Optimal. Leaf size=112 \[ -\frac{i b d (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac{b d^2 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{a (c+d x)^3}{3 d}+\frac{b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i b (c+d x)^3}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.209533, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3722, 3717, 2190, 2531, 2282, 6589} \[ -\frac{i b d (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac{b d^2 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{a (c+d x)^3}{3 d}+\frac{b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i b (c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3722
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^2 (a+b \cot (e+f x)) \, dx &=\int \left (a (c+d x)^2+b (c+d x)^2 \cot (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^3}{3 d}+b \int (c+d x)^2 \cot (e+f x) \, dx\\ &=\frac{a (c+d x)^3}{3 d}-\frac{i b (c+d x)^3}{3 d}-(2 i b) \int \frac{e^{2 i (e+f x)} (c+d x)^2}{1-e^{2 i (e+f x)}} \, dx\\ &=\frac{a (c+d x)^3}{3 d}-\frac{i b (c+d x)^3}{3 d}+\frac{b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{(2 b d) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=\frac{a (c+d x)^3}{3 d}-\frac{i b (c+d x)^3}{3 d}+\frac{b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i b d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{\left (i b d^2\right ) \int \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=\frac{a (c+d x)^3}{3 d}-\frac{i b (c+d x)^3}{3 d}+\frac{b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i b d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}\\ &=\frac{a (c+d x)^3}{3 d}-\frac{i b (c+d x)^3}{3 d}+\frac{b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i b d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{b d^2 \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}\\ \end{align*}
Mathematica [B] time = 2.48925, size = 406, normalized size = 3.62 \[ \frac{-3 i b c d f \text{PolyLog}\left (2,e^{2 i \left (\tan ^{-1}(\tan (e))+f x\right )}\right )+6 i b d^2 f x \text{PolyLog}\left (2,-e^{-i (e+f x)}\right )+6 i b d^2 f x \text{PolyLog}\left (2,e^{-i (e+f x)}\right )+6 b d^2 \text{PolyLog}\left (3,-e^{-i (e+f x)}\right )+6 b d^2 \text{PolyLog}\left (3,e^{-i (e+f x)}\right )+3 a c^2 f^3 x+3 a c d f^3 x^2+a d^2 f^3 x^3+3 b c^2 f^2 \log (\sin (e+f x))+3 b c d f^3 x^2 \cot (e)-3 b c d f^3 x^2 e^{i \tan ^{-1}(\tan (e))} \cot (e) \sqrt{\sec ^2(e)}-6 i b c d f^2 x \tan ^{-1}(\tan (e))+6 b c d f^2 x \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (e))+f x\right )}\right )+6 b c d f \tan ^{-1}(\tan (e)) \log \left (1-e^{2 i \left (\tan ^{-1}(\tan (e))+f x\right )}\right )-6 b c d f \tan ^{-1}(\tan (e)) \log \left (\sin \left (\tan ^{-1}(\tan (e))+f x\right )\right )+3 i \pi b c d f^2 x+3 \pi b c d f \log \left (1+e^{-2 i f x}\right )-3 \pi b c d f \log (\cos (f x))+3 b d^2 f^2 x^2 \log \left (1-e^{-i (e+f x)}\right )+3 b d^2 f^2 x^2 \log \left (1+e^{-i (e+f x)}\right )+i b d^2 f^3 x^3}{3 f^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.339, size = 516, normalized size = 4.6 \begin{align*} acd{x}^{2}-2\,{\frac{b{c}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}+{\frac{b{c}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}-1 \right ) }{f}}+2\,{\frac{bcd\ln \left ( 1-{{\rm e}^{i \left ( fx+e \right ) }} \right ) e}{{f}^{2}}}+{\frac{a{d}^{2}{x}^{3}}{3}}+a{c}^{2}x-ibcd{x}^{2}-2\,{\frac{b{d}^{2}{e}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}+{\frac{b{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( fx+e \right ) }} \right ){x}^{2}}{f}}-{\frac{b{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( fx+e \right ) }} \right ){e}^{2}}{{f}^{3}}}+{\frac{b{d}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ){x}^{2}}{f}}+{\frac{{\frac{4\,i}{3}}b{d}^{2}{e}^{3}}{{f}^{3}}}+{\frac{b{c}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) }{f}}+2\,{\frac{b{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}+2\,{\frac{b{d}^{2}{\it polylog} \left ( 3,{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}+2\,{\frac{bcd\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) x}{f}}+2\,{\frac{bcd\ln \left ( 1-{{\rm e}^{i \left ( fx+e \right ) }} \right ) x}{f}}-2\,{\frac{bcde\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}-1 \right ) }{{f}^{2}}}+4\,{\frac{bcde\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-{\frac{i}{3}}b{d}^{2}{x}^{3}+ib{c}^{2}x+{\frac{b{d}^{2}{e}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}-1 \right ) }{{f}^{3}}}-{\frac{2\,ib{d}^{2}{\it polylog} \left ( 2,{{\rm e}^{i \left ( fx+e \right ) }} \right ) x}{{f}^{2}}}-{\frac{2\,ibcd{e}^{2}}{{f}^{2}}}+{\frac{2\,ib{d}^{2}{e}^{2}x}{{f}^{2}}}-{\frac{2\,ib{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{i \left ( fx+e \right ) }} \right ) x}{{f}^{2}}}-{\frac{2\,ibcd{\it polylog} \left ( 2,{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-{\frac{2\,ibcd{\it polylog} \left ( 2,-{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}-{\frac{4\,ibcdex}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.59511, size = 702, normalized size = 6.27 \begin{align*} \frac{6 \,{\left (f x + e\right )} a c^{2} + \frac{2 \,{\left (f x + e\right )}^{3} a d^{2}}{f^{2}} - \frac{6 \,{\left (f x + e\right )}^{2} a d^{2} e}{f^{2}} + \frac{6 \,{\left (f x + e\right )} a d^{2} e^{2}}{f^{2}} + \frac{6 \,{\left (f x + e\right )}^{2} a c d}{f} - \frac{12 \,{\left (f x + e\right )} a c d e}{f} + 6 \, b c^{2} \log \left (\sin \left (f x + e\right )\right ) + \frac{6 \, b d^{2} e^{2} \log \left (\sin \left (f x + e\right )\right )}{f^{2}} - \frac{12 \, b c d e \log \left (\sin \left (f x + e\right )\right )}{f} + \frac{-2 i \,{\left (f x + e\right )}^{3} b d^{2} + 12 \, b d^{2}{\rm Li}_{3}(-e^{\left (i \, f x + i \, e\right )}) + 12 \, b d^{2}{\rm Li}_{3}(e^{\left (i \, f x + i \, e\right )}) +{\left (6 i \, b d^{2} e - 6 i \, b 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 (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) +{\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 (\sin \left (f x + e\right ), -\cos \left (f x + e\right ) + 1\right ) +{\left (-12 i \,{\left (f x + e\right )} b d^{2} + 12 i \, b d^{2} e - 12 i \, b c d f\right )}{\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) +{\left (-12 i \,{\left (f x + e\right )} b d^{2} + 12 i \, b d^{2} e - 12 i \, b c d f\right )}{\rm Li}_2\left (e^{\left (i \, f x + i \, e\right )}\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 (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\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 (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )}{f^{2}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 1.81055, size = 1025, normalized size = 9.15 \begin{align*} \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, \cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + 3 \, b d^{2}{\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) +{\left (-6 i \, b d^{2} f x - 6 i \, b c d f\right )}{\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) +{\left (6 i \, b d^{2} f x + 6 i \, b c d f\right )}{\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) - i \, \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} \, \cos \left (2 \, f x + 2 \, e\right ) + \frac{1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac{1}{2}\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} \, \cos \left (2 \, f x + 2 \, e\right ) - \frac{1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac{1}{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 (-\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\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 (-\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right )}{12 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \cot{\left (e + f x \right )}\right ) \left (c + d x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2}{\left (b \cot \left (f x + e\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]