Optimal. Leaf size=326 \[ -\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-5 b c^3 d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{27} b^2 c^6 d^3 x^3+\frac{50}{9} b^2 c^4 d^3 x-\frac{b^2 c^2 d^3}{3 x} \]
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Rubi [A] time = 1.02137, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {5739, 5684, 5653, 5717, 8, 5744, 5742, 5760, 4182, 2279, 2391, 270} \[ -\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-5 b c^3 d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{27} b^2 c^6 d^3 x^3+\frac{50}{9} b^2 c^4 d^3 x-\frac{b^2 c^2 d^3}{3 x} \]
Antiderivative was successfully verified.
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Rule 5739
Rule 5684
Rule 5653
Rule 5717
Rule 8
Rule 5744
Rule 5742
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (2 c^2 d\right ) \int \frac{\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{1}{3} \left (2 b c d^3\right ) \int \frac{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx\\ &=-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (8 c^4 d^2\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{1}{3} \left (b^2 c^2 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^2}{x^2} \, dx+\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx\\ &=\frac{17}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d^3\right ) \int \left (2 c^2+\frac{1}{x^2}+c^4 x^2\right ) \, dx+\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac{1}{3} \left (16 c^4 d^3\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{9} \left (5 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{3} \left (4 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{3} \left (16 b c^5 d^3\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{11}{9} b^2 c^4 d^3 x-\frac{14}{27} b^2 c^6 d^3 x^3+\frac{17}{3} b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx+\left (4 b c^3 d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx-\frac{1}{3} \left (5 b^2 c^4 d^3\right ) \int 1 \, dx+\frac{1}{9} \left (16 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (4 b^2 c^4 d^3\right ) \int 1 \, dx-\frac{1}{3} \left (32 b c^5 d^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{46}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (5 b c^3 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\left (4 b c^3 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (32 b^2 c^4 d^3\right ) \int 1 \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )-\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )-\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{17}{3} b^2 c^3 d^3 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac{17}{3} b^2 c^3 d^3 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 1.11144, size = 461, normalized size = 1.41 \[ \frac{d^3 \left (153 b^2 c^3 x^3 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-153 b^2 c^3 x^3 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+9 a^2 c^6 x^6+81 a^2 c^4 x^4-81 a^2 c^2 x^2-9 a^2-6 a b c^5 x^5 \sqrt{c^2 x^2+1}-150 a b c^3 x^3 \sqrt{c^2 x^2+1}-9 a b c x \sqrt{c^2 x^2+1}+18 a b c^6 x^6 \sinh ^{-1}(c x)+162 a b c^4 x^4 \sinh ^{-1}(c x)-162 a b c^2 x^2 \sinh ^{-1}(c x)-153 a b c^3 x^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-18 a b \sinh ^{-1}(c x)+2 b^2 c^6 x^6+150 b^2 c^4 x^4-9 b^2 c^2 x^2+9 b^2 c^6 x^6 \sinh ^{-1}(c x)^2-6 b^2 c^5 x^5 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+81 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-150 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-81 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-9 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-9 b^2 \sinh ^{-1}(c x)^2\right )}{27 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.331, size = 528, normalized size = 1.6 \begin{align*} -{\frac{17\,{c}^{3}{d}^{3}ab}{3}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) }-{\frac{2\,{d}^{3}ab{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{3}}}+{\frac{{c}^{6}{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{x}^{3}}{3}}+3\,{c}^{4}{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}x-3\,{\frac{{d}^{3}{b}^{2}{c}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{50\,{d}^{3}{b}^{2}{c}^{3}{\it Arcsinh} \left ( cx \right ) }{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{17\,{d}^{3}{b}^{2}{c}^{3}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{17\,{d}^{3}{b}^{2}{c}^{3}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{{d}^{3}{b}^{2}{c}^{2}}{3\,x}}+{\frac{50\,{b}^{2}{c}^{4}{d}^{3}x}{9}}-{\frac{2\,{c}^{5}{d}^{3}ab{x}^{2}}{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{c{d}^{3}ab}{3\,{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{{d}^{3}{b}^{2}c{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{5}{d}^{3}{b}^{2}{\it Arcsinh} \left ( cx \right ){x}^{2}}{9}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,{c}^{6}{d}^{3}ab{\it Arcsinh} \left ( cx \right ){x}^{3}}{3}}+6\,{c}^{4}{d}^{3}ab{\it Arcsinh} \left ( cx \right ) x-6\,{\frac{{d}^{3}ab{c}^{2}{\it Arcsinh} \left ( cx \right ) }{x}}+{\frac{2\,{b}^{2}{c}^{6}{d}^{3}{x}^{3}}{27}}-{\frac{17\,{d}^{3}{b}^{2}{c}^{3}}{3}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{17\,{d}^{3}{b}^{2}{c}^{3}}{3}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{50\,{c}^{3}{d}^{3}ab}{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{{d}^{3}{a}^{2}}{3\,{x}^{3}}}+{\frac{{c}^{6}{d}^{3}{a}^{2}{x}^{3}}{3}}+3\,{c}^{4}{d}^{3}{a}^{2}x-3\,{\frac{{c}^{2}{d}^{3}{a}^{2}}{x}}-{\frac{{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{3\,{x}^{3}}} \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}{3} \, a^{2} c^{6} d^{3} x^{3} + \frac{2}{9} \,{\left (3 \, x^{3} \operatorname{arsinh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{6} d^{3} + 3 \, b^{2} c^{4} d^{3} x \operatorname{arsinh}\left (c x\right )^{2} + 6 \, b^{2} c^{4} d^{3}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + 3 \, a^{2} c^{4} d^{3} x + 6 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b c^{3} d^{3} - 6 \,{\left (c \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arsinh}\left (c x\right )}{x}\right )} a b c^{2} d^{3} + \frac{1}{3} \,{\left ({\left (c^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac{2 \, \operatorname{arsinh}\left (c x\right )}{x^{3}}\right )} a b d^{3} - \frac{3 \, a^{2} c^{2} d^{3}}{x} - \frac{a^{2} d^{3}}{3 \, x^{3}} + \frac{{\left (b^{2} c^{6} d^{3} x^{6} - 9 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{3 \, x^{3}} - \int \frac{2 \,{\left (b^{2} c^{9} d^{3} x^{8} + b^{2} c^{7} d^{3} x^{6} - 9 \, b^{2} c^{5} d^{3} x^{4} - 10 \, b^{2} c^{3} d^{3} x^{2} - b^{2} c d^{3} +{\left (b^{2} c^{8} d^{3} x^{7} - 9 \, b^{2} c^{4} d^{3} x^{3} - b^{2} c^{2} d^{3} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{3 \,{\left (c^{3} x^{6} + c x^{4} +{\left (c^{2} x^{5} + x^{3}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{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^{2} c^{6} d^{3} x^{6} + 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} +{\left (b^{2} c^{6} d^{3} x^{6} + 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} + b^{2} d^{3}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{6} d^{3} x^{6} + 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} + a b d^{3}\right )} \operatorname{arsinh}\left (c x\right )}{x^{4}}, 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*} d^{3} \left (\int 3 a^{2} c^{4}\, dx + \int \frac{a^{2}}{x^{4}}\, dx + \int \frac{3 a^{2} c^{2}}{x^{2}}\, dx + \int a^{2} c^{6} x^{2}\, dx + \int 3 b^{2} c^{4} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 6 a b c^{4} \operatorname{asinh}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{3 b^{2} c^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{6} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{6 a b c^{2} \operatorname{asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{6} x^{2} \operatorname{asinh}{\left (c x \right )}\, dx\right ) \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 (c^{2} d x^{2} + d\right )}^{3}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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