Optimal. Leaf size=177 \[ \frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b d^2 (c x-1)^{7/2} (c x+1)^{7/2}}{49 c^3}-\frac{b d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{175 c^3}+\frac{4 b d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{315 c^3}-\frac{8 b d^2 \sqrt{c x-1} \sqrt{c x+1}}{105 c^3} \]
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Rubi [A] time = 0.248498, antiderivative size = 223, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {270, 5731, 12, 520, 1251, 771} \[ \frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{5} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b d^2 \left (1-c^2 x^2\right )^4}{49 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 \left (1-c^2 x^2\right )^3}{175 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b d^2 \left (1-c^2 x^2\right )^2}{315 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^2 \left (1-c^2 x^2\right )}{105 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5731
Rule 12
Rule 520
Rule 1251
Rule 771
Rubi steps
\begin{align*} \int x^2 \left (d-c^2 d 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} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^2 x^3 \left (35-42 c^2 x^2+15 c^4 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} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{105} \left (b c d^2\right ) \int \frac{x^3 \left (35-42 c^2 x^2+15 c^4 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} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^3 \left (35-42 c^2 x^2+15 c^4 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} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x \left (35-42 c^2 x+15 c^4 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} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{c^2 \sqrt{-1+c^2 x}}-\frac{4 \sqrt{-1+c^2 x}}{c^2}+\frac{3 \left (-1+c^2 x\right )^{3/2}}{c^2}+\frac{15 \left (-1+c^2 x\right )^{5/2}}{c^2}\right ) \, dx,x,x^2\right )}{210 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{8 b d^2 \left (1-c^2 x^2\right )}{105 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b d^2 \left (1-c^2 x^2\right )^2}{315 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \left (1-c^2 x^2\right )^3}{175 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^2 \left (1-c^2 x^2\right )^4}{49 c^3 \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} c^2 d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} c^4 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.1657, size = 116, normalized size = 0.66 \[ \frac{d^2 \left (105 a c^3 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right )-b \sqrt{c x-1} \sqrt{c x+1} \left (225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right )+105 b c^3 x^3 \left (15 c^4 x^4-42 c^2 x^2+35\right ) \cosh ^{-1}(c x)\right )}{11025 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 120, normalized size = 0.7 \begin{align*}{\frac{1}{{c}^{3}} \left ({d}^{2}a \left ({\frac{{c}^{7}{x}^{7}}{7}}-{\frac{2\,{c}^{5}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{3}}{3}} \right ) +{d}^{2}b \left ({\frac{{\rm arccosh} \left (cx\right ){c}^{7}{x}^{7}}{7}}-{\frac{2\,{\rm arccosh} \left (cx\right ){c}^{5}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{3}{\rm arccosh} \left (cx\right )}{3}}-{\frac{225\,{c}^{6}{x}^{6}-612\,{c}^{4}{x}^{4}+409\,{c}^{2}{x}^{2}+818}{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 = 1.16108, size = 352, normalized size = 1.99 \begin{align*} \frac{1}{7} \, a c^{4} d^{2} x^{7} - \frac{2}{5} \, a c^{2} d^{2} x^{5} + \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 c^{4} 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 c^{2} d^{2} + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77013, size = 351, normalized size = 1.98 \begin{align*} \frac{1575 \, a c^{7} d^{2} x^{7} - 4410 \, a c^{5} d^{2} x^{5} + 3675 \, a c^{3} d^{2} x^{3} + 105 \,{\left (15 \, b c^{7} d^{2} x^{7} - 42 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (225 \, b c^{6} d^{2} x^{6} - 612 \, b c^{4} d^{2} x^{4} + 409 \, b c^{2} d^{2} x^{2} + 818 \, b d^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{11025 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.92045, size = 209, normalized size = 1.18 \begin{align*} \begin{cases} \frac{a c^{4} d^{2} x^{7}}{7} - \frac{2 a c^{2} d^{2} x^{5}}{5} + \frac{a d^{2} x^{3}}{3} + \frac{b c^{4} d^{2} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} - \frac{b c^{3} d^{2} x^{6} \sqrt{c^{2} x^{2} - 1}}{49} - \frac{2 b c^{2} d^{2} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{68 b c d^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{1225} + \frac{b d^{2} x^{3} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{409 b d^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{11025 c} - \frac{818 b d^{2} \sqrt{c^{2} x^{2} - 1}}{11025 c^{3}} & \text{for}\: c \neq 0 \\\frac{d^{2} x^{3} \left (a + \frac{i \pi b}{2}\right )}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.48075, size = 347, normalized size = 1.96 \begin{align*} \frac{1}{7} \, a c^{4} d^{2} x^{7} - \frac{2}{5} \, a c^{2} d^{2} x^{5} + \frac{1}{245} \,{\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 c^{4} d^{2} - \frac{2}{75} \,{\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 c^{2} d^{2} + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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