Optimal. Leaf size=196 \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right )}{2 d \log (f)}+\frac{i \text{PolyLog}\left (2,1-\frac{2 b f^{c+d x}}{(-a+i) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right )}{2 d \log (f)}-\frac{\log \left (\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right ) \cot ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)}+\frac{\log \left (\frac{2 b f^{c+d x}}{(-a+i) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right ) \cot ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)} \]
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Rubi [A] time = 0.154011, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2282, 5048, 4857, 2402, 2315, 2447} \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right )}{2 d \log (f)}+\frac{i \text{PolyLog}\left (2,1-\frac{2 b f^{c+d x}}{(-a+i) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right )}{2 d \log (f)}-\frac{\log \left (\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right ) \cot ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)}+\frac{\log \left (\frac{2 b f^{c+d x}}{(-a+i) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right ) \cot ^{-1}\left (a+b f^{c+d x}\right )}{d \log (f)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 5048
Rule 4857
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(a+b x)}{x} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b f^{c+d x}\right )}{b d \log (f)}\\ &=-\frac{\cot ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac{\cot ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac{2 b f^{c+d x}}{(i-a) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right )}{d \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-i x}\right )}{1+x^2} \, dx,x,a+b f^{c+d x}\right )}{d \log (f)}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (-\frac{a}{b}+\frac{x}{b}\right )}{\left (\frac{i}{b}-\frac{a}{b}\right ) (1-i x)}\right )}{1+x^2} \, dx,x,a+b f^{c+d x}\right )}{d \log (f)}\\ &=-\frac{\cot ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac{\cot ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac{2 b f^{c+d x}}{(i-a) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right )}{d \log (f)}+\frac{i \text{Li}_2\left (1-\frac{2 b f^{c+d x}}{(i-a) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right )}{2 d \log (f)}-\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i \left (a+b f^{c+d x}\right )}\right )}{d \log (f)}\\ &=-\frac{\cot ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right )}{d \log (f)}+\frac{\cot ^{-1}\left (a+b f^{c+d x}\right ) \log \left (\frac{2 b f^{c+d x}}{(i-a) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right )}{d \log (f)}-\frac{i \text{Li}_2\left (1-\frac{2}{1-i \left (a+b f^{c+d x}\right )}\right )}{2 d \log (f)}+\frac{i \text{Li}_2\left (1-\frac{2 b f^{c+d x}}{(i-a) \left (1-i \left (a+b f^{c+d x}\right )\right )}\right )}{2 d \log (f)}\\ \end{align*}
Mathematica [A] time = 0.181573, size = 167, normalized size = 0.85 \[ \frac{b \left (\text{PolyLog}\left (2,-\frac{b^2 f^{c+d x}}{a b-\sqrt{-b^2}}\right )-\text{PolyLog}\left (2,-\frac{b^2 f^{c+d x}}{a b+\sqrt{-b^2}}\right )+d x \log (f) \left (\log \left (\frac{b^2 f^{c+d x}}{a b-\sqrt{-b^2}}+1\right )-\log \left (\frac{b^2 f^{c+d x}}{a b+\sqrt{-b^2}}+1\right )\right )\right )}{2 \sqrt{-b^2} d \log (f)}+x \cot ^{-1}\left (a+b f^{c+d x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 186, normalized size = 1. \begin{align*}{\frac{\ln \left ( b{f}^{dx+c} \right ){\rm arccot} \left (a+b{f}^{dx+c}\right )}{d\ln \left ( f \right ) }}-{\frac{{\frac{i}{2}}\ln \left ( b{f}^{dx+c} \right ) }{d\ln \left ( f \right ) }\ln \left ({\frac{-b{f}^{dx+c}-a+i}{i-a}} \right ) }+{\frac{{\frac{i}{2}}\ln \left ( b{f}^{dx+c} \right ) }{d\ln \left ( f \right ) }\ln \left ({\frac{b{f}^{dx+c}+a+i}{i+a}} \right ) }-{\frac{{\frac{i}{2}}}{d\ln \left ( f \right ) }{\it dilog} \left ({\frac{-b{f}^{dx+c}-a+i}{i-a}} \right ) }+{\frac{{\frac{i}{2}}}{d\ln \left ( f \right ) }{\it dilog} \left ({\frac{b{f}^{dx+c}+a+i}{i+a}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71152, size = 302, normalized size = 1.54 \begin{align*} \frac{\operatorname{arccot}\left (b f^{d x + c} + a\right ) \log \left (f^{d x + c}\right )}{d \log \left (f\right )} + \frac{\arctan \left (\frac{b f^{d x + c}}{a^{2} + 1}, -\frac{a b f^{d x + c}}{a^{2} + 1}\right ) \log \left (b^{2} f^{2 \, d x + 2 \, c} + 2 \, a b f^{d x + c} + a^{2} + 1\right ) - \arctan \left (b f^{d x + c} + a\right ) \log \left (\frac{b^{2} f^{2 \, d x + 2 \, c}}{a^{2} + 1}\right ) + 2 \, \arctan \left (\frac{b^{2} f^{d x + c} + a b}{b}\right ) \log \left (f^{d x + c}\right ) + i \,{\rm Li}_2\left (\frac{i \, b f^{d x + c} + i \, a + 1}{i \, a + 1}\right ) - i \,{\rm Li}_2\left (\frac{i \, b f^{d x + c} + i \, a - 1}{i \, a - 1}\right )}{2 \, d \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69097, size = 554, normalized size = 2.83 \begin{align*} \frac{2 \, d x \operatorname{arccot}\left (b f^{d x + c} + a\right ) \log \left (f\right ) - i \, c \log \left (b f^{d x + c} + a + i\right ) \log \left (f\right ) + i \, c \log \left (b f^{d x + c} + a - i\right ) \log \left (f\right ) +{\left (-i \, d x - i \, c\right )} \log \left (f\right ) \log \left (\frac{a^{2} +{\left (a b + i \, b\right )} f^{d x + c} + 1}{a^{2} + 1}\right ) +{\left (i \, d x + i \, c\right )} \log \left (f\right ) \log \left (\frac{a^{2} +{\left (a b - i \, b\right )} f^{d x + c} + 1}{a^{2} + 1}\right ) - i \,{\rm Li}_2\left (-\frac{a^{2} +{\left (a b + i \, b\right )} f^{d x + c} + 1}{a^{2} + 1} + 1\right ) + i \,{\rm Li}_2\left (-\frac{a^{2} +{\left (a b - i \, b\right )} f^{d x + c} + 1}{a^{2} + 1} + 1\right )}{2 \, d \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arccot}\left (b f^{d x + c} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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