Optimal. Leaf size=321 \[ -\frac{i b d e \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+2 d e \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b d e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}+\frac{b c d^2 \sqrt{c x-1} \sqrt{c x+1}}{2 x}-\frac{b e^2 x \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]
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Rubi [A] time = 0.814215, antiderivative size = 321, 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 = {266, 43, 5790, 12, 6742, 95, 90, 52, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac{i b d e \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+2 d e \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b d e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}+\frac{b c d^2 \sqrt{c x-1} \sqrt{c x+1}}{2 x}-\frac{b e^2 x \sqrt{c x-1} \sqrt{c x+1}}{4 c} \]
Antiderivative was successfully verified.
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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 \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac{-\frac{d^2}{x^2}+e^2 x^2+4 d e \log (x)}{2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{2} (b c) \int \frac{-\frac{d^2}{x^2}+e^2 x^2+4 d e \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{2} (b c) \int \left (-\frac{d^2}{x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{e^2 x^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 d e \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx\\ &=-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)+\frac{1}{2} \left (b c d^2\right ) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx-(2 b c d e) \int \frac{\log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{2} \left (b c e^2\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{b c d^2 \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{\left (b e^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c}-\frac{\left (2 b c d e \sqrt{1-c^2 x^2}\right ) \int \frac{\log (x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d^2 \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b d e \sqrt{1-c^2 x^2}\right ) \int \frac{\sin ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d^2 \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b d e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d^2 \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{-1+c x} \sqrt{1+c x}}+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 i b d e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d^2 \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b d e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d^2 \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b d e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d^2 \sqrt{-1+c x} \sqrt{1+c x}}{2 x}-\frac{b e^2 x \sqrt{-1+c x} \sqrt{1+c x}}{4 c}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}-\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac{i b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+2 d e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b d e \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.437015, size = 173, normalized size = 0.54 \[ \frac{1}{4} \left (4 b d e \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )-\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2+\frac{b e^2 \left (2 c^2 x^2 \cosh ^{-1}(c x)-c x \sqrt{c x-1} \sqrt{c x+1}-2 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{c^2}+\frac{2 b d^2 \left (c x \sqrt{c x-1} \sqrt{c x+1}-\cosh ^{-1}(c x)\right )}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.316, size = 198, normalized size = 0.6 \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 arccosh} \left (cx\right ) \right ) ^{2}de+{\frac{b{\rm arccosh} \left (cx\right ){x}^{2}{e}^{2}}{2}}-{\frac{b{e}^{2}x}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{e}^{2}{\rm arccosh} \left (cx\right )}{4\,{c}^{2}}}+{\frac{bc{d}^{2}}{2\,x}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{c}^{2}b{d}^{2}}{2}}-{\frac{b{d}^{2}{\rm arccosh} \left (cx\right )}{2\,{x}^{2}}}+2\,bde{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) +bde{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \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}{2} \, b d^{2}{\left (\frac{\sqrt{c^{2} x^{2} - 1} c}{x} - \frac{\operatorname{arcosh}\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 (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + \frac{2 \, b d e \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\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{arcosh}\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{acosh}{\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{arcosh}\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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