Optimal. Leaf size=109 \[ d x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x \left (6 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{6 c^2 \sqrt{c^2 x^2}}-\frac{b e x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.0513458, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5228, 12, 388, 217, 206} \[ d x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b x \left (6 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{6 c^2 \sqrt{c^2 x^2}}-\frac{b e x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5228
Rule 12
Rule 388
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right ) \, dx &=d x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{3 d+e x^2}{3 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=d x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{(b c x) \int \frac{3 d+e x^2}{\sqrt{-1+c^2 x^2}} \, dx}{3 \sqrt{c^2 x^2}}\\ &=-\frac{b e x^2 \sqrt{-1+c^2 x^2}}{6 c \sqrt{c^2 x^2}}+d x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{\left (b \left (-6 c^2 d-e\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{6 c \sqrt{c^2 x^2}}\\ &=-\frac{b e x^2 \sqrt{-1+c^2 x^2}}{6 c \sqrt{c^2 x^2}}+d x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \sec ^{-1}(c x)\right )+\frac{\left (b \left (-6 c^2 d-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 )}{6 c \sqrt{c^2 x^2}}\\ &=-\frac{b e x^2 \sqrt{-1+c^2 x^2}}{6 c \sqrt{c^2 x^2}}+d x \left (a+b \sec ^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \sec ^{-1}(c x)\right )-\frac{b \left (6 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 = 0.26434, size = 150, normalized size = 1.38 \[ a d x+\frac{1}{3} a e x^3-\frac{b d x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{\sqrt{c^2 x^2-1}}-\frac{b e x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{6 c}-\frac{b e \log \left (x \left (\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{6 c^3}+b d x \sec ^{-1}(c x)+\frac{1}{3} b e x^3 \sec ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 195, normalized size = 1.8 \begin{align*}{\frac{a{x}^{3}e}{3}}+adx+{\frac{b{\rm arcsec} \left (cx\right ){x}^{3}e}{3}}+b{\rm arcsec} \left (cx\right )xd-{\frac{bd}{{c}^{2}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}}}}}}}-{\frac{be{x}^{2}}{6\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{be}{6\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{be}{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 = 0.96545, size = 208, normalized size = 1.91 \begin{align*} \frac{1}{3} \, a e x^{3} + \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 + a d 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}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54092, size = 327, normalized size = 3. \begin{align*} \frac{2 \, a c^{3} e x^{3} + 6 \, a c^{3} d x - \sqrt{c^{2} x^{2} - 1} b c e x + 2 \,{\left (b c^{3} e x^{3} + 3 \, b c^{3} d x - 3 \, b c^{3} d - b c^{3} e\right )} \operatorname{arcsec}\left (c x\right ) + 4 \,{\left (3 \, b c^{3} d + b c^{3} e\right )} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (6 \, b c^{2} d + b e\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{6 \, c^{3}} \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 \left (a + b \operatorname{asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, 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 )}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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