Optimal. Leaf size=187 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b}-\frac{\sqrt{(a+b x)^2+1}}{2 b}+\frac{i \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right ) \tan ^{-1}(a+b x)}{b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \tan ^{-1}(a+b x)}{2 b} \]
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Rubi [A] time = 0.215784, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {5057, 4952, 261, 4886} \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b}-\frac{\sqrt{(a+b x)^2+1}}{2 b}+\frac{i \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right ) \tan ^{-1}(a+b x)}{b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \tan ^{-1}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5057
Rule 4952
Rule 261
Rule 4886
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \tan ^{-1}(x)}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sqrt{1+(a+b x)^2} \tan ^{-1}(a+b x)}{2 b}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{2 b}-\frac{\operatorname{Subst}\left (\int \frac{\tan ^{-1}(x)}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{\sqrt{1+(a+b x)^2}}{2 b}+\frac{(a+b x) \sqrt{1+(a+b x)^2} \tan ^{-1}(a+b x)}{2 b}+\frac{i \tan ^{-1}(a+b x) \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b}-\frac{i \text{Li}_2\left (-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b}+\frac{i \text{Li}_2\left (\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.657008, size = 145, normalized size = 0.78 \[ \frac{-i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a+b x)}\right )+i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a+b x)}\right )-\sqrt{(a+b x)^2+1}+(a+b x) \sqrt{(a+b x)^2+1} \tan ^{-1}(a+b x)+\tan ^{-1}(a+b x) \left (-\log \left (1-i e^{i \tan ^{-1}(a+b x)}\right )\right )+\tan ^{-1}(a+b x) \log \left (1+i e^{i \tan ^{-1}(a+b x)}\right )}{2 b} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.619, size = 187, normalized size = 1. \begin{align*}{\frac{\arctan \left ( bx+a \right ) xb+\arctan \left ( bx+a \right ) a-1}{2\,b}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{\arctan \left ( bx+a \right ) }{2\,b}\ln \left ( 1+{i \left ( 1+i \left ( bx+a \right ) \right ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) }-{\frac{\arctan \left ( bx+a \right ) }{2\,b}\ln \left ( 1-{i \left ( 1+i \left ( bx+a \right ) \right ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) }-{\frac{{\frac{i}{2}}}{b}{\it dilog} \left ( 1+{i \left ( 1+i \left ( bx+a \right ) \right ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) }+{\frac{{\frac{i}{2}}}{b}{\it dilog} \left ( 1-{i \left ( 1+i \left ( bx+a \right ) \right ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \arctan \left (b x + a\right )}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{2} \operatorname{atan}{\left (a + b x \right )}}{\sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{2} \arctan \left (b x + a\right )}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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