Optimal. Leaf size=147 \[ -\frac {2}{a x}+e^{\text {sech}^{-1}\left (a x^2\right )} x-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{a x}-\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} E\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}}+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{\sqrt {a}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6465, 30, 265,
331, 313, 227, 1213, 435} \begin {gather*} -\frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{a x}+\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 )}{\sqrt {a}}-\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 )}{\sqrt {a}}+x e^{\text {sech}^{-1}\left (a x^2\right )}-\frac {2}{a x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 227
Rule 265
Rule 313
Rule 331
Rule 435
Rule 1213
Rule 6465
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} \, dx &=e^{\text {sech}^{-1}\left (a x^2\right )} x+\frac {2 \int \frac {1}{x^2} \, dx}{a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x^2 \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a}\\ &=-\frac {2}{a x}+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^2 \sqrt {1-a^2 x^4}} \, dx}{a}\\ &=-\frac {2}{a x}+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}}{a x}-\left (2 a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^2}{\sqrt {1-a^2 x^4}} \, dx\\ &=-\frac {2}{a x}+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}}{a x}+\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-\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\\ &=-\frac {2}{a x}+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}}{a x}+\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 )}{\sqrt {a}}-\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\\ &=-\frac {2}{a x}+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}}{a x}-\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 )}{\sqrt {a}}+\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 )}{\sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 135, normalized size = 0.92 \begin {gather*} -\frac {1}{a x}+\left (-\frac {1}{a x}-x\right ) \sqrt {\frac {1-a x^2}{1+a x^2}}-\frac {2 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 )}{\sqrt {-a} \left (-1+a x^2\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 132, normalized size = 0.90
method | result | size |
default | \(-\frac {1}{a x}-\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{2} x^{4}+2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {a}-2 \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}\, x \EllipticE \left (x \sqrt {a}, i\right ) \sqrt {a}-1\right )}{a^{2} x^{4}-1}\) | \(132\) |
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 [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{x^{2}}\, dx + \int a \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]
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 \sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________