Optimal. Leaf size=313 \[ -\frac {i \text {Li}_4\left (\frac {b f^{c+d x}}{i-a}\right )}{d^3 \log ^3(f)}+\frac {i \text {Li}_4\left (-\frac {b f^{c+d x}}{a+i}\right )}{d^3 \log ^3(f)}+\frac {i x \text {Li}_3\left (\frac {b f^{c+d x}}{i-a}\right )}{d^2 \log ^2(f)}-\frac {i x \text {Li}_3\left (-\frac {b f^{c+d x}}{a+i}\right )}{d^2 \log ^2(f)}-\frac {i x^2 \text {Li}_2\left (\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x^2 \text {Li}_2\left (-\frac {b f^{c+d x}}{a+i}\right )}{2 d \log (f)}-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{-a+i}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{a+i}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right ) \]
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Rubi [A] time = 2.45, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5144, 2551, 12, 6742, 2190, 2531, 6609, 2282, 6589} \[ \frac {i x \text {PolyLog}\left (3,\frac {b f^{c+d x}}{-a+i}\right )}{d^2 \log ^2(f)}-\frac {i x \text {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+i}\right )}{d^2 \log ^2(f)}-\frac {i \text {PolyLog}\left (4,\frac {b f^{c+d x}}{-a+i}\right )}{d^3 \log ^3(f)}+\frac {i \text {PolyLog}\left (4,-\frac {b f^{c+d x}}{a+i}\right )}{d^3 \log ^3(f)}-\frac {i x^2 \text {PolyLog}\left (2,\frac {b f^{c+d x}}{-a+i}\right )}{2 d \log (f)}+\frac {i x^2 \text {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+i}\right )}{2 d \log (f)}-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{-a+i}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{a+i}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2190
Rule 2282
Rule 2531
Rule 2551
Rule 5144
Rule 6589
Rule 6609
Rule 6742
Rubi steps
\begin {align*} \int x^2 \cot ^{-1}\left (a+b f^{c+d x}\right ) \, dx &=\frac {1}{2} i \int x^2 \log \left (1-\frac {i}{a+b f^{c+d x}}\right ) \, dx-\frac {1}{2} i \int x^2 \log \left (1+\frac {i}{a+b f^{c+d x}}\right ) \, dx\\ &=\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{6} \int \frac {b d f^{c+d x} x^3 \log (f)}{\left (i (1-i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx+\frac {1}{6} \int \frac {b d f^{c+d x} x^3 \log (f)}{\left (-i (1+i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx\\ &=\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{6} (b d \log (f)) \int \frac {f^{c+d x} x^3}{\left (i (1-i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx+\frac {1}{6} (b d \log (f)) \int \frac {f^{c+d x} x^3}{\left (-i (1+i a)+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx\\ &=\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{6} (b d \log (f)) \int \left (\frac {i f^{c+d x} x^3}{a+b f^{c+d x}}-\frac {i f^{c+d x} x^3}{-i+a+b f^{c+d x}}\right ) \, dx+\frac {1}{6} (b d \log (f)) \int \left (-\frac {i f^{c+d x} x^3}{a+b f^{c+d x}}+\frac {i f^{c+d x} x^3}{i+a+b f^{c+d x}}\right ) \, dx\\ &=\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} (i b d \log (f)) \int \frac {f^{c+d x} x^3}{-i+a+b f^{c+d x}} \, dx+\frac {1}{6} (i b d \log (f)) \int \frac {f^{c+d x} x^3}{i+a+b f^{c+d x}} \, dx\\ &=-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )+\frac {1}{2} i \int x^2 \log \left (1+\frac {b f^{c+d x}}{-i+a}\right ) \, dx-\frac {1}{2} i \int x^2 \log \left (1+\frac {b f^{c+d x}}{i+a}\right ) \, dx\\ &=-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x^2 \text {Li}_2\left (\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x^2 \text {Li}_2\left (-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i \int x \text {Li}_2\left (-\frac {b f^{c+d x}}{-i+a}\right ) \, dx}{d \log (f)}-\frac {i \int x \text {Li}_2\left (-\frac {b f^{c+d x}}{i+a}\right ) \, dx}{d \log (f)}\\ &=-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x^2 \text {Li}_2\left (\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x^2 \text {Li}_2\left (-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i x \text {Li}_3\left (\frac {b f^{c+d x}}{i-a}\right )}{d^2 \log ^2(f)}-\frac {i x \text {Li}_3\left (-\frac {b f^{c+d x}}{i+a}\right )}{d^2 \log ^2(f)}-\frac {i \int \text {Li}_3\left (-\frac {b f^{c+d x}}{-i+a}\right ) \, dx}{d^2 \log ^2(f)}+\frac {i \int \text {Li}_3\left (-\frac {b f^{c+d x}}{i+a}\right ) \, dx}{d^2 \log ^2(f)}\\ &=-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x^2 \text {Li}_2\left (\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x^2 \text {Li}_2\left (-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i x \text {Li}_3\left (\frac {b f^{c+d x}}{i-a}\right )}{d^2 \log ^2(f)}-\frac {i x \text {Li}_3\left (-\frac {b f^{c+d x}}{i+a}\right )}{d^2 \log ^2(f)}-\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{i-a}\right )}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}+\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{i+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}\\ &=-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{i-a}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{i+a}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right )-\frac {i x^2 \text {Li}_2\left (\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x^2 \text {Li}_2\left (-\frac {b f^{c+d x}}{i+a}\right )}{2 d \log (f)}+\frac {i x \text {Li}_3\left (\frac {b f^{c+d x}}{i-a}\right )}{d^2 \log ^2(f)}-\frac {i x \text {Li}_3\left (-\frac {b f^{c+d x}}{i+a}\right )}{d^2 \log ^2(f)}-\frac {i \text {Li}_4\left (\frac {b f^{c+d x}}{i-a}\right )}{d^3 \log ^3(f)}+\frac {i \text {Li}_4\left (-\frac {b f^{c+d x}}{i+a}\right )}{d^3 \log ^3(f)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 313, normalized size = 1.00 \[ -\frac {i \text {Li}_4\left (\frac {b f^{c+d x}}{i-a}\right )}{d^3 \log ^3(f)}+\frac {i \text {Li}_4\left (-\frac {b f^{c+d x}}{a+i}\right )}{d^3 \log ^3(f)}+\frac {i x \text {Li}_3\left (\frac {b f^{c+d x}}{i-a}\right )}{d^2 \log ^2(f)}-\frac {i x \text {Li}_3\left (-\frac {b f^{c+d x}}{a+i}\right )}{d^2 \log ^2(f)}-\frac {i x^2 \text {Li}_2\left (\frac {b f^{c+d x}}{i-a}\right )}{2 d \log (f)}+\frac {i x^2 \text {Li}_2\left (-\frac {b f^{c+d x}}{a+i}\right )}{2 d \log (f)}-\frac {1}{6} i x^3 \log \left (1-\frac {b f^{c+d x}}{-a+i}\right )+\frac {1}{6} i x^3 \log \left (1+\frac {b f^{c+d x}}{a+i}\right )+\frac {1}{6} i x^3 \log \left (1-\frac {i}{a+b f^{c+d x}}\right )-\frac {1}{6} i x^3 \log \left (1+\frac {i}{a+b f^{c+d x}}\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.79, size = 378, normalized size = 1.21 \[ \frac {2 \, d^{3} x^{3} \operatorname {arccot}\left (b f^{d x + c} + a\right ) \log \relax (f)^{3} - 3 i \, d^{2} x^{2} {\rm Li}_2\left (-\frac {a^{2} + {\left (a b + i \, b\right )} f^{d x + c} + 1}{a^{2} + 1} + 1\right ) \log \relax (f)^{2} + 3 i \, d^{2} x^{2} {\rm Li}_2\left (-\frac {a^{2} + {\left (a b - i \, b\right )} f^{d x + c} + 1}{a^{2} + 1} + 1\right ) \log \relax (f)^{2} - i \, c^{3} \log \left (b f^{d x + c} + a + i\right ) \log \relax (f)^{3} + i \, c^{3} \log \left (b f^{d x + c} + a - i\right ) \log \relax (f)^{3} + {\left (-i \, d^{3} x^{3} - i \, c^{3}\right )} \log \relax (f)^{3} \log \left (\frac {a^{2} + {\left (a b + i \, b\right )} f^{d x + c} + 1}{a^{2} + 1}\right ) + {\left (i \, d^{3} x^{3} + i \, c^{3}\right )} \log \relax (f)^{3} \log \left (\frac {a^{2} + {\left (a b - i \, b\right )} f^{d x + c} + 1}{a^{2} + 1}\right ) + 6 i \, d x \log \relax (f) {\rm polylog}\left (3, -\frac {{\left (a b + i \, b\right )} f^{d x + c}}{a^{2} + 1}\right ) - 6 i \, d x \log \relax (f) {\rm polylog}\left (3, -\frac {{\left (a b - i \, b\right )} f^{d x + c}}{a^{2} + 1}\right ) - 6 i \, {\rm polylog}\left (4, -\frac {{\left (a b + i \, b\right )} f^{d x + c}}{a^{2} + 1}\right ) + 6 i \, {\rm polylog}\left (4, -\frac {{\left (a b - i \, b\right )} f^{d x + c}}{a^{2} + 1}\right )}{6 \, d^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arccot}\left (b f^{d x + c} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 764, normalized size = 2.44 \[ -\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) x \,c^{2}}{2 d^{2}}-\frac {i c^{2} \ln \left (\frac {b \,f^{d x} f^{c}+a -i}{-i+a}\right ) x}{2 d^{2}}+\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) x \,c^{2}}{2 d^{2}}+\frac {i c^{2} \ln \left (\frac {b \,f^{d x} f^{c}+i+a}{i+a}\right ) x}{2 d^{2}}+\frac {i \polylog \left (2, \frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) c^{2}}{2 d^{3} \ln \relax (f )}-\frac {i c^{2} \dilog \left (\frac {b \,f^{d x} f^{c}+a -i}{-i+a}\right )}{2 d^{3} \ln \relax (f )}+\frac {i \polylog \left (2, \frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) x^{2}}{2 d \ln \relax (f )}-\frac {i \polylog \left (2, \frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) c^{2}}{2 d^{3} \ln \relax (f )}+\frac {i c^{2} \dilog \left (\frac {b \,f^{d x} f^{c}+i+a}{i+a}\right )}{2 d^{3} \ln \relax (f )}-\frac {i \polylog \left (4, \frac {i b \,f^{d x} f^{c}}{-i a -1}\right )}{d^{3} \ln \relax (f )^{3}}+\frac {i \polylog \left (4, \frac {i b \,f^{d x} f^{c}}{-i a +1}\right )}{d^{3} \ln \relax (f )^{3}}+\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) c^{3}}{3 d^{3}}-\frac {i c^{3} \ln \left (\frac {b \,f^{d x} f^{c}+a -i}{-i+a}\right )}{2 d^{3}}+\frac {i c^{3} \ln \left (\frac {b \,f^{d x} f^{c}+i+a}{i+a}\right )}{2 d^{3}}-\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) c^{3}}{3 d^{3}}-\frac {i c^{3} \ln \left (1-i a -i f^{d x} f^{c} b \right )}{6 d^{3}}+\frac {i c^{3} \ln \left (i f^{d x} f^{c} b +i a +1\right )}{6 d^{3}}+\frac {\pi \,x^{3}}{6}-\frac {i \polylog \left (3, \frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) x}{d^{2} \ln \relax (f )^{2}}+\frac {i \polylog \left (3, \frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) x}{d^{2} \ln \relax (f )^{2}}-\frac {i \polylog \left (2, \frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) x^{2}}{2 d \ln \relax (f )}+\frac {i x^{3} \ln \left (1+i \left (a +b \,f^{d x +c}\right )\right )}{6}-\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a -1}\right ) x^{3}}{6}+\frac {i \ln \left (1-\frac {i b \,f^{d x} f^{c}}{-i a +1}\right ) x^{3}}{6}-\frac {i x^{3} \ln \left (1-i \left (a +b \,f^{d x +c}\right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b d f^{c} \int \frac {f^{d x} x^{3}}{3 \, {\left (b^{2} f^{2 \, d x} f^{2 \, c} + 2 \, a b f^{d x} f^{c} + a^{2} + 1\right )}}\,{d x} \log \relax (f) + \frac {1}{3} \, x^{3} \arctan \left (\frac {1}{b f^{d x} f^{c} + a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\mathrm {acot}\left (a+b\,f^{c+d\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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