Integrand size = 18, antiderivative size = 152 \[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}+\frac {i b (c+d x)^4}{4 d}-\frac {b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 i b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4} \]
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Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3803, 3800, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i b (c+d x)^4}{4 d}-\frac {3 i b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 3803
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {\left (3 i b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-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 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {\left (3 b d^3\right ) \int \operatorname {PolyLog}\left (3,-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 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {\left (3 i b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 i b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(152)=304\).
Time = 0.15 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.25 \[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=a c^3 x+\frac {3}{2} a c^2 d x^2+\frac {3}{2} i b c^2 d x^2+a c d^2 x^3+i b c d^2 x^3+\frac {1}{4} a d^3 x^4+\frac {1}{4} i b d^3 x^4-\frac {3 b c^2 d x \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 b c d^2 x^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b d^3 x^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b c^3 \log (\cos (e+f x))}{f}+\frac {3 i b c^2 d \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 i b c d^2 x \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 i b d^3 x^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b c d^2 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 b d^3 x \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 i b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (131 ) = 262\).
Time = 0.64 (sec) , antiderivative size = 500, normalized size of antiderivative = 3.29
method | result | size |
risch | \(-\frac {i b \,c^{4}}{4 d}-i b \,c^{3} x +\frac {i d^{3} b \,x^{4}}{4}-\frac {b \,c^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f}+\frac {2 b \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}-\frac {b \,d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{3}}{f}-\frac {3 b \,d^{3} \operatorname {Li}_{3}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{2 f^{3}}-\frac {2 b \,e^{3} d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{4}}-\frac {3 b \,d^{2} c \,\operatorname {Li}_{3}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{3}}+\frac {3 i b \,d^{3} e^{4}}{2 f^{4}}+\frac {3 i d b \,c^{2} x^{2}}{2}-\frac {3 i b \,d^{3} \operatorname {Li}_{4}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{4 f^{4}}+i d^{2} b c \,x^{3}+d^{2} a c \,x^{3}+\frac {3 d a \,c^{2} x^{2}}{2}+a \,c^{3} x +\frac {d^{3} a \,x^{4}}{4}+\frac {a \,c^{4}}{4 d}-\frac {3 b \,d^{2} c \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{2}}{f}-\frac {3 b d \,c^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x}{f}+\frac {3 i b \,d^{3} \operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right ) x^{2}}{2 f^{2}}+\frac {3 i b d \,c^{2} e^{2}}{f^{2}}-\frac {4 i b \,d^{2} c \,e^{3}}{f^{3}}+\frac {2 i b \,d^{3} e^{3} x}{f^{3}}+\frac {3 i b d \,c^{2} \operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{2}}+\frac {6 b \,e^{2} d^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {6 b e d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {3 i b \,d^{2} c \,\operatorname {Li}_{2}\left (-{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{2}}-\frac {6 i b \,d^{2} c \,e^{2} x}{f^{2}}+\frac {6 i b d \,c^{2} e x}{f}\) | \(500\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (127) = 254\).
Time = 0.27 (sec) , antiderivative size = 500, normalized size of antiderivative = 3.29 \[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=\frac {2 \, a d^{3} f^{4} x^{4} + 8 \, a c d^{2} f^{4} x^{3} + 12 \, a c^{2} d f^{4} x^{2} + 8 \, a c^{3} f^{4} x + 3 i \, b d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 i \, b d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \, {\left (i \, b d^{3} f^{2} x^{2} + 2 i \, b c d^{2} f^{2} x + i \, b c^{2} d f^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 6 \, {\left (-i \, b d^{3} f^{2} x^{2} - 2 i \, b c d^{2} f^{2} x - i \, b c^{2} d f^{2}\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b c^{3} f^{3}\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right )}{8 \, f^{4}} \]
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\[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (127) = 254\).
Time = 0.72 (sec) , antiderivative size = 672, normalized size of antiderivative = 4.42 \[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=\frac {12 \, {\left (f x + e\right )} a c^{3} + \frac {3 \, {\left (f x + e\right )}^{4} a d^{3}}{f^{3}} - \frac {12 \, {\left (f x + e\right )}^{3} a d^{3} e}{f^{3}} + \frac {18 \, {\left (f x + e\right )}^{2} a d^{3} e^{2}}{f^{3}} - \frac {12 \, {\left (f x + e\right )} a d^{3} e^{3}}{f^{3}} + \frac {12 \, {\left (f x + e\right )}^{3} a c d^{2}}{f^{2}} - \frac {36 \, {\left (f x + e\right )}^{2} a c d^{2} e}{f^{2}} + \frac {36 \, {\left (f x + e\right )} a c d^{2} e^{2}}{f^{2}} + \frac {18 \, {\left (f x + e\right )}^{2} a c^{2} d}{f} - \frac {36 \, {\left (f x + e\right )} a c^{2} d e}{f} + 12 \, b c^{3} \log \left (\sec \left (f x + e\right )\right ) - \frac {12 \, b d^{3} e^{3} \log \left (\sec \left (f x + e\right )\right )}{f^{3}} + \frac {36 \, b c d^{2} e^{2} \log \left (\sec \left (f x + e\right )\right )}{f^{2}} - \frac {36 \, b c^{2} d e \log \left (\sec \left (f x + e\right )\right )}{f} - \frac {-3 i \, {\left (f x + e\right )}^{4} b d^{3} + 12 i \, b d^{3} {\rm Li}_{4}(-e^{\left (2 i \, f x + 2 i \, e\right )}) - 12 \, {\left (-i \, b d^{3} e + i \, b c d^{2} f\right )} {\left (f x + e\right )}^{3} - 18 \, {\left (i \, b d^{3} e^{2} - 2 i \, b c d^{2} e f + i \, b c^{2} d f^{2}\right )} {\left (f x + e\right )}^{2} - 4 \, {\left (-4 i \, {\left (f x + e\right )}^{3} b d^{3} + 9 \, {\left (i \, b d^{3} e - i \, b c d^{2} f\right )} {\left (f x + e\right )}^{2} + 9 \, {\left (-i \, b d^{3} e^{2} + 2 i \, b c d^{2} e f - i \, b c^{2} d f^{2}\right )} {\left (f x + e\right )}\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 6 \, {\left (4 i \, {\left (f x + e\right )}^{2} b d^{3} + 3 i \, b d^{3} e^{2} - 6 i \, b c d^{2} e f + 3 i \, b c^{2} d f^{2} + 6 \, {\left (-i \, b d^{3} e + i \, b c d^{2} f\right )} {\left (f x + e\right )}\right )} {\rm Li}_2\left (-e^{\left (2 i \, f x + 2 i \, e\right )}\right ) + 2 \, {\left (4 \, {\left (f x + e\right )}^{3} b d^{3} - 9 \, {\left (b d^{3} e - b c d^{2} f\right )} {\left (f x + e\right )}^{2} + 9 \, {\left (b d^{3} e^{2} - 2 \, b c d^{2} e f + b c^{2} d f^{2}\right )} {\left (f x + e\right )}\right )} \log \left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) + 6 \, {\left (4 \, {\left (f x + e\right )} b d^{3} - 3 \, b d^{3} e + 3 \, b c d^{2} f\right )} {\rm Li}_{3}(-e^{\left (2 i \, f x + 2 i \, e\right )})}{f^{3}}}{12 \, f} \]
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\[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (c+d x)^3 (a+b \tan (e+f x)) \, dx=\int \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3 \,d x \]
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