Optimal. Leaf size=189 \[ i b d e \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac{1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)+\frac{b e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}+i b d e \csc ^{-1}(c x)^2-2 b d e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+2 b d e \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x) \]
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Rubi [A] time = 0.429612, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {5241, 266, 43, 4731, 12, 6742, 264, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ i b d e \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac{1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)+\frac{b e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}+i b d e \csc ^{-1}(c x)^2-2 b d e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+2 b d e \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5241
Rule 266
Rule 43
Rule 4731
Rule 12
Rule 6742
Rule 264
Rule 321
Rule 216
Rule 2326
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right )^2 \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{e^2}{x^2}+d^2 x^2+4 d e \log (x)}{2 \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{e^2}{x^2}+d^2 x^2+4 d e \log (x)}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \left (-\frac{e^2}{x^2 \sqrt{1-\frac{x^2}{c^2}}}+\frac{d^2 x^2}{\sqrt{1-\frac{x^2}{c^2}}}+\frac{4 d e \log (x)}{\sqrt{1-\frac{x^2}{c^2}}}\right ) \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}+\frac{(2 b d e) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}{2 c}-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )+2 b d e \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{1}{4} \left (b c d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )-(2 b d e) \operatorname{Subst}\left (\int \frac{\sin ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )+2 b d e \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-(2 b d e) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)+i b d e \csc ^{-1}(c x)^2-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )+2 b d e \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+(4 i b d e) \operatorname{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^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)+i b d e \csc ^{-1}(c x)^2-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-2 b d e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+2 b d e \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+(2 b d e) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)+i b d e \csc ^{-1}(c x)^2-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-2 b d e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+2 b d e \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-(i b d e) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{b e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}{2 c}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)+i b d e \csc ^{-1}(c x)^2-\frac{d^2 \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-2 b d e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+2 b d e \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )-2 d e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+i b d e \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.640438, size = 187, normalized size = 0.99 \[ \frac{1}{4} \left (4 i b d e \left (\csc ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}} \left (\frac{c^2 x^2 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}}+1\right )}{x}+\frac{2 b e^2 x \left (\sqrt{1-\frac{1}{c^2 x^2}}+c x \csc ^{-1}(c x)\right )}{c}-\frac{2 b d^2 \csc ^{-1}(c x)}{x^2}-8 b d e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.7, size = 276, normalized size = 1.5 \begin{align*}{\frac{a{x}^{2}{e}^{2}}{2}}-{\frac{a{d}^{2}}{2\,{x}^{2}}}+2\,aed\ln \left ( cx \right ) +ibde \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}-{\frac{cb{d}^{2}}{4\,x}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{b{c}^{2}{d}^{2}{\rm arccsc} \left (cx\right )}{4}}-{\frac{b{\rm arccsc} \left (cx\right ){d}^{2}}{2\,{x}^{2}}}+{\frac{b{\rm arccsc} \left (cx\right ){x}^{2}{e}^{2}}{2}}+{\frac{bx{e}^{2}}{2\,c}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}-{\frac{{\frac{i}{2}}b{e}^{2}}{{c}^{2}}}-2\,bed{\rm arccsc} \left (cx\right )\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) -2\,bed{\rm arccsc} \left (cx\right )\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +2\,ibed{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +2\,ibed{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a e^{2} x^{2} + \frac{1}{4} \, b d^{2}{\left (\frac{\frac{c^{4} x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} - 1} - c^{3} \arctan \left (c x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}\right )}{c} - \frac{2 \, \operatorname{arccsc}\left (c x\right )}{x^{2}}\right )} + 2 \, a d e \log \left (x\right ) - \frac{a d^{2}}{2 \, x^{2}} + \frac{4 i \, b c^{2} d e \log \left (-c x + 1\right ) \log \left (x\right ) + 4 i \, b c^{2} d e \log \left (x\right )^{2} + 4 i \, b c^{2} d e{\rm Li}_2\left (c x\right ) + 4 i \, b c^{2} d e{\rm Li}_2\left (-c x\right ) + 2 \,{\left (b c^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) + i \, b c^{2} \log \left (c\right )\right )} e^{2} x^{2} + i \, b e^{2} \log \left (c x - 1\right ) - i \,{\left (4 \,{\left ({\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right ) - 2 \, \log \left (x\right )\right )} \log \left (x\right ) - \log \left (c x - 1\right ) \log \left (x\right ) + \log \left (-c x + 1\right ) \log \left (x\right ) + \log \left (x\right )^{2} +{\rm Li}_2\left (c x\right ) +{\rm Li}_2\left (-c x\right )\right )} b d e + b e^{2}{\left (\frac{\log \left (c x + 1\right )}{c^{2}} + \frac{\log \left (c x - 1\right )}{c^{2}}\right )}\right )} c^{2} + 2 \,{\left (4 \, b d e \int \frac{\sqrt{c x + 1} \sqrt{c x - 1} \log \left (x\right )}{c^{2} x^{3} - x}\,{d x} + \frac{\sqrt{c x + 1} \sqrt{c x - 1} b e^{2}}{c^{2}}\right )} c^{2} +{\left (-i \, b c^{2} e^{2} x^{2} - 4 i \, b c^{2} d e \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right ) +{\left (4 i \, b c^{2} d e \log \left (x\right ) + i \, b e^{2}\right )} \log \left (c x + 1\right ) +{\left (2 i \, b c^{2} e^{2} x^{2} + 8 \,{\left (b c^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) + i \, b c^{2} \log \left (c\right )\right )} d e\right )} \log \left (x\right )}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname{arccsc}\left (c x\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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