Optimal. Leaf size=260 \[ \frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^2 \sqrt{-c^2 x^2-1} \left (280 c^4 d^2-252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt{-c^2 x^2}}+\frac{b x \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{1680 c^6 \sqrt{-c^2 x^2}}+\frac{b e x^4 \sqrt{-c^2 x^2-1} \left (84 c^2 d-25 e\right )}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.25813, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {270, 6302, 12, 1267, 459, 321, 217, 203} \[ \frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^2 \sqrt{-c^2 x^2-1} \left (280 c^4 d^2-252 c^2 d e+75 e^2\right )}{1680 c^5 \sqrt{-c^2 x^2}}+\frac{b x \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{1680 c^6 \sqrt{-c^2 x^2}}+\frac{b e x^4 \sqrt{-c^2 x^2-1} \left (84 c^2 d-25 e\right )}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 6302
Rule 12
Rule 1267
Rule 459
Rule 321
Rule 217
Rule 203
Rubi steps
\begin{align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{x^2 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt{-1-c^2 x^2}} \, dx}{105 \sqrt{-c^2 x^2}}\\ &=\frac{b e^2 x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(b x) \int \frac{x^2 \left (-210 c^2 d^2-3 \left (84 c^2 d-25 e\right ) e x^2\right )}{\sqrt{-1-c^2 x^2}} \, dx}{630 c \sqrt{-c^2 x^2}}\\ &=\frac{b \left (84 c^2 d-25 e\right ) e x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{\left (b \left (-840 c^4 d^2+9 \left (84 c^2 d-25 e\right ) e\right ) x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2}} \, dx}{2520 c^3 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1-c^2 x^2}}{1680 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (84 c^2 d-25 e\right ) e x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b \left (-840 c^4 d^2+9 \left (84 c^2 d-25 e\right ) e\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2}} \, dx}{5040 c^5 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1-c^2 x^2}}{1680 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (84 c^2 d-25 e\right ) e x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b \left (-840 c^4 d^2+9 \left (84 c^2 d-25 e\right ) e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1-c^2 x^2}}\right )}{5040 c^5 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x^2 \sqrt{-1-c^2 x^2}}{1680 c^5 \sqrt{-c^2 x^2}}+\frac{b \left (84 c^2 d-25 e\right ) e x^4 \sqrt{-1-c^2 x^2}}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b e^2 x^6 \sqrt{-1-c^2 x^2}}{42 c \sqrt{-c^2 x^2}}+\frac{1}{3} d^2 x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{2}{5} d e x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{7} e^2 x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \left (280 c^4 d^2-252 c^2 d e+75 e^2\right ) x \tan ^{-1}\left (\frac{c x}{\sqrt{-1-c^2 x^2}}\right )}{1680 c^6 \sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.352764, size = 182, normalized size = 0.7 \[ \frac{c^2 x^2 \left (16 a c^5 x \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \sqrt{\frac{1}{c^2 x^2}+1} \left (8 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )-2 c^2 e \left (126 d+25 e x^2\right )+75 e^2\right )\right )+b \left (-280 c^4 d^2+252 c^2 d e-75 e^2\right ) \log \left (x \left (\sqrt{\frac{1}{c^2 x^2}+1}+1\right )\right )+16 b c^7 x^3 \text{csch}^{-1}(c x) \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{1680 c^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.19, size = 286, normalized size = 1.1 \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 arccsch} \left (cx\right ){e}^{2}{c}^{7}{x}^{7}}{7}}+{\frac{2\,{\rm arccsch} \left (cx\right ){c}^{7}de{x}^{5}}{5}}+{\frac{{\rm arccsch} \left (cx\right ){c}^{7}{x}^{3}{d}^{2}}{3}}-{\frac{1}{1680\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( -40\,{e}^{2}{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}-168\,{c}^{5}de{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}-280\,{d}^{2}{c}^{5}x\sqrt{{c}^{2}{x}^{2}+1}+280\,{d}^{2}{c}^{4}{\it Arcsinh} \left ( cx \right ) +50\,{e}^{2}{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+252\,{c}^{3}dex\sqrt{{c}^{2}{x}^{2}+1}-252\,{c}^{2}de{\it Arcsinh} \left ( cx \right ) -75\,{e}^{2}cx\sqrt{{c}^{2}{x}^{2}+1}+75\,{e}^{2}{\it Arcsinh} \left ( cx \right ) \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00898, size = 535, normalized size = 2.06 \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}{12} \,{\left (4 \, x^{3} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d^{2} + \frac{1}{40} \,{\left (16 \, x^{5} \operatorname{arcsch}\left (c x\right ) - \frac{\frac{2 \,{\left (3 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d e + \frac{1}{672} \,{\left (96 \, x^{7} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \,{\left (15 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{6}} - \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} + \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.20877, size = 891, normalized size = 3.43 \begin{align*} \frac{240 \, a c^{7} e^{2} x^{7} + 672 \, a c^{7} d e x^{5} + 560 \, a c^{7} d^{2} x^{3} + 16 \,{\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) +{\left (280 \, b c^{4} d^{2} - 252 \, b c^{2} d e + 75 \, b e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 16 \,{\left (35 \, b c^{7} d^{2} + 42 \, b c^{7} d e + 15 \, b c^{7} e^{2}\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 16 \,{\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} - 35 \, b c^{7} d^{2} - 42 \, b c^{7} d e - 15 \, b c^{7} e^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (40 \, b c^{6} e^{2} x^{6} + 2 \,{\left (84 \, b c^{6} d e - 25 \, b c^{4} e^{2}\right )} x^{4} +{\left (280 \, b c^{6} d^{2} - 252 \, b c^{4} d e + 75 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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