Optimal. Leaf size=115 \[ \frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{3 a x^3}-\frac {2}{3} \sqrt {a} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right ) \]
[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,
331, 227} \begin {gather*} \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{3 a x^3}-\frac {2}{3} \sqrt {a} \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 )+\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 227
Rule 265
Rule 331
Rule 6470
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x^2} \, dx &=-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}-\frac {2 \int \frac {1}{x^4} \, dx}{a}-\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^4 \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a}\\ &=\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}-\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^4 \sqrt {1-a^2 x^4}} \, dx}{a}\\ &=\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{3 a x^3}-\frac {1}{3} \left (2 a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx\\ &=\frac {2}{3 a x^3}-\frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{3 a x^3}-\frac {2}{3} \sqrt {a} \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 )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 123, normalized size = 1.07 \begin {gather*} -\frac {1}{3 a x^3}-\frac {\sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )}{3 a x^3}+\frac {2 i \sqrt {-a} \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 )}{-3+3 a x^2} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 104, normalized size = 0.90
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, \EllipticF \left (x \sqrt {a}, i\right ) x^{3} a^{\frac {3}{2}}-a^{2} x^{4}+1\right )}{3 x \left (a^{2} x^{4}-1\right )}-\frac {1}{3 a \,x^{3}}\) | \(104\) |
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 = 63, normalized size = 0.55 \begin {gather*} -\frac {2 \, a^{\frac {3}{2}} x^{3} {\rm ellipticF}\left (\sqrt {a} x, -1\right ) + a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1}{3 \, a x^{3}} \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 \frac {1}{x^{4}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}}{x^{2}}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________