Optimal. Leaf size=196 \[ d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right ) \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b e \left (1-c^2 x^2\right )^2 \left (5 c^2 d+3 e\right )}{45 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.203285, antiderivative size = 196, 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, 520, 1247, 698} \[ d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b \left (1-c^2 x^2\right ) \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{15 c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b e \left (1-c^2 x^2\right )^2 \left (5 c^2 d+3 e\right )}{45 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 194
Rule 5705
Rule 12
Rule 520
Rule 1247
Rule 698
Rubi steps
\begin{align*} \int \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{15 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{15} (b c) \int \frac{x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{15 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{15 d^2+10 d e x+3 e^2 x^2}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{30 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \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{15 c^4 d^2+10 c^2 d e+3 e^2}{c^4 \sqrt{-1+c^2 x}}+\frac{2 e \left (5 c^2 d+3 e\right ) \sqrt{-1+c^2 x}}{c^4}+\frac{3 e^2 \left (-1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{30 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \left (1-c^2 x^2\right )}{15 c^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b e \left (5 c^2 d+3 e\right ) \left (1-c^2 x^2\right )^2}{45 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^2 \left (1-c^2 x^2\right )^3}{25 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+d^2 x \left (a+b \cosh ^{-1}(c x)\right )+\frac{2}{3} d e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.191845, size = 130, normalized size = 0.66 \[ \frac{1}{225} \left (15 a x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )+4 c^2 e \left (25 d+3 e x^2\right )+24 e^2\right )}{c^5}+15 b x \cosh ^{-1}(c x) \left (15 d^2+10 d e x^2+3 e^2 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 157, normalized size = 0.8 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{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}{{c}^{4}} \left ({\frac{{\rm arccosh} \left (cx\right ){e}^{2}{c}^{5}{x}^{5}}{5}}+{\frac{2\,{\rm arccosh} \left (cx\right ){c}^{5}de{x}^{3}}{3}}+{\rm arccosh} \left (cx\right ){c}^{5}x{d}^{2}-{\frac{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}}{225}\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 = 1.18332, size = 243, normalized size = 1.24 \begin{align*} \frac{1}{5} \, a e^{2} x^{5} + \frac{2}{3} \, a d e x^{3} + \frac{2}{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 e + \frac{1}{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 e^{2} + a d^{2} x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b d^{2}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42728, size = 370, normalized size = 1.89 \begin{align*} \frac{45 \, a c^{5} e^{2} x^{5} + 150 \, a c^{5} d e x^{3} + 225 \, a c^{5} d^{2} x + 15 \,{\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (9 \, b c^{4} e^{2} x^{4} + 225 \, b c^{4} d^{2} + 100 \, b c^{2} d e + 24 \, b e^{2} + 2 \,{\left (25 \, b c^{4} d e + 6 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{225 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.7601, size = 246, normalized size = 1.26 \begin{align*} \begin{cases} a d^{2} x + \frac{2 a d e x^{3}}{3} + \frac{a e^{2} x^{5}}{5} + b d^{2} x \operatorname{acosh}{\left (c x \right )} + \frac{2 b d e x^{3} \operatorname{acosh}{\left (c x \right )}}{3} + \frac{b e^{2} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} - \frac{b d^{2} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{2 b d e x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{b e^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{4 b d e \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} - \frac{4 b e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{3}} - \frac{8 b e^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{5}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \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 = 1.29177, size = 262, normalized size = 1.34 \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^{2} + a d^{2} x + \frac{1}{75} \,{\left (15 \, a 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\right )} e^{2} + \frac{2}{9} \,{\left (3 \, a d 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\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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