Optimal. Leaf size=103 \[ -\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{c^{3/2}}-\frac{a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20189, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4966, 4944, 266, 63, 208, 4894} \[ -\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{c^{3/2}}-\frac{a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4966
Rule 4944
Rule 266
Rule 63
Rule 208
Rule 4894
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=-\frac{a}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}+\frac{a \int \frac{1}{x \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=-\frac{a}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )}{2 c}\\ &=-\frac{a}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )}{a c^2}\\ &=-\frac{a}{c \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.181418, size = 122, normalized size = 1.18 \[ -\frac{a \sqrt{c \left (a^2 x^2+1\right )}}{c^2 \left (a^2 x^2+1\right )}-\frac{a \log \left (\sqrt{c} \sqrt{c \left (a^2 x^2+1\right )}+c\right )}{c^{3/2}}-\frac{\left (2 a^2 x^2+1\right ) \sqrt{c \left (a^2 x^2+1\right )} \tan ^{-1}(a x)}{c^2 x \left (a^2 x^2+1\right )}+\frac{a \log (x)}{c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.295, size = 231, normalized size = 2.2 \begin{align*} -{\frac{a \left ( \arctan \left ( ax \right ) +i \right ) \left ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( ax+i \right ) \left ( \arctan \left ( ax \right ) -i \right ) a}{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{\arctan \left ( ax \right ) }{{c}^{2}x}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{a}{{c}^{2}}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{a}{{c}^{2}}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-1 \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.69173, size = 234, normalized size = 2.27 \begin{align*} \frac{{\left (a^{3} x^{3} + a x\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{a^{2} c x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt{a^{2} c x^{2} + c}{\left (a x +{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )}}{2 \,{\left (a^{2} c^{2} x^{3} + c^{2} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]