Optimal. Leaf size=137 \[ -\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} i b e \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {5349, 14,
4815, 12, 6874, 327, 222, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} i b e \csc ^{-1}(c x)^2-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 222
Rule 327
Rule 2221
Rule 2317
Rule 2363
Rule 2438
Rule 3798
Rule 4721
Rule 4815
Rule 5349
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right ) \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {d x^2+2 e \log (x)}{2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {d x^2+2 e \log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \left (\frac {d x^2}{\sqrt {1-\frac {x^2}{c^2}}}+\frac {2 e \log (x)}{\sqrt {1-\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {(b d) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}+\frac {(b e) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{4} (b c d) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )-(b e) \text {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(b e) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(2 i b e) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(b e) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} (i b e) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} i b e \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 125, normalized size = 0.91 \begin {gather*} -\frac {a d}{2 x^2}-\frac {b c d \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{4 x}-\frac {b d \csc ^{-1}(c x)}{2 x^2}+\frac {1}{4} b c^2 d \text {ArcSin}\left (\frac {1}{c x}\right )-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+a e \log (x)+\frac {1}{2} i b e \left (\csc ^{-1}(c x)^2+\text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.11, size = 204, normalized size = 1.49
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {i b \mathrm {arccsc}\left (c x \right )^{2} e}{2 c^{2}}-\frac {b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}-\frac {b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}+\frac {i b e \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}+\frac {i b e \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}+\frac {b d \,\mathrm {arccsc}\left (c x \right ) \cos \left (2 \,\mathrm {arccsc}\left (c x \right )\right )}{4}-\frac {b d \sin \left (2 \,\mathrm {arccsc}\left (c x \right )\right )}{8}\right )\) | \(204\) |
default | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {i b \mathrm {arccsc}\left (c x \right )^{2} e}{2 c^{2}}-\frac {b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}-\frac {b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}+\frac {i b e \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}+\frac {i b e \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c^{2}}+\frac {b d \,\mathrm {arccsc}\left (c x \right ) \cos \left (2 \,\mathrm {arccsc}\left (c x \right )\right )}{4}-\frac {b d \sin \left (2 \,\mathrm {arccsc}\left (c x \right )\right )}{8}\right )\) | \(204\) |
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 {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{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 [B]
time = 0.99, size = 124, normalized size = 0.91 \begin {gather*} -a\,e\,\ln \left (\frac {1}{x}\right )-\frac {a\,d}{2\,x^2}-b\,e\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{c\,x}\right )-\frac {b\,c\,d\,\sqrt {1-\frac {1}{c^2\,x^2}}}{4\,x}-\frac {b\,c^2\,d\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\left (\frac {2}{c^2\,x^2}-1\right )}{4}+\frac {b\,e\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {b\,e\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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