Optimal. Leaf size=185 \[ -\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e^2 \text {ArcSin}(c x)}{4 c^2}-i b d e \text {ArcSin}(c x)^2-\frac {d^2 (a+b \text {ArcSin}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {ArcSin}(c x))+2 b d e \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-2 b d e \text {ArcSin}(c x) \log (x)+2 d e (a+b \text {ArcSin}(c x)) \log (x)-i b d e \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 185, normalized size of antiderivative = 1.00, number
of steps used = 13, number of rules used = 14, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules
used = {272, 45, 4815, 12, 6874, 270, 327, 222, 2363, 4721, 3798, 2221, 2317, 2438}
\begin {gather*} -\frac {d^2 (a+b \text {ArcSin}(c x))}{2 x^2}+2 d e \log (x) (a+b \text {ArcSin}(c x))+\frac {1}{2} e^2 x^2 (a+b \text {ArcSin}(c x))-\frac {b e^2 \text {ArcSin}(c x)}{4 c^2}-i b d e \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )-i b d e \text {ArcSin}(c x)^2+2 b d e \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )-2 b d e \log (x) \text {ArcSin}(c x)-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 222
Rule 270
Rule 272
Rule 327
Rule 2221
Rule 2317
Rule 2363
Rule 2438
Rule 3798
Rule 4721
Rule 4815
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {-\frac {d^2}{x^2}+e^2 x^2+4 d e \log (x)}{2 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{2} (b c) \int \frac {-\frac {d^2}{x^2}+e^2 x^2+4 d e \log (x)}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-\frac {1}{2} (b c) \int \left (-\frac {d^2}{x^2 \sqrt {1-c^2 x^2}}+\frac {e^2 x^2}{\sqrt {1-c^2 x^2}}+\frac {4 d e \log (x)}{\sqrt {1-c^2 x^2}}\right ) \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+\frac {1}{2} \left (b c d^2\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx-(2 b c d e) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} \left (b c e^2\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+(2 b d e) \int \frac {\sin ^{-1}(c x)}{x} \, dx-\frac {\left (b e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c}\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e^2 \sin ^{-1}(c x)}{4 c^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+(2 b d e) \text {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(4 i b d e) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-(2 b d e) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)+(i b d e) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{2 x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{4 c}-\frac {b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \sin ^{-1}(c x)^2-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )+2 b d e \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2 b d e \sin ^{-1}(c x) \log (x)+2 d e \left (a+b \sin ^{-1}(c x)\right ) \log (x)-i b d e \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 184, normalized size = 0.99 \begin {gather*} \frac {1}{2} \left (-\frac {a d^2}{x^2}+a e^2 x^2-\frac {b c d^2 \sqrt {1-c^2 x^2}}{x}+\frac {b e^2 x \sqrt {1-c^2 x^2}}{2 c}-2 i b d e \text {ArcSin}(c x)^2+\frac {b e^2 \text {ArcTan}\left (\frac {c x}{1-\sqrt {1-c^2 x^2}}\right )}{c^2}+b \text {ArcSin}(c x) \left (-\frac {d^2}{x^2}+e^2 x^2+4 d e \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )+4 a d e \log (x)-2 i b d e \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 281, normalized size = 1.52
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {i b \arcsin \left (c x \right )^{2} d e}{c^{2}}+\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}+\frac {b \arcsin \left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}+\frac {i d^{2} b}{2}-\frac {b \,d^{2} \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {b \arcsin \left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b d e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}+\frac {2 b d e \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {2 i b d e \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {2 i b d e \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}\right )\) | \(281\) |
default | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {i b \arcsin \left (c x \right )^{2} d e}{c^{2}}+\frac {b \,e^{2} x \sqrt {-c^{2} x^{2}+1}}{4 c^{3}}+\frac {b \arcsin \left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \,e^{2} \arcsin \left (c x \right )}{4 c^{4}}+\frac {i d^{2} b}{2}-\frac {b \,d^{2} \sqrt {-c^{2} x^{2}+1}}{2 c x}-\frac {b \arcsin \left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b d e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}+\frac {2 b d e \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {2 i b d e \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}-\frac {2 i b d e \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}\right )\) | \(281\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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