Optimal. Leaf size=373 \[ \frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b c d^2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{4 x}-\frac{b e^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{2 c} \]
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Rubi [A] time = 1.04749, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 16, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.762, Rules used = {6303, 266, 43, 5790, 12, 6742, 95, 90, 52, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}-2 d e \log \left (\frac{1}{x}\right ) \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \log \left (\frac{1}{x}\right ) \csc ^{-1}(c x)}{\sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b c d^2 \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{4 x}-\frac{b e^2 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}{2 c} \]
Antiderivative was successfully verified.
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Rule 6303
Rule 266
Rule 43
Rule 5790
Rule 12
Rule 6742
Rule 95
Rule 90
Rule 52
Rule 2328
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 \text{sech}^{-1}(c x)\right )}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (e+d x^2\right )^2 \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-2 d e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-2 d e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-2 d e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}}+\frac{d^2 x^2}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}}+\frac{4 d e \log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}}\right ) \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-2 d e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{2 c}+\frac{(2 b d e) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}}} \, 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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=\frac{b c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-2 d e \left (a+b \text{sech}^{-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}{c}} \sqrt{1+\frac{x}{c}}} \, dx,x,\frac{1}{x}\right )+\frac{\left (2 b d e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-2 d e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (2 b d e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sin ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-2 d e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (2 b d e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-2 d e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (4 i b d e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-2 d e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{\left (2 b d e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-2 d e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )-\frac{\left (i b d e \sqrt{1-\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=\frac{b c d^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}{4 x}-\frac{b e^2 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}{2 c}+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{1}{4} b c^2 d^2 \text{sech}^{-1}(c x)-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\frac{2 b d e \sqrt{1-\frac{1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac{1}{x}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-2 d e \left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{1}{x}\right )+\frac{i b d e \sqrt{1-\frac{1}{c^2 x^2}} \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ \end{align*}
Mathematica [A] time = 0.808271, size = 212, normalized size = 0.57 \[ \frac{1}{4} \left (4 b d e \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(c x)}\right )-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2-\frac{b d^2 \sqrt{\frac{1-c x}{c x+1}} \left (-c^2 x^2+c^2 x^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+1\right )}{x^2 (c x-1)}-\frac{2 b e^2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c^2}-\frac{2 b d^2 \text{sech}^{-1}(c x)}{x^2}-4 b d e \text{sech}^{-1}(c x) \left (\text{sech}^{-1}(c x)+2 \log \left (e^{-2 \text{sech}^{-1}(c x)}+1\right )\right )+2 b e^2 x^2 \text{sech}^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.417, size = 252, normalized size = 0.7 \begin{align*}{\frac{a{x}^{2}{e}^{2}}{2}}+2\,ade\ln \left ( cx \right ) -{\frac{a{d}^{2}}{2\,{x}^{2}}}+b \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}de+{\frac{bc{d}^{2}}{4\,x}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{b{c}^{2}{d}^{2}{\rm arcsech} \left (cx\right )}{4}}-{\frac{b{\rm arcsech} \left (cx\right ){d}^{2}}{2\,{x}^{2}}}-{\frac{bx{e}^{2}}{2\,c}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{b{\rm arcsech} \left (cx\right ){x}^{2}{e}^{2}}{2}}+{\frac{b{e}^{2}}{2\,{c}^{2}}}-2\,bde{\rm arcsech} \left (cx\right )\ln \left ( 1+ \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) ^{2} \right ) -bde{\it polylog} \left ( 2,- \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) ^{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}{8} \, b d^{2}{\left (\frac{\frac{2 \, 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} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} + 1\right ) + c^{3} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} - 1\right )}{c} + \frac{4 \, \operatorname{arsech}\left (c x\right )}{x^{2}}\right )} + 2 \, a d e \log \left (x\right ) - \frac{a d^{2}}{2 \, x^{2}} + \int b e^{2} x \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right ) + \frac{2 \, b d e \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{x}\,{d x} \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{arsech}\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{asech}{\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{arsech}\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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