Integrand size = 18, antiderivative size = 137 \[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4} \]
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Time = 0.18 (sec) , antiderivative size = 137, 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, 3799, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {b (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b (c+d x)^4}{4 d}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 3803
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)^3+b (c+d x)^3 \tanh (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^4}{4 d}+b \int (c+d x)^3 \tanh (e+f x) \, dx \\ & = \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+(2 b) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1+e^{2 (e+f x)}} \, dx \\ & = \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {(3 b d) \int (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f} \\ & = \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (3 b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right ) \, dx}{f^2} \\ & = \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (3 b d^3\right ) \int \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right ) \, dx}{2 f^3} \\ & = \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (3 b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4} \\ & = \frac {a (c+d x)^4}{4 d}-\frac {b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18 \[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\frac {1}{4} \left (\frac {2 b (c+d x)^4}{d \left (1+e^{2 e}\right )}+\frac {4 b (c+d x)^3 \log \left (1+e^{-2 (e+f x)}\right )}{f}-\frac {3 b d \left (2 f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{-2 (e+f x)}\right )+d \left (2 f (c+d x) \operatorname {PolyLog}\left (3,-e^{-2 (e+f x)}\right )+d \operatorname {PolyLog}\left (4,-e^{-2 (e+f x)}\right )\right )\right )}{f^4}+x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) (a+b \tanh (e))\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(469\) vs. \(2(127)=254\).
Time = 0.26 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.43
method | result | size |
risch | \(-\frac {2 b \,d^{3} e^{3} x}{f^{3}}-\frac {3 b d \,c^{2} e^{2}}{f^{2}}+\frac {4 b \,d^{2} c \,e^{3}}{f^{3}}+\frac {b \,d^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{3}}{f}+\frac {3 b \,d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{2 f^{2}}-\frac {3 b \,d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{2 f x +2 e}\right ) x}{2 f^{3}}+\frac {2 b \,d^{3} e^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}+\frac {3 b d \,c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}-\frac {3 b \,d^{2} c \operatorname {polylog}\left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}+a \,d^{2} c \,x^{3}+\frac {3 a d \,c^{2} x^{2}}{2}+a \,c^{3} x +\frac {b \,c^{4}}{4 d}-\frac {d^{3} b \,x^{4}}{4}+\frac {a \,d^{3} x^{4}}{4}+\frac {a \,c^{4}}{4 d}+\frac {6 b \,d^{2} c \,e^{2} x}{f^{2}}-d^{2} b c \,x^{3}-\frac {3 d b \,c^{2} x^{2}}{2}+b \,c^{3} x -\frac {6 b d \,c^{2} e x}{f}+\frac {b \,c^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 b \,c^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {3 b \,d^{3} \operatorname {polylog}\left (4, -{\mathrm e}^{2 f x +2 e}\right )}{4 f^{4}}-\frac {3 b \,d^{3} e^{4}}{2 f^{4}}-\frac {6 b \,d^{2} c \,e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {3 b \,d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}+\frac {6 b d \,c^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 b d \,c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {3 b \,d^{2} c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f}\) | \(470\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 583, normalized size of antiderivative = 4.26 \[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\frac {{\left (a - b\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a - b\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a - b\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a - b\right )} c^{3} f^{4} x + 24 \, b d^{3} {\rm polylog}\left (4, i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) + 24 \, b d^{3} {\rm polylog}\left (4, -i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) + 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right ) - 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + i\right ) - 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - i\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 d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right ) + 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 d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right ) + 1\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) - 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, -i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right )}{4 \, f^{4}} \]
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\[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\int \left (a + b \tanh {\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (126) = 252\).
Time = 0.26 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.22 \[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\frac {1}{4} \, a d^{3} x^{4} + \frac {1}{4} \, b d^{3} x^{4} + a c d^{2} x^{3} + b c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + \frac {3}{2} \, b c^{2} d x^{2} + a c^{3} x + \frac {b c^{3} \log \left (\cosh \left (f x + e\right )\right )}{f} + \frac {3 \, {\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )} b c^{2} d}{2 \, f^{2}} + \frac {3 \, {\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} b c d^{2}}{2 \, f^{3}} + \frac {{\left (4 \, f^{3} x^{3} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} b d^{3}}{3 \, f^{4}} - \frac {b d^{3} f^{4} x^{4} + 4 \, b c d^{2} f^{4} x^{3} + 6 \, b c^{2} d f^{4} x^{2}}{2 \, f^{4}} \]
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\[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \tanh \left (f x + e\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (c+d x)^3 (a+b \tanh (e+f x)) \, dx=\int \left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3 \,d x \]
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