Optimal. Leaf size=115 \[ \frac {2 x^5}{35 a}+\frac {1}{7} e^{\text {sech}^{-1}\left (a x^2\right )} x^7-\frac {2 x \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{21 a^3}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{21 a^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, 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 = {6470, 30, 265,
327, 227} \begin {gather*} \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\text {ArcSin}\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 )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 227
Rule 265
Rule 327
Rule 6470
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*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 139, normalized size = 1.21 \begin {gather*} \frac {x^5}{5 a}+\frac {x \sqrt {\frac {1-a x^2}{1+a x^2}} \left (-2-2 a x^2+3 a^2 x^4+3 a^3 x^6\right )}{21 a^3}-\frac {2 i \sqrt {\frac {1-a x^2}{1+a x^2}} \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 (-1+a x^2\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 114, normalized size = 0.99
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (3 a^{\frac {9}{2}} x^{9}-5 a^{\frac {5}{2}} x^{5}-2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}+2 x \sqrt {a}\right )}{21 a^{\frac {5}{2}} \left (a^{2} x^{4}-1\right )}+\frac {x^{5}}{5 a}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.08, size = 61, normalized size = 0.53 \begin {gather*} \frac {21 \, a x^{5} + 5 \, {\left (3 \, a^{2} x^{7} - 2 \, x^{3}\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}}}{105 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________