Integrand size = 26, antiderivative size = 268 \[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{3 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{18 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{30 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{3 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x} \]
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Time = 1.50 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {272, 45, 5354, 12, 6853, 6874, 862, 52, 65, 214, 797} \[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^8}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )}{3 c^9 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{30 c^9 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{18 c^9 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{3 c^9 x \sqrt {1-\frac {1}{c^2 x^2}}} \]
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Rule 12
Rule 45
Rule 52
Rule 65
Rule 214
Rule 272
Rule 797
Rule 862
Rule 5354
Rule 6853
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{3 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.59 \[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {-15 a \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right )+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1-c^4 x^4} \left (28+c^2 x^2+3 c^4 x^4\right )}{-1+c^2 x^2}-15 b \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right ) \sec ^{-1}(c x)+30 b \arctan \left (\frac {c \sqrt {1-\frac {1}{c^2 x^2}} x}{\sqrt {1-c^4 x^4}}\right )}{90 c^8} \]
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\[\int \frac {x^{7} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}d x\]
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Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.68 \[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {{\left (3 \, b c^{4} x^{4} + b c^{2} x^{2} + 28 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c^{4} x^{4} + 1}}{\sqrt {c^{2} x^{2} - 1}}\right ) - 15 \, {\left (a c^{6} x^{6} - a c^{4} x^{4} + 2 \, a c^{2} x^{2} + {\left (b c^{6} x^{6} - b c^{4} x^{4} + 2 \, b c^{2} x^{2} - 2 \, b\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{90 \, {\left (c^{10} x^{2} - c^{8}\right )}} \]
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Timed out. \[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{7}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \]
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Exception generated. \[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^7 \left (a+b \sec ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^7\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]
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