Integrand size = 14, antiderivative size = 250 \[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=-\frac {1}{4} i x^2 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{i-a}\right )}{2 d^2 \log ^2(f)}-\frac {i \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{i+a}\right )}{2 d^2 \log ^2(f)} \]
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Time = 1.93 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5252, 2631, 12, 6874, 2221, 2611, 2320, 6724} \[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {i \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{i-a}\right )}{2 d^2 \log ^2(f)}-\frac {i \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+i}\right )}{2 d^2 \log ^2(f)}-\frac {i x \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+i}\right )}{2 d \log (f)}-\frac {1}{4} i x^2 \log \left (1-\frac {b f^{c+d x}}{-a+i}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {b f^{c+d x}}{a+i}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right ) \]
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 2631
Rule 5252
Rule 6724
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} i \int x \log \left (1-\frac {i}{a+b f^{c+d x}}\right ) \, dx-\frac {1}{2} i \int x \log \left (1+\frac {i}{a+b f^{c+d x}}\right ) \, dx \\ & = \frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{4} \int \frac {b d f^{c+d x} x^2 \log (f)}{\left (i (1-i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx+\frac {1}{4} \int \frac {b d f^{c+d x} x^2 \log (f)}{\left (-i (1+i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx \\ & = \frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{4} (b d \log (f)) \int \frac {f^{c+d x} x^2}{\left (i (1-i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx+\frac {1}{4} (b d \log (f)) \int \frac {f^{c+d x} x^2}{\left (-i (1+i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx \\ & = \frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{4} (b d \log (f)) \int \left (\frac {i f^{c+d x} x^2}{a+b f^{c+d x}}-\frac {i f^{c+d x} x^2}{-i+a+b f^{c+d x}}\right ) \, dx+\frac {1}{4} (b d \log (f)) \int \left (-\frac {i f^{c+d x} x^2}{a+b f^{c+d x}}+\frac {i f^{c+d x} x^2}{i+a+b f^{c+d x}}\right ) \, dx \\ & = \frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} (i b d \log (f)) \int \frac {f^{c+d x} x^2}{-i+a+b f^{c+d x}} \, dx+\frac {1}{4} (i b d \log (f)) \int \frac {f^{c+d x} x^2}{i+a+b f^{c+d x}} \, dx \\ & = -\frac {1}{4} i x^2 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{2} i \int x \log \left (1+\frac {b f^{c+d x}}{-i+a}\right ) \, dx-\frac {1}{2} i \int x \log \left (1+\frac {b f^{c+d x}}{i+a}\right ) \, dx \\ & = -\frac {1}{4} i x^2 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i \int \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{-i+a}\right ) \, dx}{2 d \log (f)}-\frac {i \int \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{i+a}\right ) \, dx}{2 d \log (f)} \\ & = -\frac {1}{4} i x^2 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{i-a}\right )}{x} \, dx,x,f^{c+d x}\right )}{2 d^2 \log ^2(f)}-\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{i+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{2 d^2 \log ^2(f)} \\ & = -\frac {1}{4} i x^2 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{i-a}\right )}{2 d^2 \log ^2(f)}-\frac {i \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{i+a}\right )}{2 d^2 \log ^2(f)} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00 \[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=-\frac {1}{4} i x^2 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{4} i x^2 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{4} i x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{4} i x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{i-a}\right )}{2 d^2 \log ^2(f)}-\frac {i \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{i+a}\right )}{2 d^2 \log ^2(f)} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (218 ) = 436\).
Time = 1.04 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.71
method | result | size |
risch | \(-\frac {i x^{2} \ln \left (1-i \left (a +b \,f^{d x +c}\right )\right )}{4}+\frac {\pi \,x^{2}}{4}-\frac {i c \ln \left (\frac {b \,f^{d x} f^{c}+a +i}{i+a}\right ) x}{2 d}-\frac {i \operatorname {polylog}\left (2, \frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) c}{2 \ln \left (f \right ) d^{2}}+\frac {i c \ln \left (\frac {b \,f^{d x} f^{c}+a -i}{a -i}\right ) x}{2 d}+\frac {i c^{2} \ln \left (1-i a -i f^{d x} f^{c} b \right )}{4 d^{2}}-\frac {i c \operatorname {dilog}\left (\frac {b \,f^{d x} f^{c}+a +i}{i+a}\right )}{2 \ln \left (f \right ) d^{2}}+\frac {i x^{2} \ln \left (1+i \left (a +b \,f^{d x +c}\right )\right )}{4}+\frac {i \operatorname {polylog}\left (2, \frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) c}{2 \ln \left (f \right ) d^{2}}-\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) x^{2}}{4}-\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) c^{2}}{4 d^{2}}+\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) c x}{2 d}+\frac {i \operatorname {polylog}\left (3, \frac {i b \,f^{d x} f^{c}}{-i a -1}\right )}{2 \ln \left (f \right )^{2} d^{2}}+\frac {i \operatorname {polylog}\left (2, \frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) x}{2 \ln \left (f \right ) d}-\frac {i c^{2} \ln \left (i f^{d x} f^{c} b +i a +1\right )}{4 d^{2}}-\frac {i c^{2} \ln \left (\frac {b \,f^{d x} f^{c}+a +i}{i+a}\right )}{2 d^{2}}+\frac {i c^{2} \ln \left (\frac {b \,f^{d x} f^{c}+a -i}{a -i}\right )}{2 d^{2}}-\frac {i \operatorname {polylog}\left (2, \frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) x}{2 \ln \left (f \right ) d}-\frac {i \operatorname {polylog}\left (3, \frac {i b \,f^{d x} f^{c}}{-i a +1}\right )}{2 \ln \left (f \right )^{2} d^{2}}-\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) c x}{2 d}+\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) c^{2}}{4 d^{2}}+\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) x^{2}}{4}+\frac {i c \operatorname {dilog}\left (\frac {b \,f^{d x} f^{c}+a -i}{a -i}\right )}{2 \ln \left (f \right ) d^{2}}\) | \(678\) |
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Time = 0.30 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.22 \[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {2 \, d^{2} x^{2} \operatorname {arccot}\left (b f^{d x + c} + a\right ) \log \left (f\right )^{2} + i \, c^{2} \log \left (b f^{d x + c} + a + i\right ) \log \left (f\right )^{2} - i \, c^{2} \log \left (b f^{d x + c} + a - i\right ) \log \left (f\right )^{2} - 2 i \, d x {\rm Li}_2\left (-\frac {a^{2} + {\left (a b + i \, b\right )} f^{d x + c} + 1}{a^{2} + 1} + 1\right ) \log \left (f\right ) + 2 i \, d x {\rm Li}_2\left (-\frac {a^{2} + {\left (a b - i \, b\right )} f^{d x + c} + 1}{a^{2} + 1} + 1\right ) \log \left (f\right ) + {\left (-i \, d^{2} x^{2} + i \, c^{2}\right )} \log \left (f\right )^{2} \log \left (\frac {a^{2} + {\left (a b + i \, b\right )} f^{d x + c} + 1}{a^{2} + 1}\right ) + {\left (i \, d^{2} x^{2} - i \, c^{2}\right )} \log \left (f\right )^{2} \log \left (\frac {a^{2} + {\left (a b - i \, b\right )} f^{d x + c} + 1}{a^{2} + 1}\right ) + 2 i \, {\rm polylog}\left (3, -\frac {{\left (a b + i \, b\right )} f^{d x + c}}{a^{2} + 1}\right ) - 2 i \, {\rm polylog}\left (3, -\frac {{\left (a b - i \, b\right )} f^{d x + c}}{a^{2} + 1}\right )}{4 \, d^{2} \log \left (f\right )^{2}} \]
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\[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int x \operatorname {acot}{\left (a + b f^{c + d x} \right )}\, dx \]
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\[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int { x \operatorname {arccot}\left (b f^{d x + c} + a\right ) \,d x } \]
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\[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int { x \operatorname {arccot}\left (b f^{d x + c} + a\right ) \,d x } \]
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Timed out. \[ \int x \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int x\,\mathrm {acot}\left (a+b\,f^{c+d\,x}\right ) \,d x \]
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