Optimal. Leaf size=137 \[ \frac{1}{2} i b e \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac{1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac{b c d \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 d \csc ^{-1}(c x)+\frac{1}{2} i b e \csc ^{-1}(c x)^2-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x) \]
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Rubi [A] time = 0.302397, antiderivative size = 137, normalized size of antiderivative = 1., 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 = {5241, 14, 4731, 12, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac{1}{2} i b e \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac{1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac{b c d \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 d \csc ^{-1}(c x)+\frac{1}{2} i b e \csc ^{-1}(c x)^2-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 5241
Rule 14
Rule 4731
Rule 12
Rule 6742
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 ) \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx &=-\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{2 c}+\frac{(b e) \operatorname{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) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )-(b e) \operatorname{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) \operatorname{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) \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 \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) \operatorname{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) \operatorname{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*}
Mathematica [A] time = 0.413998, size = 137, normalized size = 1. \[ \frac{1}{4} \left (2 i b e \left (\csc ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )-\frac{2 a d}{x^2}+4 a e \log (x)-\frac{b c d \sqrt{1-\frac{1}{c^2 x^2}} \left (\frac{c^2 x^2 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{\sqrt{1-c^2 x^2}}+1\right )}{x}-\frac{2 b d \csc ^{-1}(c x)}{x^2}-4 b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.384, size = 201, normalized size = 1.5 \begin{align*} -{\frac{ad}{2\,{x}^{2}}}+ae\ln \left ( cx \right ) +{\frac{i}{2}}be \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}-{\frac{bcd}{4\,x}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{b{c}^{2}d{\rm arccsc} \left (cx\right )}{4}}-{\frac{b{\rm arccsc} \left (cx\right )d}{2\,{x}^{2}}}-be{\rm arccsc} \left (cx\right )\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) -be{\rm arccsc} \left (cx\right )\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +ibe{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +ibe{\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*}{\left (c^{2} \int \frac{\sqrt{c x + 1} \sqrt{c x - 1} \log \left (x\right )}{c^{4} x^{3} - c^{2} x}\,{d x} + \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) \log \left (x\right )\right )} b e + \frac{1}{4} \, b d{\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 )} + a e \log \left (x\right ) - \frac{a d}{2 \, x^{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 x^{2} + a d +{\left (b e x^{2} + b d\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 )}{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 )}{\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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