Optimal. Leaf size=152 \[ -\frac{3 b d^2 (c+d x) \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac{3 i b d (c+d x)^2 \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 i b d^3 \text{PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}+\frac{a (c+d x)^4}{4 d}-\frac{b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i b (c+d x)^4}{4 d} \]
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Rubi [A] time = 0.251341, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3722, 3719, 2190, 2531, 6609, 2282, 6589} \[ \frac{a (c+d x)^4}{4 d}-\frac{3 b d^2 (c+d x) \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac{3 i b d (c+d x)^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i b (c+d x)^4}{4 d}-\frac{3 i b d^3 \text{Li}_4\left (-e^{2 i (e+f x)}\right )}{4 f^4} \]
Antiderivative was successfully verified.
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Rule 3722
Rule 3719
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^3 (a+b \tan (e+f x)) \, dx &=\int \left (a (c+d x)^3+b (c+d x)^3 \tan (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+b \int (c+d x)^3 \tan (e+f x) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i b (c+d x)^4}{4 d}-(2 i b) \int \frac{e^{2 i (e+f x)} (c+d x)^3}{1+e^{2 i (e+f x)}} \, dx\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i b (c+d x)^4}{4 d}-\frac{b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{(3 b d) \int (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i b (c+d x)^4}{4 d}-\frac{b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i b d (c+d x)^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{\left (3 i b d^2\right ) \int (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i b (c+d x)^4}{4 d}-\frac{b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i b d (c+d x)^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 b d^2 (c+d x) \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac{\left (3 b d^3\right ) \int \text{Li}_3\left (-e^{2 i (e+f x)}\right ) \, dx}{2 f^3}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i b (c+d x)^4}{4 d}-\frac{b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i b d (c+d x)^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 b d^2 (c+d x) \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{\left (3 i b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 f^4}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i b (c+d x)^4}{4 d}-\frac{b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i b d (c+d x)^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 b d^2 (c+d x) \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 i b d^3 \text{Li}_4\left (-e^{2 i (e+f x)}\right )}{4 f^4}\\ \end{align*}
Mathematica [A] time = 0.327143, size = 255, normalized size = 1.68 \[ \frac{1}{4} \left (-\frac{6 b d^2 (c+d x) \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}+\frac{6 i b d (c+d x)^2 \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 i b d^3 \text{PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{f^4}+6 a c^2 d x^2+4 a c^3 x+4 a c d^2 x^3+a d^3 x^4-\frac{12 b c^2 d x \log \left (1+e^{2 i (e+f x)}\right )}{f}+6 i b c^2 d x^2-\frac{4 b c^3 \log (\cos (e+f x))}{f}-\frac{12 b c d^2 x^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+4 i b c d^2 x^3-\frac{4 b d^3 x^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+i b d^3 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.143, size = 481, normalized size = 3.2 \begin{align*}{\frac{3\,i}{2}}b{c}^{2}d{x}^{2}-{\frac{{\frac{3\,i}{4}}b{d}^{3}{\it polylog} \left ( 4,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{4}}}+a{c}^{3}x+ac{d}^{2}{x}^{3}+{\frac{a{d}^{3}{x}^{4}}{4}}-ib{c}^{3}x-{\frac{6\,ibc{d}^{2}{e}^{2}x}{{f}^{2}}}+{\frac{6\,ib{c}^{2}dex}{f}}+{\frac{3\,ibc{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) x}{{f}^{2}}}+2\,{\frac{b{c}^{3}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{f}}-{\frac{b{c}^{3}\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) }{f}}+{\frac{i}{4}}b{d}^{3}{x}^{4}-2\,{\frac{b{d}^{3}{e}^{3}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{4}}}-{\frac{3\,bc{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{2\,{f}^{3}}}-{\frac{3\,b{d}^{3}{\it polylog} \left ( 3,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) x}{2\,{f}^{3}}}+{\frac{{\frac{3\,i}{2}}b{d}^{3}{e}^{4}}{{f}^{4}}}+{\frac{3\,a{c}^{2}d{x}^{2}}{2}}+ibc{d}^{2}{x}^{3}-{\frac{b{d}^{3}\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ){x}^{3}}{f}}+6\,{\frac{bc{d}^{2}{e}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}}}-6\,{\frac{b{c}^{2}de\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+{\frac{3\,ib{c}^{2}d{e}^{2}}{{f}^{2}}}-{\frac{4\,ibc{d}^{2}{e}^{3}}{{f}^{3}}}+{\frac{2\,ib{d}^{3}{e}^{3}x}{{f}^{3}}}+{\frac{{\frac{3\,i}{2}}b{c}^{2}d{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ) }{{f}^{2}}}+{\frac{{\frac{3\,i}{2}}b{d}^{3}{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( fx+e \right ) }} \right ){x}^{2}}{{f}^{2}}}-3\,{\frac{b{c}^{2}d\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ) x}{f}}-3\,{\frac{bc{d}^{2}\ln \left ({{\rm e}^{2\,i \left ( fx+e \right ) }}+1 \right ){x}^{2}}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.80296, size = 898, normalized size = 5.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.73715, size = 1257, normalized size = 8.27 \begin{align*} \frac{2 \, a d^{3} f^{4} x^{4} + 8 \, a c d^{2} f^{4} x^{3} + 12 \, a c^{2} d f^{4} x^{2} + 8 \, a c^{3} f^{4} x + 3 i \, b d^{3}{\rm polylog}\left (4, \frac{\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 i \, b d^{3}{\rm polylog}\left (4, \frac{\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) +{\left (-6 i \, b d^{3} f^{2} x^{2} - 12 i \, b c d^{2} f^{2} x - 6 i \, b c^{2} d f^{2}\right )}{\rm Li}_2\left (\frac{2 \,{\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) +{\left (6 i \, b d^{3} f^{2} x^{2} + 12 i \, b c d^{2} f^{2} x + 6 i \, b c^{2} d f^{2}\right )}{\rm Li}_2\left (\frac{2 \,{\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 4 \,{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\right )} \log \left (-\frac{2 \,{\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 4 \,{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\right )} \log \left (-\frac{2 \,{\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \,{\left (b d^{3} f x + b c d^{2} f\right )}{\rm polylog}\left (3, \frac{\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \,{\left (b d^{3} f x + b c d^{2} f\right )}{\rm polylog}\left (3, \frac{\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right )}{8 \, f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \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{\left (d x + c\right )}^{3}{\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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