Optimal. Leaf size=197 \[ -\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}+\frac{8 b c^5 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 \sqrt{c^2 x^2}}+\frac{4 b c^3 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 x^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{1225 x^4 \sqrt{c^2 x^2}}+\frac{b c d \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.117144, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 5238, 12, 453, 271, 264} \[ -\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}+\frac{8 b c^5 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 \sqrt{c^2 x^2}}+\frac{4 b c^3 \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 x^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+49 e\right )}{1225 x^4 \sqrt{c^2 x^2}}+\frac{b c d \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5238
Rule 12
Rule 453
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac{(b c x) \int \frac{-5 d-7 e x^2}{35 x^8 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac{(b c x) \int \frac{-5 d-7 e x^2}{x^8 \sqrt{-1+c^2 x^2}} \, dx}{35 \sqrt{c^2 x^2}}\\ &=\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac{\left (b c \left (-30 c^2 d-49 e\right ) x\right ) \int \frac{1}{x^6 \sqrt{-1+c^2 x^2}} \, dx}{245 \sqrt{c^2 x^2}}\\ &=\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{1225 x^4 \sqrt{c^2 x^2}}-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac{\left (4 b c^3 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac{1}{x^4 \sqrt{-1+c^2 x^2}} \, dx}{1225 \sqrt{c^2 x^2}}\\ &=\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{1225 x^4 \sqrt{c^2 x^2}}+\frac{4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{3675 x^2 \sqrt{c^2 x^2}}-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac{\left (8 b c^5 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac{1}{x^2 \sqrt{-1+c^2 x^2}} \, dx}{3675 \sqrt{c^2 x^2}}\\ &=\frac{8 b c^5 \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{3675 \sqrt{c^2 x^2}}+\frac{b c d \sqrt{-1+c^2 x^2}}{49 x^6 \sqrt{c^2 x^2}}+\frac{b c \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{1225 x^4 \sqrt{c^2 x^2}}+\frac{4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt{-1+c^2 x^2}}{3675 x^2 \sqrt{c^2 x^2}}-\frac{d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}\\ \end{align*}
Mathematica [A] time = 0.144323, size = 110, normalized size = 0.56 \[ \frac{-105 a \left (5 d+7 e x^2\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (15 d \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )+49 e x^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )\right )-105 b \sec ^{-1}(c x) \left (5 d+7 e x^2\right )}{3675 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 158, normalized size = 0.8 \begin{align*}{c}^{7} \left ({\frac{a}{{c}^{2}} \left ( -{\frac{d}{7\,{c}^{5}{x}^{7}}}-{\frac{e}{5\,{c}^{5}{x}^{5}}} \right ) }+{\frac{b}{{c}^{2}} \left ( -{\frac{{\rm arcsec} \left (cx\right )d}{7\,{c}^{5}{x}^{7}}}-{\frac{{\rm arcsec} \left (cx\right )e}{5\,{c}^{5}{x}^{5}}}+{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 240\,{c}^{8}d{x}^{6}+392\,{c}^{6}e{x}^{6}+120\,{c}^{6}d{x}^{4}+196\,{c}^{4}e{x}^{4}+90\,{c}^{4}d{x}^{2}+147\,{c}^{2}e{x}^{2}+75\,{c}^{2}d \right ) }{3675\,{c}^{8}{x}^{8}}{\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 = 0.990804, size = 232, normalized size = 1.18 \begin{align*} -\frac{1}{245} \, b d{\left (\frac{5 \, c^{8}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{7}{2}} - 21 \, c^{8}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 35 \, c^{8}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 35 \, c^{8} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} + \frac{35 \, \operatorname{arcsec}\left (c x\right )}{x^{7}}\right )} + \frac{1}{75} \, b e{\left (\frac{3 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 10 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, c^{6} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{15 \, \operatorname{arcsec}\left (c x\right )}{x^{5}}\right )} - \frac{a e}{5 \, x^{5}} - \frac{a d}{7 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05828, size = 273, normalized size = 1.39 \begin{align*} -\frac{735 \, a e x^{2} + 525 \, a d + 105 \,{\left (7 \, b e x^{2} + 5 \, b d\right )} \operatorname{arcsec}\left (c x\right ) -{\left (8 \,{\left (30 \, b c^{6} d + 49 \, b c^{4} e\right )} x^{6} + 4 \,{\left (30 \, b c^{4} d + 49 \, b c^{2} e\right )} x^{4} + 3 \,{\left (30 \, b c^{2} d + 49 \, b e\right )} x^{2} + 75 \, b d\right )} \sqrt{c^{2} x^{2} - 1}}{3675 \, x^{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 \frac{\left (a + b \operatorname{asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{8}}\, 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 )}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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