Optimal. Leaf size=162 \[ -\frac {2 x \sqrt {a^2 x^4+1}}{a x^2+1}-\frac {\left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {a^2 x^4+1}}+\frac {2 \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {a^2 x^4+1}}+x \sinh ^{-1}\left (a x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5900, 12, 305, 220, 1196} \[ -\frac {2 x \sqrt {a^2 x^4+1}}{a x^2+1}-\frac {\left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {a^2 x^4+1}}+\frac {2 \left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {a^2 x^4+1}}+x \sinh ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 220
Rule 305
Rule 1196
Rule 5900
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (a x^2\right ) \, dx &=x \sinh ^{-1}\left (a x^2\right )-\int \frac {2 a x^2}{\sqrt {1+a^2 x^4}} \, dx\\ &=x \sinh ^{-1}\left (a x^2\right )-(2 a) \int \frac {x^2}{\sqrt {1+a^2 x^4}} \, dx\\ &=x \sinh ^{-1}\left (a x^2\right )-2 \int \frac {1}{\sqrt {1+a^2 x^4}} \, dx+2 \int \frac {1-a x^2}{\sqrt {1+a^2 x^4}} \, dx\\ &=-\frac {2 x \sqrt {1+a^2 x^4}}{1+a x^2}+x \sinh ^{-1}\left (a x^2\right )+\frac {2 \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {1+a^2 x^4}}-\frac {\left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {1+a^2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 35, normalized size = 0.22 \[ x \sinh ^{-1}\left (a x^2\right )-\frac {2}{3} a x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-a^2 x^4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arsinh}\left (a x^{2}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arsinh}\left (a x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.01, size = 77, normalized size = 0.48 \[ x \arcsinh \left (a \,x^{2}\right )-\frac {2 i \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {i a}, i\right )-\EllipticE \left (x \sqrt {i a}, i\right )\right )}{\sqrt {i a}\, \sqrt {a^{2} x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, a \int \frac {x^{2}}{a^{3} x^{6} + a x^{2} + {\left (a^{2} x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} + x \log \left (a x^{2} + \sqrt {a^{2} x^{4} + 1}\right ) - 2 \, x - \frac {i \, \sqrt {2} {\left (\log \left (\frac {i \, \sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right ) - \log \left (-\frac {i \, \sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right )\right )}}{4 \, \sqrt {a}} - \frac {i \, \sqrt {2} {\left (\log \left (\frac {i \, \sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right ) - \log \left (-\frac {i \, \sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}} + 1\right )\right )}}{4 \, \sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{4 \, \sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{4 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {asinh}\left (a\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {asinh}{\left (a x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________