Optimal. Leaf size=268 \[ \frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{30 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{18 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}} \]
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Rubi [A] time = 2.01752, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {266, 43, 5246, 12, 6721, 6742, 848, 50, 63, 208, 783} \[ \frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{30 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{18 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5246
Rule 12
Rule 6721
Rule 6742
Rule 848
Rule 50
Rule 63
Rule 208
Rule 783
Rubi steps
\begin{align*} \int \frac{x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt{1-c^4 x^4}} \, dx &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac{b \int \frac{\left (-2-c^4 x^4\right ) \sqrt{1-c^4 x^4}}{6 c^8 \sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac{b \int \frac{\left (-2-c^4 x^4\right ) \sqrt{1-c^4 x^4}}{\sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{6 c^9}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\left (-2-c^4 x^4\right ) \sqrt{1-c^4 x^4}}{x \sqrt{1-c^2 x^2}} \, dx}{6 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2} \left (2+c^4 x^2\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2 \sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}}+\frac{c^4 x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}}\right ) \, dx,x,x^2\right )}{12 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{6 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+c^2 x}}{x} \, dx,x,x^2\right )}{6 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x \sqrt{1+c^2 x} \, dx,x,x^2\right )}{12 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{3 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{6 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{1+c^2 x}}{c^2}+\frac{\left (1+c^2 x\right )^{3/2}}{c^2}\right ) \, dx,x,x^2\right )}{12 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{3 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{18 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{30 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{3 c^{11} \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{3 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{18 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{30 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{3 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.316417, size = 159, normalized size = 0.59 \[ \frac{-15 a \sqrt{1-c^4 x^4} \left (c^4 x^4+2\right )+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^4 x^4} \left (3 c^4 x^4+c^2 x^2+28\right )}{c^2 x^2-1}+30 b \tan ^{-1}\left (\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right )-15 b \sqrt{1-c^4 x^4} \left (c^4 x^4+2\right ) \sec ^{-1}(c x)}{90 c^8} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.461, size = 0, normalized size = 0. \begin{align*} \int{{x}^{7} \left ( a+b{\rm arcsec} \left (cx\right ) \right ){\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{7}}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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