Optimal. Leaf size=359 \[ -\frac{8 b d e \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{8 b e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^3}-\frac{16 b e^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^5}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 b d^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{4 b d e x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 b e^2 x^4 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+\frac{8 b^2 d e x}{9 c^2}+\frac{8 b^2 e^2 x^3}{225 c^2}+\frac{16 b^2 e^2 x}{75 c^4}+2 b^2 d^2 x+\frac{4}{27} b^2 d e x^3+\frac{2}{125} b^2 e^2 x^5 \]
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Rubi [A] time = 1.19862, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {5707, 5654, 5718, 8, 5662, 5759, 30} \[ -\frac{8 b d e \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{8 b e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^3}-\frac{16 b e^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^5}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 b d^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{4 b d e x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 b e^2 x^4 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+\frac{8 b^2 d e x}{9 c^2}+\frac{8 b^2 e^2 x^3}{225 c^2}+\frac{16 b^2 e^2 x}{75 c^4}+2 b^2 d^2 x+\frac{4}{27} b^2 d e x^3+\frac{2}{125} b^2 e^2 x^5 \]
Antiderivative was successfully verified.
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Rule 5707
Rule 5654
Rule 5718
Rule 8
Rule 5662
Rule 5759
Rule 30
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 \left (a+b \cosh ^{-1}(c x)\right )^2+2 d e x^2 \left (a+b \cosh ^{-1}(c x)\right )^2+e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^2 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx+e^2 \int x^4 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2-\left (2 b c d^2\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{3} (4 b c d e) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx-\frac{1}{5} \left (2 b c e^2\right ) \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{2 b d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{4 b d e x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac{2 b e^2 x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\left (2 b^2 d^2\right ) \int 1 \, dx+\frac{1}{9} \left (4 b^2 d e\right ) \int x^2 \, dx-\frac{(8 b d e) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c}+\frac{1}{25} \left (2 b^2 e^2\right ) \int x^4 \, dx-\frac{\left (8 b e^2\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{25 c}\\ &=2 b^2 d^2 x+\frac{4}{27} b^2 d e x^3+\frac{2}{125} b^2 e^2 x^5-\frac{2 b d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{8 b d e \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{4 b d e x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac{8 b e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^3}-\frac{2 b e^2 x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (8 b^2 d e\right ) \int 1 \, dx}{9 c^2}-\frac{\left (16 b e^2\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{75 c^3}+\frac{\left (8 b^2 e^2\right ) \int x^2 \, dx}{75 c^2}\\ &=2 b^2 d^2 x+\frac{8 b^2 d e x}{9 c^2}+\frac{4}{27} b^2 d e x^3+\frac{8 b^2 e^2 x^3}{225 c^2}+\frac{2}{125} b^2 e^2 x^5-\frac{2 b d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{8 b d e \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{16 b e^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^5}-\frac{4 b d e x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac{8 b e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^3}-\frac{2 b e^2 x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (16 b^2 e^2\right ) \int 1 \, dx}{75 c^4}\\ &=2 b^2 d^2 x+\frac{8 b^2 d e x}{9 c^2}+\frac{16 b^2 e^2 x}{75 c^4}+\frac{4}{27} b^2 d e x^3+\frac{8 b^2 e^2 x^3}{225 c^2}+\frac{2}{125} b^2 e^2 x^5-\frac{2 b d^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac{8 b d e \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac{16 b e^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^5}-\frac{4 b d e x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac{8 b e^2 x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{75 c^3}-\frac{2 b e^2 x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end{align*}
Mathematica [A] time = 0.535749, size = 299, normalized size = 0.83 \[ \frac{225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-30 a b \sqrt{c x-1} \sqrt{c x+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )-30 b \cosh ^{-1}(c x) \left (b \sqrt{c x-1} \sqrt{c x+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )-15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )\right )+2 b^2 c x \left (c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )+60 c^2 e \left (25 d+e x^2\right )+360 e^2\right )+225 b^2 c^5 x \cosh ^{-1}(c x)^2 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{3375 c^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 402, normalized size = 1.1 \begin{align*}{\frac{1}{c} \left ({\frac{{a}^{2}}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,{c}^{5}de{x}^{3}}{3}}+x{c}^{5}{d}^{2} \right ) }+{\frac{{b}^{2}}{{c}^{4}} \left ({\frac{{e}^{2}}{1125} \left ( 225\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{5}{x}^{5}-90\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{c}^{4}{x}^{4}-120\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{c}^{2}{x}^{2}+18\,{c}^{5}{x}^{5}-240\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}+40\,{c}^{3}{x}^{3}+240\,cx \right ) }+{\frac{2\,{c}^{2}de}{27} \left ( 9\, \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}{c}^{3}{x}^{3}-6\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}{c}^{2}{x}^{2}-12\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}+2\,{c}^{3}{x}^{3}+12\,cx \right ) }+{d}^{2}{c}^{4} \left ( \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}cx-2\,{\rm arccosh} \left (cx\right )\sqrt{cx-1}\sqrt{cx+1}+2\,cx \right ) \right ) }+2\,{\frac{ab}{{c}^{4}} \left ( 1/5\,{\rm arccosh} \left (cx\right ){e}^{2}{c}^{5}{x}^{5}+2/3\,{\rm arccosh} \left (cx\right ){c}^{5}de{x}^{3}+{\rm arccosh} \left (cx\right ){c}^{5}x{d}^{2}-{\frac{\sqrt{cx-1}\sqrt{cx+1} \left ( 9\,{c}^{4}{e}^{2}{x}^{4}+50\,{c}^{4}de{x}^{2}+225\,{d}^{2}{c}^{4}+12\,{c}^{2}{e}^{2}{x}^{2}+100\,{c}^{2}de+24\,{e}^{2} \right ) }{225}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11498, size = 579, normalized size = 1.61 \begin{align*} \frac{1}{5} \, b^{2} e^{2} x^{5} \operatorname{arcosh}\left (c x\right )^{2} + \frac{1}{5} \, a^{2} e^{2} x^{5} + \frac{2}{3} \, b^{2} d e x^{3} \operatorname{arcosh}\left (c x\right )^{2} + \frac{2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \operatorname{arcosh}\left (c x\right )^{2} + \frac{4}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\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 d e - \frac{4}{27} \,{\left (3 \, c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname{arcosh}\left (c x\right ) - \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d e + \frac{2}{75} \,{\left (15 \, x^{5} \operatorname{arcosh}\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 e^{2} - \frac{2}{1125} \,{\left (15 \,{\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 \operatorname{arcosh}\left (c x\right ) - \frac{9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} + 2 \, b^{2} d^{2}{\left (x - \frac{\sqrt{c^{2} x^{2} - 1} \operatorname{arcosh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac{2 \,{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} a b d^{2}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89801, size = 845, normalized size = 2.35 \begin{align*} \frac{27 \,{\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \,{\left (25 \,{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{5} d e + 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \,{\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 15 \,{\left (225 \,{\left (a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} + 200 \, b^{2} c^{3} d e + 48 \, b^{2} c e^{2}\right )} x + 30 \,{\left (45 \, a b c^{5} e^{2} x^{5} + 150 \, a b c^{5} d e x^{3} + 225 \, a b c^{5} d^{2} x -{\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} + 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \,{\left (25 \, b^{2} c^{4} d e + 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - 30 \,{\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} + 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \,{\left (25 \, a b c^{4} d e + 6 \, a b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{3375 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.0095, size = 602, normalized size = 1.68 \begin{align*} \begin{cases} a^{2} d^{2} x + \frac{2 a^{2} d e x^{3}}{3} + \frac{a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname{acosh}{\left (c x \right )} + \frac{4 a b d e x^{3} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{2 a b e^{2} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} - \frac{2 a b d^{2} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{4 a b d e x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{2 a b e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{8 a b d e \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} - \frac{8 a b e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{3}} - \frac{16 a b e^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname{acosh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + \frac{2 b^{2} d e x^{3} \operatorname{acosh}^{2}{\left (c x \right )}}{3} + \frac{4 b^{2} d e x^{3}}{27} + \frac{b^{2} e^{2} x^{5} \operatorname{acosh}^{2}{\left (c x \right )}}{5} + \frac{2 b^{2} e^{2} x^{5}}{125} - \frac{2 b^{2} d^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{c} - \frac{4 b^{2} d e x^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{9 c} - \frac{2 b^{2} e^{2} x^{4} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{25 c} + \frac{8 b^{2} d e x}{9 c^{2}} + \frac{8 b^{2} e^{2} x^{3}}{225 c^{2}} - \frac{8 b^{2} d e \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{9 c^{3}} - \frac{8 b^{2} e^{2} x^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{75 c^{3}} + \frac{16 b^{2} e^{2} x}{75 c^{4}} - \frac{16 b^{2} e^{2} \sqrt{c^{2} x^{2} - 1} \operatorname{acosh}{\left (c x \right )}}{75 c^{5}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right )^{2} \left (d^{2} x + \frac{2 d e x^{3}}{3} + \frac{e^{2} x^{5}}{5}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.15997, size = 656, normalized size = 1.83 \begin{align*} 2 \,{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} a b d^{2} +{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 2 \, c{\left (\frac{x}{c} - \frac{\sqrt{c^{2} x^{2} - 1} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{2}}\right )}\right )} b^{2} d^{2} + a^{2} d^{2} x + \frac{1}{1125} \,{\left (225 \, a^{2} x^{5} + 30 \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} a b +{\left (225 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 2 \, c{\left (\frac{9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{5}} - \frac{15 \,{\left (3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{6}}\right )}\right )} b^{2}\right )} e^{2} + \frac{2}{27} \,{\left (9 \, a^{2} d x^{3} + 6 \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} a b d +{\left (9 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )^{2} + 2 \, c{\left (\frac{c^{2} x^{3} + 6 \, x}{c^{3}} - \frac{3 \,{\left ({\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{4}}\right )}\right )} b^{2} d\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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