Optimal. Leaf size=82 \[ -i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.146934, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4852, 4924, 4868, 2447} \[ -i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4852
Rule 4924
Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x}+(2 b c) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx\\ &=-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x}+(2 i b c) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx\\ &=-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )-\left (2 b^2 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )-i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )\\ \end{align*}
Mathematica [A] time = 0.14126, size = 102, normalized size = 1.24 \[ \frac{-i b^2 c x \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )-a \left (a+b c x \log \left (c^2 x^2+1\right )-2 b c x \log (c x)\right )+2 b \tan ^{-1}(c x) \left (-a+b c x \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )+b^2 (-1-i c x) \tan ^{-1}(c x)^2}{x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 323, normalized size = 3.9 \begin{align*} -{\frac{{a}^{2}}{x}}-{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{x}}-c{b}^{2}\arctan \left ( cx \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) +2\,c{b}^{2}\ln \left ( cx \right ) \arctan \left ( cx \right ) +{\frac{i}{2}}c{b}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) +ic{b}^{2}{\it dilog} \left ( 1+icx \right ) +{\frac{i}{2}}c{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) \ln \left ( cx+i \right ) -ic{b}^{2}\ln \left ( cx \right ) \ln \left ( 1-icx \right ) -{\frac{i}{2}}c{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) \ln \left ( cx-i \right ) -{\frac{i}{2}}c{b}^{2}\ln \left ( cx+i \right ) \ln \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) -{\frac{i}{4}}c{b}^{2} \left ( \ln \left ( cx+i \right ) \right ) ^{2}-{\frac{i}{2}}c{b}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) +ic{b}^{2}\ln \left ( cx \right ) \ln \left ( 1+icx \right ) +{\frac{i}{2}}c{b}^{2}\ln \left ( cx-i \right ) \ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) -ic{b}^{2}{\it dilog} \left ( 1-icx \right ) +{\frac{i}{4}}c{b}^{2} \left ( \ln \left ( cx-i \right ) \right ) ^{2}-2\,{\frac{ab\arctan \left ( cx \right ) }{x}}-cab\ln \left ({c}^{2}{x}^{2}+1 \right ) +2\,cab\ln \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{x^{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{\left (a + b \operatorname{atan}{\left (c x \right )}\right )^{2}}{x^{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{{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]