Optimal. Leaf size=68 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \sqrt {d+e x^2}}{x^2},x\right )-\frac {a \sqrt {d+e x^2}}{x}+a \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=a \int \frac {\sqrt {d+e x^2}}{x^2} \, dx+b \int \frac {\sqrt {d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx\\ &=-\frac {a \sqrt {d+e x^2}}{x}+b \int \frac {\sqrt {d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx+(a e) \int \frac {1}{\sqrt {d+e x^2}} \, dx\\ &=-\frac {a \sqrt {d+e x^2}}{x}+b \int \frac {\sqrt {d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx+(a e) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )\\ &=-\frac {a \sqrt {d+e x^2}}{x}+a \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+b \int \frac {\sqrt {d+e x^2} \tan ^{-1}(c x)}{x^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 8.95, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (\sqrt {e} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right ) - \frac {\sqrt {e x^{2} + d}}{x}\right )} a + b \int \frac {\sqrt {e x^{2} + d} \arctan \left (c x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________