \sin ^{-1}(c x)}{4 c^2}-\frac {1}{2} i b d \sin ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )-b d \sin ^{-1}(c x) \log (x)+d \left (a+b \sin ^{-1}(c x)\right ) \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 {b e \sin ^{-1}(c x)}{4 c^2}-\frac {1}{2} i b d \sin ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )+b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d \sin ^{-1}(c x) \log (x)+d \left (a+b \sin ^{-1}(c x)\right ) \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 {b e \sin ^{-1}(c x)}{4 c^2}-\frac {1}{2} i b d \sin ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )+b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d \sin ^{-1}(c x) \log (x)+d \left (a+b \sin ^{-1}(c x)\right ) \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 {b e \sin ^{-1}(c x)}{4 c^2}-\frac {1}{2} i b d \sin ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \sin ^{-1}(c x)\right )+b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b d \sin ^{-1}(c x) \log (x)+d \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{2} i b d \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 108, normalized size = 0.82 \[ \frac {1}{2} \left (2 a d \log (x)+a e x^2+\frac {b e \left (c x \sqrt {1-c^2 x^2}-\sin ^{-1}(c x)\right )}{2 c^2}-i b d \left (\sin ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\right )+2 b d \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b e x^2 \sin ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \arcsin \left (c x\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 167, normalized size = 1.27 \[ \frac {a \,x^{2} e}{2}+d a \ln \left (c x \right )-\frac {i b d \arcsin \left (c x \right )^{2}}{2}+d b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+d b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {b \arcsin \left (c x \right ) e \cos \left (2 \arcsin \left (c x \right )\right )}{4 c^{2}}+\frac {b e \sin \left (2 \arcsin \left (c x \right )\right )}{8 c^{2}} \]
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 \[ \frac {1}{2} \, a e x^{2} + a d \log \relax (x) + \int \frac {{\left (b e x^{2} + b d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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