Optimal. Leaf size=115 \[ \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{21 a^{7/2}}-\frac {2 x \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{21 a^3}+\frac {2 x^5}{35 a}+\frac {1}{7} x^7 e^{\text {sech}^{-1}\left (a x^2\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6335, 30, 259, 321, 221} \[ -\frac {2 x \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{21 a^3}+\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{21 a^{7/2}}+\frac {2 x^5}{35 a}+\frac {1}{7} x^7 e^{\text {sech}^{-1}\left (a x^2\right )} \]
Antiderivative was successfully verified.
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Rule 30
Rule 221
Rule 259
Rule 321
Rule 6335
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} x^6 \, dx &=\frac {1}{7} e^{\text {sech}^{-1}\left (a x^2\right )} x^7+\frac {2 \int x^4 \, dx}{7 a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^4}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{7 a}\\ &=\frac {2 x^5}{35 a}+\frac {1}{7} e^{\text {sech}^{-1}\left (a x^2\right )} x^7+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^4}{\sqrt {1-a^2 x^4}} \, dx}{7 a}\\ &=\frac {2 x^5}{35 a}+\frac {1}{7} e^{\text {sech}^{-1}\left (a x^2\right )} x^7-\frac {2 x \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{21 a^3}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx}{21 a^3}\\ &=\frac {2 x^5}{35 a}+\frac {1}{7} e^{\text {sech}^{-1}\left (a x^2\right )} x^7-\frac {2 x \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{21 a^3}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{21 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 139, normalized size = 1.21 \[ -\frac {2 i \sqrt {\frac {1-a x^2}{a x^2+1}} \sqrt {1-a^2 x^4} F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )}{21 (-a)^{7/2} \left (a x^2-1\right )}+\frac {x \sqrt {\frac {1-a x^2}{a x^2+1}} \left (3 a^3 x^6+3 a^2 x^4-2 a x^2-2\right )}{21 a^3}+\frac {x^5}{5 a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{6} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + x^{4}}{a}, x\right ) \]
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 [A] time = 0.06, size = 114, normalized size = 0.99 \[ \frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (3 x^{9} a^{\frac {9}{2}}-5 x^{5} a^{\frac {5}{2}}-2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}+2 x \sqrt {a}\right )}{21 a^{\frac {5}{2}} \left (a^{2} x^{4}-1\right )}+\frac {x^{5}}{5 a} \]
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 \[ \frac {x^{5}}{5 \, a} + \frac {\int \sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1} x^{4}\,{d x}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,\left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int x^{4}\, dx + \int a x^{6} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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