Optimal. Leaf size=395 \[ \frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2 b e^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b e \left (1-c^2 x^2\right )^2 \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (378 c^4 d^2 e^2+420 c^6 d^3 e+315 c^8 d^4+180 c^2 d e^3+35 e^4\right )}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^4 \left (9 c^2 d+7 e\right )}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.47495, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {194, 5705, 12, 1610, 1799, 1850} \[ \frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2 b e^2 \left (1-c^2 x^2\right )^3 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b e \left (1-c^2 x^2\right )^2 \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (378 c^4 d^2 e^2+420 c^6 d^3 e+315 c^8 d^4+180 c^2 d e^3+35 e^4\right )}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b e^3 \left (1-c^2 x^2\right )^4 \left (9 c^2 d+7 e\right )}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 194
Rule 5705
Rule 12
Rule 1610
Rule 1799
Rule 1850
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{315 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{315} (b c) \int \frac{x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt{-1+c^2 x^2}} \, dx}{315 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{315 d^4+420 d^3 e x+378 d^2 e^2 x^2+180 d e^3 x^3+35 e^4 x^4}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4}{c^8 \sqrt{-1+c^2 x}}+\frac{4 e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \sqrt{-1+c^2 x}}{c^8}+\frac{6 e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (-1+c^2 x\right )^{3/2}}{c^8}+\frac{20 e^3 \left (9 c^2 d+7 e\right ) \left (-1+c^2 x\right )^{5/2}}{c^8}+\frac{35 e^4 \left (-1+c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right ) \left (1-c^2 x^2\right )}{315 c^9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4 b e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )^2}{945 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )^3}{525 c^9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{4 b e^3 \left (9 c^2 d+7 e\right ) \left (1-c^2 x^2\right )^4}{441 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^4 \left (1-c^2 x^2\right )^5}{81 c^9 \sqrt{-1+c x} \sqrt{1+c x}}+d^4 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.390997, size = 265, normalized size = 0.67 \[ \frac{315 a x \left (378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^8 \left (23814 d^2 e^2 x^4+44100 d^3 e x^2+99225 d^4+8100 d e^3 x^6+1225 e^4 x^8\right )+8 c^6 e \left (3969 d^2 e x^2+11025 d^3+1215 d e^2 x^4+175 e^3 x^6\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )+320 c^2 e^3 \left (81 d+7 e x^2\right )+4480 e^4\right )}{c^9}+315 b x \cosh ^{-1}(c x) \left (378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4+180 d e^3 x^6+35 e^4 x^8\right )}{99225} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 331, normalized size = 0.8 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{8}} \left ({\frac{{e}^{4}{c}^{9}{x}^{9}}{9}}+{\frac{4\,{c}^{9}d{e}^{3}{x}^{7}}{7}}+{\frac{6\,{c}^{9}{d}^{2}{e}^{2}{x}^{5}}{5}}+{\frac{4\,{c}^{9}{d}^{3}e{x}^{3}}{3}}+{c}^{9}{d}^{4}x \right ) }+{\frac{b}{{c}^{8}} \left ({\frac{{\rm arccosh} \left (cx\right ){e}^{4}{c}^{9}{x}^{9}}{9}}+{\frac{4\,{\rm arccosh} \left (cx\right ){c}^{9}d{e}^{3}{x}^{7}}{7}}+{\frac{6\,{\rm arccosh} \left (cx\right ){c}^{9}{d}^{2}{e}^{2}{x}^{5}}{5}}+{\frac{4\,{\rm arccosh} \left (cx\right ){c}^{9}{d}^{3}e{x}^{3}}{3}}+{\rm arccosh} \left (cx\right ){c}^{9}{d}^{4}x-{\frac{1225\,{c}^{8}{e}^{4}{x}^{8}+8100\,{c}^{8}d{e}^{3}{x}^{6}+23814\,{c}^{8}{d}^{2}{e}^{2}{x}^{4}+1400\,{c}^{6}{e}^{4}{x}^{6}+44100\,{c}^{8}{d}^{3}e{x}^{2}+9720\,{c}^{6}d{e}^{3}{x}^{4}+99225\,{c}^{8}{d}^{4}+31752\,{c}^{6}{d}^{2}{e}^{2}{x}^{2}+1680\,{c}^{4}{e}^{4}{x}^{4}+88200\,{c}^{6}{d}^{3}e+12960\,{c}^{4}d{e}^{3}{x}^{2}+63504\,{c}^{4}{d}^{2}{e}^{2}+2240\,{c}^{2}{e}^{4}{x}^{2}+25920\,{c}^{2}d{e}^{3}+4480\,{e}^{4}}{99225}\sqrt{cx-1}\sqrt{cx+1}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11512, size = 560, normalized size = 1.42 \begin{align*} \frac{1}{9} \, a e^{4} x^{9} + \frac{4}{7} \, a d e^{3} x^{7} + \frac{6}{5} \, a d^{2} e^{2} x^{5} + \frac{4}{3} \, a d^{3} e x^{3} + \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 )} b d^{3} e + \frac{2}{25} \,{\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 )} b d^{2} e^{2} + \frac{4}{245} \,{\left (35 \, x^{7} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d e^{3} + \frac{1}{2835} \,{\left (315 \, x^{9} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{35 \, \sqrt{c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac{40 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac{64 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b e^{4} + a d^{4} x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b d^{4}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69421, size = 803, normalized size = 2.03 \begin{align*} \frac{11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \,{\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} + 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} + 25920 \, b c^{2} d e^{3} + 100 \,{\left (81 \, b c^{8} d e^{3} + 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \,{\left (3969 \, b c^{8} d^{2} e^{2} + 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \,{\left (11025 \, b c^{8} d^{3} e + 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} + 560 \, b c^{2} e^{4}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{99225 \, c^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 41.6936, size = 600, normalized size = 1.52 \begin{align*} \begin{cases} a d^{4} x + \frac{4 a d^{3} e x^{3}}{3} + \frac{6 a d^{2} e^{2} x^{5}}{5} + \frac{4 a d e^{3} x^{7}}{7} + \frac{a e^{4} x^{9}}{9} + b d^{4} x \operatorname{acosh}{\left (c x \right )} + \frac{4 b d^{3} e x^{3} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{6 b d^{2} e^{2} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{4 b d e^{3} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{b e^{4} x^{9} \operatorname{acosh}{\left (c x \right )}}{9} - \frac{b d^{4} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{4 b d^{3} e x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{6 b d^{2} e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{4 b d e^{3} x^{6} \sqrt{c^{2} x^{2} - 1}}{49 c} - \frac{b e^{4} x^{8} \sqrt{c^{2} x^{2} - 1}}{81 c} - \frac{8 b d^{3} e \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} - \frac{8 b d^{2} e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{25 c^{3}} - \frac{24 b d e^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{245 c^{3}} - \frac{8 b e^{4} x^{6} \sqrt{c^{2} x^{2} - 1}}{567 c^{3}} - \frac{16 b d^{2} e^{2} \sqrt{c^{2} x^{2} - 1}}{25 c^{5}} - \frac{32 b d e^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{5}} - \frac{16 b e^{4} x^{4} \sqrt{c^{2} x^{2} - 1}}{945 c^{5}} - \frac{64 b d e^{3} \sqrt{c^{2} x^{2} - 1}}{245 c^{7}} - \frac{64 b e^{4} x^{2} \sqrt{c^{2} x^{2} - 1}}{2835 c^{7}} - \frac{128 b e^{4} \sqrt{c^{2} x^{2} - 1}}{2835 c^{9}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (d^{4} x + \frac{4 d^{3} e x^{3}}{3} + \frac{6 d^{2} e^{2} x^{5}}{5} + \frac{4 d e^{3} x^{7}}{7} + \frac{e^{4} x^{9}}{9}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4616, size = 547, normalized size = 1.38 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} b d^{4} + a d^{4} x + \frac{1}{2835} \,{\left (315 \, a x^{9} +{\left (315 \, x^{9} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{9}{2}} + 180 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 378 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 420 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 315 \, \sqrt{c^{2} x^{2} - 1}}{c^{9}}\right )} b\right )} e^{4} + \frac{4}{245} \,{\left (35 \, a d x^{7} +{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b d\right )} e^{3} + \frac{2}{25} \,{\left (15 \, a d^{2} x^{5} +{\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 )} b d^{2}\right )} e^{2} + \frac{4}{9} \,{\left (3 \, a d^{3} x^{3} +{\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 )} b d^{3}\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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