Optimal. Leaf size=186 \[ \frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} i b d^2 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.30, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5349, 272,
45, 4815, 6874, 464, 270, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -d^2 \log \left (\frac {1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e^2 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}+\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (6 c^2 d+e\right )}{6 c^3}+\frac {1}{2} i b d^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2-b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d^2 \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 270
Rule 272
Rule 464
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 )^2 \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right )^2 \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {-\frac {e \left (e+4 d x^2\right )}{4 x^4}+d^2 \log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \left (-\frac {e \left (e+4 d x^2\right )}{4 x^4 \sqrt {1-\frac {x^2}{c^2}}}+\frac {d^2 \log (x)}{\sqrt {1-\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {(b e) \text {Subst}\left (\int \frac {e+4 d x^2}{x^4 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (b d^2\right ) \text {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )-\frac {\left (b e \left (6 c^2 d+e\right )\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}\\ &=\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (b d^2\right ) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (2 i b d^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (b d^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} \left (i b d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \csc ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} i b d^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 157, normalized size = 0.84 \begin {gather*} a d e x^2+\frac {1}{4} a e^2 x^4+\frac {b d e x \left (\sqrt {1-\frac {1}{c^2 x^2}}+c x \csc ^{-1}(c x)\right )}{c}+\frac {b e^2 x \left (\sqrt {1-\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )+3 c^3 x^3 \csc ^{-1}(c x)\right )}{12 c^3}+a d^2 \log (x)+\frac {1}{2} i b d^2 \left (\csc ^{-1}(c x) \left (\csc ^{-1}(c x)+2 i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )+\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 = 8.67, size = 301, normalized size = 1.62
method | result | size |
derivativedivides | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {i b \,d^{2} \mathrm {arccsc}\left (c x \right )^{2}}{2}+b \,\mathrm {arccsc}\left (c x \right ) d e \,x^{2}+\frac {b \,\mathrm {arccsc}\left (c x \right ) e^{2} x^{4}}{4}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d e x}{c}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e^{2} x^{3}}{12 c}-\frac {i b d e}{c^{2}}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e^{2} x}{6 c^{3}}-\frac {i b \,e^{2}}{6 c^{4}}-b \,d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b \,d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b \,d^{2} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b \,d^{2} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\) | \(301\) |
default | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {i b \,d^{2} \mathrm {arccsc}\left (c x \right )^{2}}{2}+b \,\mathrm {arccsc}\left (c x \right ) d e \,x^{2}+\frac {b \,\mathrm {arccsc}\left (c x \right ) e^{2} x^{4}}{4}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, d e x}{c}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e^{2} x^{3}}{12 c}-\frac {i b d e}{c^{2}}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e^{2} x}{6 c^{3}}-\frac {i b \,e^{2}}{6 c^{4}}-b \,d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b \,d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b \,d^{2} \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b \,d^{2} \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\) | \(301\) |
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 )^{2}}{x}\, 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: RuntimeError} \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 (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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