Optimal. Leaf size=162 \[ -\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{b c d^2 \sqrt{c^2 x^2-1}}{\sqrt{c^2 x^2}}-\frac{b e x \left (12 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{6 c^2 \sqrt{c^2 x^2}}-\frac{b e^2 x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.127189, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {270, 5238, 12, 1265, 388, 217, 206} \[ -\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{b c d^2 \sqrt{c^2 x^2-1}}{\sqrt{c^2 x^2}}-\frac{b e x \left (12 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{6 c^2 \sqrt{c^2 x^2}}-\frac{b e^2 x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 5238
Rule 12
Rule 1265
Rule 388
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx &=-\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{-3 d^2+6 d e x^2+e^2 x^4}{3 x^2 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{-3 d^2+6 d e x^2+e^2 x^4}{x^2 \sqrt{-1+c^2 x^2}} \, dx}{3 \sqrt{c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1+c^2 x^2}}{\sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{6 d e+e^2 x^2}{\sqrt{-1+c^2 x^2}} \, dx}{3 \sqrt{c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1+c^2 x^2}}{\sqrt{c^2 x^2}}-\frac{b e^2 x^2 \sqrt{-1+c^2 x^2}}{6 c \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )--\frac{\left (b \left (-12 c^2 d e-e^2\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{6 c \sqrt{c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1+c^2 x^2}}{\sqrt{c^2 x^2}}-\frac{b e^2 x^2 \sqrt{-1+c^2 x^2}}{6 c \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )--\frac{\left (b \left (-12 c^2 d e-e^2\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 )}{6 c \sqrt{c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1+c^2 x^2}}{\sqrt{c^2 x^2}}-\frac{b e^2 x^2 \sqrt{-1+c^2 x^2}}{6 c \sqrt{c^2 x^2}}-\frac{d^2 \left (a+b \sec ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e^2 x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b e \left (12 c^2 d+e\right ) x \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{6 c^2 \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.194875, size = 136, normalized size = 0.84 \[ \frac{c^2 \left (2 a c \left (-3 d^2+6 d e x^2+e^2 x^4\right )+b x \sqrt{1-\frac{1}{c^2 x^2}} \left (6 c^2 d^2-e^2 x^2\right )\right )+2 b c^3 \sec ^{-1}(c x) \left (-3 d^2+6 d e x^2+e^2 x^4\right )-b e x \left (12 c^2 d+e\right ) \log \left (x \left (\sqrt{1-\frac{1}{c^2 x^2}}+1\right )\right )}{6 c^3 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.174, size = 286, normalized size = 1.8 \begin{align*}{\frac{a{x}^{3}{e}^{2}}{3}}+2\,aedx-{\frac{a{d}^{2}}{x}}+{\frac{b{\rm arcsec} \left (cx\right ){x}^{3}{e}^{2}}{3}}+2\,b{\rm arcsec} \left (cx\right )edx-{\frac{b{\rm arcsec} \left (cx\right ){d}^{2}}{x}}+{cb{d}^{2}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{d}^{2}}{c{x}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-2\,{\frac{b\sqrt{{c}^{2}{x}^{2}-1}ed\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ) }{{c}^{2}x}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{e}^{2}{x}^{2}}{6\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{e}^{2}}{6\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b{e}^{2}}{6\,{c}^{4}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993497, size = 267, normalized size = 1.65 \begin{align*} \frac{1}{3} \, a e^{2} x^{3} +{\left (c \sqrt{-\frac{1}{c^{2} x^{2}} + 1} - \frac{\operatorname{arcsec}\left (c x\right )}{x}\right )} b d^{2} + \frac{1}{12} \,{\left (4 \, x^{3} \operatorname{arcsec}\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 e^{2} + 2 \, a d e x + \frac{{\left (2 \, c x \operatorname{arcsec}\left (c x\right ) - \log \left (\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right ) + \log \left (-\sqrt{-\frac{1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d e}{c} - \frac{a d^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.57997, size = 502, normalized size = 3.1 \begin{align*} \frac{2 \, a c^{3} e^{2} x^{4} + 6 \, b c^{4} d^{2} x + 12 \, a c^{3} d e x^{2} - 6 \, a c^{3} d^{2} - 4 \,{\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (12 \, b c^{2} d e + b e^{2}\right )} x \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + 2 \,{\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} +{\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \operatorname{arcsec}\left (c x\right ) +{\left (6 \, b c^{3} d^{2} - b c e^{2} x^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{6 \, c^{3} x} \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 \frac{\left (a + b \operatorname{asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{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 \frac{{\left (e x^{2} + d\right )}^{2}{\left (b \operatorname{arcsec}\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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