Optimal. Leaf size=307 \[ -2 b^2 c d^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{6}{5} c^2 d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{5} c^2 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2}{25} b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{22}{5} b c d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{125} b^2 c^6 d^3 x^5+\frac{14}{75} b^2 c^4 d^3 x^3+\frac{122}{25} b^2 c^2 d^3 x \]
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Rubi [A] time = 0.739251, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 24, 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, 194, 5744, 5742, 5760, 4182, 2279, 2391} \[ -2 b^2 c d^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{6}{5} c^2 d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{5} c^2 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2}{25} b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{22}{5} b c d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{125} b^2 c^6 d^3 x^5+\frac{14}{75} b^2 c^4 d^3 x^3+\frac{122}{25} b^2 c^2 d^3 x \]
Antiderivative was successfully verified.
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Rule 5739
Rule 5684
Rule 5653
Rule 5717
Rule 8
Rule 194
Rule 5744
Rule 5742
Rule 5760
Rule 4182
Rule 2279
Rule 2391
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^2} \, dx &=-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (6 c^2 d\right ) \int \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\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} \, dx\\ &=\frac{2}{5} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\frac{1}{5} \left (24 c^2 d^2\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\left (2 b c 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{1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac{1}{5} \left (12 b c^3 d^3\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=\frac{2}{3} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac{1}{5} \left (16 c^2 d^3\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{5} \left (2 b^2 c^2 d^3\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac{1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac{1}{3} \left (2 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{5} \left (16 b c^3 d^3\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{16}{15} b^2 c^2 d^3 x-\frac{22}{45} b^2 c^4 d^3 x^3-\frac{2}{25} b^2 c^6 d^3 x^5+2 b c d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx+\frac{1}{25} \left (12 b^2 c^2 d^3\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac{1}{15} \left (16 b^2 c^2 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (2 b^2 c^2 d^3\right ) \int 1 \, dx-\frac{1}{5} \left (32 b c^3 d^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{38}{25} b^2 c^2 d^3 x+\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+\left (2 b c d^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{5} \left (32 b^2 c^2 d^3\right ) \int 1 \, dx\\ &=\frac{122}{25} b^2 c^2 d^3 x+\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (2 b^2 c d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{122}{25} b^2 c^2 d^3 x+\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\left (2 b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (2 b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=\frac{122}{25} b^2 c^2 d^3 x+\frac{14}{75} b^2 c^4 d^3 x^3+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{22}{5} b c d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{16}{5} c^2 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-2 b^2 c d^3 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d^3 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 1.41425, size = 466, normalized size = 1.52 \[ \frac{1}{720} d^3 \left (1440 b^2 c \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-1440 b^2 c \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+144 a^2 c^6 x^5+720 a^2 c^4 x^3+2160 a^2 c^2 x-\frac{720 a^2}{x}-\frac{288}{5} a b c^5 x^4 \sqrt{c^2 x^2+1}-\frac{2016}{5} a b c^3 x^2 \sqrt{c^2 x^2+1}-\frac{17568}{5} a b c \sqrt{c^2 x^2+1}+288 a b c^6 x^5 \sinh ^{-1}(c x)+1440 a b c^4 x^3 \sinh ^{-1}(c x)-1440 a b c \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )+4320 a b c^2 x \sinh ^{-1}(c x)-\frac{1440 a b \sinh ^{-1}(c x)}{x}-3420 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+3460 b^2 c^2 x+1890 b^2 c^2 x \sinh ^{-1}(c x)^2+360 b^2 c^2 x \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+80 b^2 c^2 x \cosh \left (2 \sinh ^{-1}(c x)\right )-10 b^2 c \sinh \left (3 \sinh ^{-1}(c x)\right )-45 b^2 c \sinh ^{-1}(c x)^2 \sinh \left (3 \sinh ^{-1}(c x)\right )+\frac{18}{25} b^2 c \sinh \left (5 \sinh ^{-1}(c x)\right )+9 b^2 c \sinh ^{-1}(c x)^2 \sinh \left (5 \sinh ^{-1}(c x)\right )-\frac{720 b^2 \sinh ^{-1}(c x)^2}{x}+1440 b^2 c \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-1440 b^2 c \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-90 b^2 c \sinh ^{-1}(c x) \cosh \left (3 \sinh ^{-1}(c x)\right )-\frac{18}{5} b^2 c \sinh ^{-1}(c x) \cosh \left (5 \sinh ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.163, size = 516, normalized size = 1.7 \begin{align*} -2\,c{d}^{3}ab{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) -{\frac{2\,{d}^{3}ab{c}^{5}{x}^{4}}{25}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{14\,{d}^{3}ab{c}^{3}{x}^{2}}{25}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{14\,{d}^{3}{b}^{2}{\it Arcsinh} \left ( cx \right ){c}^{3}{x}^{2}}{25}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,{d}^{3}ab{\it Arcsinh} \left ( cx \right ){c}^{6}{x}^{5}}{5}}+2\,{d}^{3}ab{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{3}+6\,{d}^{3}ab{\it Arcsinh} \left ( cx \right ){c}^{2}x-{\frac{2\,{d}^{3}{b}^{2}{\it Arcsinh} \left ( cx \right ){c}^{5}{x}^{4}}{25}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{{d}^{3}{a}^{2}}{x}}+{\frac{122\,{b}^{2}{c}^{2}{d}^{3}x}{25}}+{\frac{14\,{b}^{2}{c}^{4}{d}^{3}{x}^{3}}{75}}+{\frac{2\,{b}^{2}{c}^{6}{d}^{3}{x}^{5}}{125}}-{\frac{122\,c{d}^{3}ab}{25}\sqrt{{c}^{2}{x}^{2}+1}}-2\,c{d}^{3}{b}^{2}{\it Arcsinh} \left ( cx \right ) \ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,c{d}^{3}{b}^{2}{\it Arcsinh} \left ( cx \right ) \ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +3\,{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{2}x+{\frac{{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{6}{x}^{5}}{5}}+{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{c}^{4}{x}^{3}-{\frac{122\,{d}^{3}{b}^{2}c{\it Arcsinh} \left ( cx \right ) }{25}\sqrt{{c}^{2}{x}^{2}+1}}-2\,{\frac{{d}^{3}ab{\it Arcsinh} \left ( cx \right ) }{x}}+{\frac{{d}^{3}{a}^{2}{c}^{6}{x}^{5}}{5}}+{d}^{3}{a}^{2}{c}^{4}{x}^{3}+3\,{d}^{3}{a}^{2}{c}^{2}x-{\frac{{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{x}}-2\,{b}^{2}c{d}^{3}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) +2\,{b}^{2}c{d}^{3}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \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}{5} \, a^{2} c^{6} d^{3} x^{5} + \frac{2}{75} \,{\left (15 \, x^{5} \operatorname{arsinh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac{4 \, \sqrt{c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{6} d^{3} + a^{2} c^{4} d^{3} x^{3} + \frac{2}{3} \,{\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^{4} d^{3} + 3 \, b^{2} c^{2} d^{3} x \operatorname{arsinh}\left (c x\right )^{2} + 6 \, b^{2} c^{2} d^{3}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + 3 \, a^{2} c^{2} d^{3} x + 6 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b c d^{3} - 2 \,{\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 d^{3} - \frac{a^{2} d^{3}}{x} + \frac{{\left (b^{2} c^{6} d^{3} x^{6} + 5 \, b^{2} c^{4} d^{3} x^{4} - 5 \, b^{2} d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{5 \, x} - \int \frac{2 \,{\left (b^{2} c^{9} d^{3} x^{8} + 6 \, b^{2} c^{7} d^{3} x^{6} + 5 \, b^{2} c^{5} d^{3} x^{4} - 5 \, b^{2} c^{3} d^{3} x^{2} - 5 \, b^{2} c d^{3} +{\left (b^{2} c^{8} d^{3} x^{7} + 5 \, b^{2} c^{6} d^{3} x^{5} - 5 \, 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 )}{5 \,{\left (c^{3} x^{4} + c x^{2} +{\left (c^{2} x^{3} + x\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^{2}}, 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^{2}\, dx + \int \frac{a^{2}}{x^{2}}\, dx + \int 3 a^{2} c^{4} x^{2}\, dx + \int a^{2} c^{6} x^{4}\, dx + \int 3 b^{2} c^{2} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 6 a b c^{2} \operatorname{asinh}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 3 b^{2} c^{4} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{4} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{2} \operatorname{asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{4} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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