Optimal. Leaf size=401 \[ -\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^{12}}+\frac {b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{9/2}}{90 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {3 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {13 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {7 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{15 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {4 b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{15 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}} \]
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Rubi [A] time = 2.48, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 16, 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} \[ -\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^{12}}+\frac {b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{9/2}}{90 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {3 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {13 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {7 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{15 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {4 b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{15 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 50
Rule 63
Rule 208
Rule 266
Rule 783
Rule 848
Rule 5246
Rule 6721
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^{11} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}-\frac {b \int \frac {\sqrt {1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{30 c^{12} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}-\frac {b \int \frac {\sqrt {1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{30 c^{13}}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \sec ^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{x \sqrt {1-c^2 x^2}} \, dx}{30 c^{13} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-c^4 x^2} \left (8+4 c^4 x^2+3 c^8 x^4\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{60 c^{13} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8 \sqrt {1-c^4 x^2}}{x \sqrt {1-c^2 x}}+\frac {4 c^4 x \sqrt {1-c^4 x^2}}{\sqrt {1-c^2 x}}+\frac {3 c^8 x^3 \sqrt {1-c^4 x^2}}{\sqrt {1-c^2 x}}\right ) \, dx,x,x^2\right )}{60 c^{13} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (2 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 )}{15 c^{13} \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 )}{15 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^3 \sqrt {1-c^4 x^2}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{20 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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+c^2 x}}{x} \, dx,x,x^2\right )}{15 c^{13} \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 )}{15 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^3 \sqrt {1+c^2 x} \, dx,x,x^2\right )}{20 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (2 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 )}{15 c^{13} \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 )}{15 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^6}+\frac {3 \left (1+c^2 x\right )^{3/2}}{c^6}-\frac {3 \left (1+c^2 x\right )^{5/2}}{c^6}+\frac {\left (1+c^2 x\right )^{7/2}}{c^6}\right ) \, dx,x,x^2\right )}{20 c^5 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {7 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {13 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {3 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (4 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 )}{15 c^{15} \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {7 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {13 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {3 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \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^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{10 c^{12}}-\frac {4 b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{15 c^{13} \sqrt {1-\frac {1}{c^2 x^2}} x}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 194, normalized size = 0.48 \[ \frac {-105 a \sqrt {1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right )-105 b \sqrt {1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right ) \sec ^{-1}(c x)+840 b \tan ^{-1}\left (\frac {c x \sqrt {1-\frac {1}{c^2 x^2}}}{\sqrt {1-c^4 x^4}}\right )+\frac {b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {1-c^4 x^4} \left (35 c^8 x^8+5 c^6 x^6+78 c^4 x^4+36 c^2 x^2+768\right )}{c^2 x^2-1}}{3150 c^{12}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 10.87, size = 0, normalized size = 0.00 \[ \int \frac {x^{11} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{11}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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