Optimal. Leaf size=181 \[ \frac {i b d (c+d x) \text {Li}_2\left (\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )}-\frac {b (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f \left (a^2+b^2\right )}-\frac {b d^2 \text {Li}_3\left (\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 f^3 \left (a^2+b^2\right )}+\frac {(c+d x)^3}{3 d (a-i b)} \]
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Rubi [A] time = 0.29, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3731, 2190, 2531, 2282, 6589} \[ \frac {i b d (c+d x) \text {PolyLog}\left (2,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f^2 \left (a^2+b^2\right )}-\frac {b d^2 \text {PolyLog}\left (3,\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 f^3 \left (a^2+b^2\right )}-\frac {b (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{f \left (a^2+b^2\right )}+\frac {(c+d x)^3}{3 d (a-i b)} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3731
Rule 6589
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{a+b \cot (e+f x)} \, dx &=\frac {(c+d x)^3}{3 (a-i b) d}+(2 i b) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{(a-i b)^2+\left (-a^2-b^2\right ) e^{2 i (e+f x)}} \, dx\\ &=\frac {(c+d x)^3}{3 (a-i b) d}-\frac {b (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {(2 b d) \int (c+d x) \log \left (1+\frac {\left (-a^2-b^2\right ) e^{2 i (e+f x)}}{(a-i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f}\\ &=\frac {(c+d x)^3}{3 (a-i b) d}-\frac {b (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {i b d (c+d x) \text {Li}_2\left (\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f^2}-\frac {\left (i b d^2\right ) \int \text {Li}_2\left (-\frac {\left (-a^2-b^2\right ) e^{2 i (e+f x)}}{(a-i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f^2}\\ &=\frac {(c+d x)^3}{3 (a-i b) d}-\frac {b (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {i b d (c+d x) \text {Li}_2\left (\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f^2}-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {(a+i b) x}{a-i b}\right )}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 \left (a^2+b^2\right ) f^3}\\ &=\frac {(c+d x)^3}{3 (a-i b) d}-\frac {b (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f}+\frac {i b d (c+d x) \text {Li}_2\left (\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{\left (a^2+b^2\right ) f^2}-\frac {b d^2 \text {Li}_3\left (\frac {(a+i b) e^{2 i (e+f x)}}{a-i b}\right )}{2 \left (a^2+b^2\right ) f^3}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 289, normalized size = 1.60 \[ \frac {x \sin (e) \left (3 c^2+3 c d x+d^2 x^2\right )}{3 (a \sin (e)+b \cos (e))}+\frac {b \left (\frac {3 d \left (b \left (1+e^{2 i e}\right )-i a \left (-1+e^{2 i e}\right )\right ) \left (2 f (c+d x) \text {Li}_2\left (\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )-i d \text {Li}_3\left (\frac {(a-i b) e^{-2 i (e+f x)}}{a+i b}\right )\right )}{f^3 \left (a^2+b^2\right )}-\frac {6 \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right ) (c+d x)^2 \log \left (1+\frac {(-a+i b) e^{-2 i (e+f x)}}{a+i b}\right )}{f \left (a^2+b^2\right )}+\frac {4 i (c+d x)^3}{d (a+i b)}\right )}{6 \left (a \left (-1+e^{2 i e}\right )+i b \left (1+e^{2 i e}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.75, size = 730, normalized size = 4.03 \[ \frac {4 \, a d^{2} f^{3} x^{3} + 12 \, a c d f^{3} x^{2} + 12 \, a c^{2} f^{3} x - 3 \, b d^{2} {\rm polylog}\left (3, \frac {{\left (a^{2} + 2 i \, a b - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) - 3 \, b d^{2} {\rm polylog}\left (3, \frac {{\left (a^{2} - 2 i \, a b - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) + {\left (6 i \, b d^{2} f x + 6 i \, b c d f\right )} {\rm Li}_2\left (-\frac {a^{2} + b^{2} - {\left (a^{2} + 2 i \, a b - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}} + 1\right ) + {\left (-6 i \, b d^{2} f x - 6 i \, b c d f\right )} {\rm Li}_2\left (-\frac {a^{2} + b^{2} - {\left (a^{2} - 2 i \, a b - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}} + 1\right ) - 6 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (\frac {1}{2} \, a^{2} + i \, a b - \frac {1}{2} \, b^{2} - \frac {1}{2} \, {\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + \frac {1}{2} \, {\left (i \, a^{2} + i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )\right ) - 6 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (-\frac {1}{2} \, a^{2} + i \, a b + \frac {1}{2} \, b^{2} + \frac {1}{2} \, {\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + \frac {1}{2} \, {\left (i \, a^{2} + i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )\right ) - 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (\frac {a^{2} + b^{2} - {\left (a^{2} + 2 i \, a b - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) - 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (\frac {a^{2} + b^{2} - {\left (a^{2} - 2 i \, a b - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right )}{12 \, {\left (a^{2} + b^{2}\right )} 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 \[ \int \frac {{\left (d x + c\right )}^{2}}{b \cot \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.27, size = 881, normalized size = 4.87 \[ \frac {d^{2} x^{3}}{3 i b +3 a}+\frac {c d \,x^{2}}{i b +a}+\frac {c^{2} x}{i b +a}-\frac {i b \,c^{2} \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{f \left (-i a +b \right ) \left (i b -a \right )}+\frac {i b \,d^{2} \ln \left (1-\frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right ) x^{2}}{f \left (-i a +b \right ) \left (-i b +a \right )}+\frac {i b \,d^{2} \polylog \left (3, \frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right )}{2 f^{3} \left (-i a +b \right ) \left (-i b +a \right )}+\frac {2 i b c d e \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{f^{2} \left (-i a +b \right ) \left (i b -a \right )}+\frac {2 b \,d^{2} x^{3}}{3 \left (-i a +b \right ) \left (-i b +a \right )}-\frac {2 b \,d^{2} e^{2} x}{f^{2} \left (-i a +b \right ) \left (-i b +a \right )}-\frac {4 b \,d^{2} e^{3}}{3 f^{3} \left (-i a +b \right ) \left (-i b +a \right )}-\frac {4 i b c d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2} \left (-i a +b \right ) \left (i b -a \right )}+\frac {2 i b c d \ln \left (1-\frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right ) x}{f \left (-i a +b \right ) \left (-i b +a \right )}+\frac {2 i b \,d^{2} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3} \left (-i a +b \right ) \left (i b -a \right )}+\frac {2 i b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f \left (-i a +b \right ) \left (i b -a \right )}-\frac {i b \,d^{2} e^{2} \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+i {\mathrm e}^{2 i \left (f x +e \right )} b -a +i b \right )}{f^{3} \left (-i a +b \right ) \left (i b -a \right )}+\frac {b \,d^{2} \polylog \left (2, \frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right ) x}{f^{2} \left (-i a +b \right ) \left (-i b +a \right )}-\frac {i b \,d^{2} \ln \left (1-\frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right ) e^{2}}{f^{3} \left (-i a +b \right ) \left (-i b +a \right )}+\frac {2 i b c d \ln \left (1-\frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right ) e}{f^{2} \left (-i a +b \right ) \left (-i b +a \right )}+\frac {2 b c d \,x^{2}}{\left (-i a +b \right ) \left (-i b +a \right )}+\frac {4 b c d e x}{f \left (-i a +b \right ) \left (-i b +a \right )}+\frac {2 b c d \,e^{2}}{f^{2} \left (-i a +b \right ) \left (-i b +a \right )}+\frac {b c d \polylog \left (2, \frac {\left (i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b +a}\right )}{f^{2} \left (-i a +b \right ) \left (-i b +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.82, size = 719, normalized size = 3.97 \[ -\frac {6 \, c d e {\left (\frac {2 \, {\left (f x + e\right )} a}{{\left (a^{2} + b^{2}\right )} f} - \frac {2 \, b \log \left (a \tan \left (f x + e\right ) + b\right )}{{\left (a^{2} + b^{2}\right )} f} + \frac {b \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{2} + b^{2}\right )} f}\right )} - 3 \, {\left (\frac {2 \, {\left (f x + e\right )} a}{a^{2} + b^{2}} - \frac {2 \, b \log \left (a \tan \left (f x + e\right ) + b\right )}{a^{2} + b^{2}} + \frac {b \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} c^{2} - \frac {2 \, {\left (f x + e\right )}^{3} {\left (a + i \, b\right )} d^{2} + 6 \, {\left (f x + e\right )} {\left (a + i \, b\right )} d^{2} e^{2} - 6 i \, b d^{2} e^{2} \arctan \left (b \cos \left (2 \, f x + 2 \, e\right ) + a \sin \left (2 \, f x + 2 \, e\right ) + b, a \cos \left (2 \, f x + 2 \, e\right ) - b \sin \left (2 \, f x + 2 \, e\right ) - a\right ) - 3 \, b d^{2} e^{2} \log \left ({\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )\right ) - 3 \, b d^{2} {\rm Li}_{3}(\frac {{\left (i \, a - b\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{i \, a + b}) - 6 \, {\left ({\left (a + i \, b\right )} d^{2} e - {\left (a + i \, b\right )} c d f\right )} {\left (f x + e\right )}^{2} - {\left (6 i \, {\left (f x + e\right )}^{2} b d^{2} + {\left (-12 i \, b d^{2} e + 12 i \, b c d f\right )} {\left (f x + e\right )}\right )} \arctan \left (-\frac {2 \, a b \cos \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} - b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, f x + 2 \, e\right ) + a^{2} + b^{2} - {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) - {\left (-6 i \, {\left (f x + e\right )} b d^{2} + 6 i \, b d^{2} e - 6 i \, b c d f\right )} {\rm Li}_2\left (\frac {{\left (i \, a - b\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{i \, a + b}\right ) - 3 \, {\left ({\left (f x + e\right )}^{2} b d^{2} - 2 \, {\left (b d^{2} e - b c d f\right )} {\left (f x + e\right )}\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right )}{{\left (a^{2} + b^{2}\right )} f^{2}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^2}{a+b\,\mathrm {cot}\left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x\right )^{2}}{a + b \cot {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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