Optimal. Leaf size=473 \[ -\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}+\frac {7 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {13 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}+\frac {3 b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {4 b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x} \]
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Rubi [A]
time = 1.09, antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 45,
6444, 12, 6874, 862, 52, 65, 214, 797} \begin {gather*} -\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-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 {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {3 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {13 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {7 b \sqrt {1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {c^2 x^2+1}}{15 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {4 b \sqrt {1-c^2 x^2} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{15 c^{13} x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 52
Rule 65
Rule 214
Rule 272
Rule 797
Rule 862
Rule 6444
Rule 6874
Rubi steps
\begin {align*} \int \frac {x^{11} \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-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 )}{30 c^{12} x \sqrt {1-c^2 x^2}} \, dx}{c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-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 x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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 x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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 x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (2 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 )}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} 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 )}{15 c^9 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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 x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+c^2 x}}{x} \, dx,x,x^2\right )}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} 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 )}{15 c^9 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^3 \sqrt {1+c^2 x} \, dx,x,x^2\right )}{20 c^5 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (2 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 )}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} 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 )}{15 c^9 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {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 x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{10 c^{12}}-\frac {\left (4 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 )}{15 c^{15} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}\\ &=-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{15 c^{13} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} 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 x}} \sqrt {1+\frac {1}{c x}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \text {sech}^{-1}(c x)\right )}{2 c^{12}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^{12}}-\frac {\left (1-c^4 x^4\right )^{5/2} \left (a+b \text {sech}^{-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 x}} \sqrt {1+\frac {1}{c x}} x}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 213, normalized size = 0.45 \begin {gather*} \frac {-105 a \sqrt {1-c^4 x^4} \left (8+4 c^4 x^4+3 c^8 x^8\right )+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^4 x^4} \left (768+36 c^2 x^2+78 c^4 x^4+5 c^6 x^6+35 c^8 x^8\right )}{-1+c x}-105 b \sqrt {1-c^4 x^4} \left (8+4 c^4 x^4+3 c^8 x^8\right ) \text {sech}^{-1}(c x)+840 b \log (x (1-c x))-840 b \log \left (1-c x-\sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^4 x^4}\right )}{3150 c^{12}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.14, size = 0, normalized size = 0.00 \[\int \frac {x^{11} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 393, normalized size = 0.83 \begin {gather*} -\frac {105 \, {\left (3 \, b c^{10} x^{10} - 3 \, b c^{8} x^{8} + 4 \, b c^{6} x^{6} - 4 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (35 \, b c^{9} x^{9} + 5 \, b c^{7} x^{7} + 78 \, b c^{5} x^{5} + 36 \, b c^{3} x^{3} + 768 \, b c x\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 420 \, {\left (b c^{2} x^{2} - b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) - 420 \, {\left (b c^{2} x^{2} - b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) + 105 \, {\left (3 \, a c^{10} x^{10} - 3 \, a c^{8} x^{8} + 4 \, a c^{6} x^{6} - 4 \, a c^{4} x^{4} + 8 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{3150 \, {\left (c^{14} x^{2} - c^{12}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x^{11}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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