Optimal. Leaf size=193 \[ -\frac {b c d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{x}-\frac {1}{2} c^2 d (a+b \text {ArcSin}(c x))^2-\frac {d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{2 x^2}+\frac {i c^2 d (a+b \text {ArcSin}(c x))^3}{3 b}-c^2 d (a+b \text {ArcSin}(c x))^2 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+b^2 c^2 d \log (x)+i b c^2 d (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )-\frac {1}{2} b^2 c^2 d \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right ) \]
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Rubi [A]
time = 0.20, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4785, 4721,
3798, 2221, 2611, 2320, 6724, 4781, 29, 4737} \begin {gather*} i b c^2 d \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))-\frac {d \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{2 x^2}-\frac {b c d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{x}+\frac {i c^2 d (a+b \text {ArcSin}(c x))^3}{3 b}-\frac {1}{2} c^2 d (a+b \text {ArcSin}(c x))^2-c^2 d \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2-\frac {1}{2} b^2 c^2 d \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )+b^2 c^2 d \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2221
Rule 2320
Rule 2611
Rule 3798
Rule 4721
Rule 4737
Rule 4781
Rule 4785
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+(b c d) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx-\left (c^2 d\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (c^2 d\right ) \text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\left (b^2 c^2 d\right ) \int \frac {1}{x} \, dx-\left (b c^3 d\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+b^2 c^2 d \log (x)+\left (2 i c^2 d\right ) \text {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 {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+\left (2 b c^2 d\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+i b c^2 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 c^2 d\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+i b c^2 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 c^2 d\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+i b c^2 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 c^2 d \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 236, normalized size = 1.22 \begin {gather*} \frac {1}{2} d \left (-\frac {a^2}{x^2}-\frac {2 a b \left (c x \sqrt {1-c^2 x^2}+\text {ArcSin}(c x)\right )}{x^2}-2 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+\text {ArcSin}(c x)^2-2 c^2 x^2 \log (c x)\right )}{x^2}+2 i a b c^2 \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )+\text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )+\frac {1}{12} i b^2 c^2 \left (\pi ^3-8 \text {ArcSin}(c x)^3+24 i \text {ArcSin}(c x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c x)}\right )-24 \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c x)}\right )+12 i \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c x)}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 526 vs. \(2 (211 ) = 422\).
time = 0.47, size = 527, normalized size = 2.73
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {d \,a^{2}}{2 c^{2} x^{2}}-d \,a^{2} \ln \left (c x \right )+\frac {i d \,b^{2} \arcsin \left (c x \right )^{3}}{3}+i d \,b^{2} \arcsin \left (c x \right )-\frac {d \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i d a b -2 d \,b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d \,b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+d \,b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-2 d \,b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+d \,b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+i d a b \arcsin \left (c x \right )^{2}-\frac {d a b \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d a b \arcsin \left (c x \right )}{c^{2} x^{2}}-2 d a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i d a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(527\) |
default | \(c^{2} \left (-\frac {d \,a^{2}}{2 c^{2} x^{2}}-d \,a^{2} \ln \left (c x \right )+\frac {i d \,b^{2} \arcsin \left (c x \right )^{3}}{3}+i d \,b^{2} \arcsin \left (c x \right )-\frac {d \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i d a b -2 d \,b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 d \,b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+d \,b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-2 d \,b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+d \,b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+i d a b \arcsin \left (c x \right )^{2}-\frac {d a b \sqrt {-c^{2} x^{2}+1}}{c x}-\frac {d a b \arcsin \left (c x \right )}{c^{2} x^{2}}-2 d a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 d a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i d a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\) | \(527\) |
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*} - d \left (\int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {a^{2} c^{2}}{x}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx\right ) \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 )}^2\,\left (d-c^2\,d\,x^2\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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