Optimal. Leaf size=216 \[ -\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c (a+b x)^2+c}}+\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c (a+b x)^2+c}}-\frac{2 i \sqrt{(a+b x)^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right ) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}} \]
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Rubi [A] time = 0.167586, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5056, 4891, 4887} \[ -\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c (a+b x)^2+c}}+\frac{i \sqrt{(a+b x)^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c (a+b x)^2+c}}-\frac{2 i \sqrt{(a+b x)^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right ) \cot ^{-1}(a+b x)}{b \sqrt{c (a+b x)^2+c}} \]
Antiderivative was successfully verified.
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Rule 5056
Rule 4891
Rule 4887
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a+b x)}{\sqrt{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{\sqrt{c+c x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\sqrt{1+(a+b x)^2} \operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{\sqrt{1+x^2}} \, dx,x,a+b x\right )}{b \sqrt{c+c (a+b x)^2}}\\ &=-\frac{2 i \sqrt{1+(a+b x)^2} \cot ^{-1}(a+b x) \tan ^{-1}\left (\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c+c (a+b x)^2}}-\frac{i \sqrt{1+(a+b x)^2} \text{Li}_2\left (-\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c+c (a+b x)^2}}+\frac{i \sqrt{1+(a+b x)^2} \text{Li}_2\left (\frac{i \sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}\right )}{b \sqrt{c+c (a+b x)^2}}\\ \end{align*}
Mathematica [A] time = 0.0765339, size = 138, normalized size = 0.64 \[ -\frac{\left ((a+b x)^2+1\right ) \left (i \text{PolyLog}\left (2,-e^{i \cot ^{-1}(a+b x)}\right )-i \text{PolyLog}\left (2,e^{i \cot ^{-1}(a+b x)}\right )+\cot ^{-1}(a+b x) \left (\log \left (1-e^{i \cot ^{-1}(a+b x)}\right )-\log \left (1+e^{i \cot ^{-1}(a+b x)}\right )\right )\right )}{b (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1} \sqrt{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.518, size = 156, normalized size = 0.7 \begin{align*}{\frac{i}{cb} \left ( i{\rm arccot} \left (bx+a\right )\ln \left ( 1-{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) -i{\rm arccot} \left (bx+a\right )\ln \left ( 1+{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) -{\it polylog} \left ( 2,-{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) +{\it polylog} \left ( 2,{(i+a+bx){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) \right ) \sqrt{c \left ( -i+a+bx \right ) \left ( i+a+bx \right ) }{\frac{1}{\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}} \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{\operatorname{arccot}\left (b x + a\right )}{\sqrt{b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c}}, 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{\operatorname{acot}{\left (a + b x \right )}}{\sqrt{c \left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )}}\, 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{\operatorname{arccot}\left (b x + a\right )}{\sqrt{b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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