Optimal. Leaf size=59 \[ \frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 c}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4930, 217, 206} \[ \frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 c}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a}\\ &=\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^2 c}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a}\\ &=\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^2 c}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2 \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 60, normalized size = 1.02 \[ \frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\sqrt {c} \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+a c x\right )}{a^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 64, normalized size = 1.08 \[ \frac {2 \, \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ) + \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{2 \, a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.92, size = 144, normalized size = 2.44 \[ \frac {\arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{2} c}-\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{2} c}+\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 61, normalized size = 1.03 \[ \frac {2 \, \sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a^{2} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {atan}\left (a\,x\right )}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________