Optimal. Leaf size=112 \[ \frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} E\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{5 a^{5/2}}-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{5 a^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6470, 30, 265,
313, 227, 1213, 435} \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 )}{5 a^{5/2}}+\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} E\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac {2 x^3}{15 a}+\frac {1}{5} x^5 e^{\text {sech}^{-1}\left (a x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 227
Rule 265
Rule 313
Rule 435
Rule 1213
Rule 6470
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} x^4 \, dx &=\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\frac {2 \int x^2 \, dx}{5 a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^2}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{5 a}\\ &=\frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^4}} \, dx}{5 a}\\ &=\frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5-\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}{5 a^2}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1+a x^2}{\sqrt {1-a^2 x^4}} \, dx}{5 a^2}\\ &=\frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5-\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 )}{5 a^{5/2}}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {\sqrt {1+a x^2}}{\sqrt {1-a x^2}} \, dx}{5 a^2}\\ &=\frac {2 x^3}{15 a}+\frac {1}{5} e^{\text {sech}^{-1}\left (a x^2\right )} x^5+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} E\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{5 a^{5/2}}-\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 )}{5 a^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.30, size = 140, normalized size = 1.25 \begin {gather*} \frac {1}{15} \left (\frac {5 x^3}{a}+\frac {3 \sqrt {\frac {1-a x^2}{1+a x^2}} \left (x^3+a x^5\right )}{a}+\frac {6 i \sqrt {\frac {1-a x^2}{1+a x^2}} \sqrt {1-a^2 x^4} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )\right )}{(-a)^{5/2} \left (-1+a x^2\right )}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.03, size = 136, normalized size = 1.21
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{\frac {7}{2}} x^{7}-x^{3} a^{\frac {3}{2}}+2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}-2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, \EllipticE \left (x \sqrt {a}, i\right )\right )}{5 \left (a^{2} x^{4}-1\right ) a^{\frac {3}{2}}}+\frac {x^{3}}{3 a}\) | \(136\) |
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 = 0.09, size = 58, normalized size = 0.52 \begin {gather*} \frac {5 \, a x^{3} + 3 \, {\left (a^{2} x^{5} - 2 \, x\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}}}{15 \, a^{2}} \end {gather*}
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 \begin {gather*} \frac {\int x^{2}\, dx + \int a x^{4} \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.
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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 = -0.01 \begin {gather*} \int x^4\,\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.
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