Optimal. Leaf size=162 \[ -\frac {2 x \sqrt {1+a^2 x^4}}{1+a x^2}+x \sinh ^{-1}\left (a x^2\right )+\frac {2 \left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} E\left (2 \text {ArcTan}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {1+a^2 x^4}}-\frac {\left (1+a x^2\right ) \sqrt {\frac {1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \text {ArcTan}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {1+a^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {5874, 12, 311,
226, 1210} \begin {gather*} -\frac {\left (a x^2+1\right ) \sqrt {\frac {a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \text {ArcTan}\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 \text {ArcTan}\left (\sqrt {a} x\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt {a^2 x^4+1}}-\frac {2 x \sqrt {a^2 x^4+1}}{a x^2+1}+x \sinh ^{-1}\left (a x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 226
Rule 311
Rule 1210
Rule 5874
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 35, normalized size = 0.22 \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 77, normalized size = 0.48
method | result | size |
default | \(x \arcsinh \left (a \,x^{2}\right )-\frac {2 i \sqrt {-i a \,x^{2}+1}\, \sqrt {i a \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {i a}, i\right )-\EllipticE \left (x \sqrt {i a}, i\right )\right )}{\sqrt {i a}\, \sqrt {a^{2} x^{4}+1}}\) | \(77\) |
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(-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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {asinh}{\left (a x^{2} \right )}\, dx \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 \mathrm {asinh}\left (a\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________