Optimal. Leaf size=301 \[ -\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}+\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {4781, 30,
4789, 4803, 4268, 2317, 2438} \begin {gather*} \frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8 x^2}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{4 x^4}+\frac {c^4 \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{4 \sqrt {1-c^2 x^2}}-\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}+\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2317
Rule 2438
Rule 4268
Rule 4781
Rule 4789
Rule 4803
Rubi steps
\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x^4} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^3 \sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (b c^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x^2} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (c^4 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (c^4 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b c^4 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c^4 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {\left (i b c^4 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}+\frac {\left (i b c^4 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {c^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}+\frac {c^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}+\frac {i b c^4 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 2.88, size = 321, normalized size = 1.07 \begin {gather*} \frac {a \left (-2+c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{8 x^4}-\frac {1}{8} a c^4 \sqrt {d} \log (x)+\frac {1}{8} a c^4 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b c^4 \sqrt {d-c^2 d x^2} \left (8 \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )+6 \text {ArcSin}(c x) \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )-c x \csc ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )-3 \text {ArcSin}(c x) \csc ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )-24 \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )+24 \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )-24 i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )+24 i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )-6 \text {ArcSin}(c x) \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+3 \text {ArcSin}(c x) \sec ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )-\frac {16 \sin ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )}{c^3 x^3}+8 \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{192 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 571, normalized size = 1.90
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 d \,x^{2}}+\frac {a \,c^{4} \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}-\frac {a \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{4}}{8 c^{2} x^{2}-8}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{3}}{8 \left (c^{2} x^{2}-1\right ) x}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{2}}{8 \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{12 \left (c^{2} x^{2}-1\right ) x^{3}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{4 \left (c^{2} x^{2}-1\right ) x^{4}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{4} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{8 c^{2} x^{2}-8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{4} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{8 c^{2} x^{2}-8}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{4} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{8 c^{2} x^{2}-8}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{4} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{8 c^{2} x^{2}-8}\) | \(571\) |
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 {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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