Optimal. Leaf size=260 \[ \frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e \left (1-c^2 x^2\right )^3 \left (14 c^2 d+15 e\right )}{175 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.318493, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {270, 5790, 12, 520, 1251, 771} \[ \frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left (1-c^2 x^2\right ) \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e \left (1-c^2 x^2\right )^3 \left (14 c^2 d+15 e\right )}{175 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5790
Rule 12
Rule 520
Rule 1251
Rule 771
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{105} (b c) \int \frac{x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{105 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x \left (35 d^2+42 d e x+15 e^2 x^2\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{210 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \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{35 c^4 d^2+42 c^2 d e+15 e^2}{c^6 \sqrt{-1+c^2 x}}+\frac{\left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \sqrt{-1+c^2 x}}{c^6}+\frac{3 e \left (14 c^2 d+15 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac{15 e^2 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{210 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )}{105 c^7 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^2}{315 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.211636, size = 163, normalized size = 0.63 \[ \frac{105 a x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^6 \left (1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right )+2 c^4 \left (1225 d^2+588 d e x^2+135 e^2 x^4\right )+24 c^2 e \left (98 d+15 e x^2\right )+720 e^2\right )}{c^7}+105 b x^3 \cosh ^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{11025} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 195, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,{c}^{7}de{x}^{5}}{5}}+{\frac{{x}^{3}{c}^{7}{d}^{2}}{3}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arccosh} \left (cx\right ){e}^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,{\rm arccosh} \left (cx\right ){c}^{7}de{x}^{5}}{5}}+{\frac{{\rm arccosh} \left (cx\right ){c}^{7}{x}^{3}{d}^{2}}{3}}-{\frac{225\,{c}^{6}{e}^{2}{x}^{6}+882\,{c}^{6}de{x}^{4}+1225\,{c}^{6}{d}^{2}{x}^{2}+270\,{c}^{4}{e}^{2}{x}^{4}+1176\,{c}^{4}de{x}^{2}+2450\,{d}^{2}{c}^{4}+360\,{c}^{2}{e}^{2}{x}^{2}+2352\,{c}^{2}de+720\,{e}^{2}}{11025}\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 = 1.14611, size = 333, normalized size = 1.28 \begin{align*} \frac{1}{7} \, a e^{2} x^{7} + \frac{2}{5} \, a d e x^{5} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{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^{2} + \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 )} b d e + \frac{1}{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 e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38416, size = 471, normalized size = 1.81 \begin{align*} \frac{1575 \, a c^{7} e^{2} x^{7} + 4410 \, a c^{7} d e x^{5} + 3675 \, a c^{7} d^{2} x^{3} + 105 \,{\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (225 \, b c^{6} e^{2} x^{6} + 2450 \, b c^{4} d^{2} + 2352 \, b c^{2} d e + 18 \,{\left (49 \, b c^{6} d e + 15 \, b c^{4} e^{2}\right )} x^{4} + 720 \, b e^{2} +{\left (1225 \, b c^{6} d^{2} + 1176 \, b c^{4} d e + 360 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{11025 \, c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.2328, size = 340, normalized size = 1.31 \begin{align*} \begin{cases} \frac{a d^{2} x^{3}}{3} + \frac{2 a d e x^{5}}{5} + \frac{a e^{2} x^{7}}{7} + \frac{b d^{2} x^{3} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{2 b d e x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{b e^{2} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} - \frac{b d^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{2 b d e x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{b e^{2} x^{6} \sqrt{c^{2} x^{2} - 1}}{49 c} - \frac{2 b d^{2} \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} - \frac{8 b d e x^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{3}} - \frac{6 b e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{245 c^{3}} - \frac{16 b d e \sqrt{c^{2} x^{2} - 1}}{75 c^{5}} - \frac{8 b e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{5}} - \frac{16 b e^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{7}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d^{2} x^{3}}{3} + \frac{2 d e x^{5}}{5} + \frac{e^{2} x^{7}}{7}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35315, size = 328, normalized size = 1.26 \begin{align*} \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{9} \,{\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^{2} + \frac{1}{245} \,{\left (35 \, a 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\right )} e^{2} + \frac{2}{75} \,{\left (15 \, a d 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\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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