Integrand size = 18, antiderivative size = 103 \[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {b d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3} \]
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Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3803, 3799, 2221, 2611, 2320, 6724} \[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\frac {a (c+d x)^3}{3 d}+\frac {b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}+\frac {b (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b (c+d x)^3}{3 d}-\frac {b d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 3803
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)^2+b (c+d x)^2 \tanh (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^3}{3 d}+b \int (c+d x)^2 \tanh (e+f x) \, dx \\ & = \frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+(2 b) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx \\ & = \frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {(2 b d) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f} \\ & = \frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {\left (b d^2\right ) \int \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right ) \, dx}{f^2} \\ & = \frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3} \\ & = \frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {b d^2 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.43 \[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\frac {1}{6} \left (\frac {b e^{2 e} \left (\frac {4 e^{-2 e} (c+d x)^3}{d}+\frac {6 \left (1+e^{-2 e}\right ) (c+d x)^2 \log \left (1+e^{-2 (e+f x)}\right )}{f}-\frac {3 d \left (1+e^{-2 e}\right ) \left (2 f (c+d x) \operatorname {PolyLog}\left (2,-e^{-2 (e+f x)}\right )+d \operatorname {PolyLog}\left (3,-e^{-2 (e+f x)}\right )\right )}{f^3}\right )}{1+e^{2 e}}+2 x \left (3 c^2+3 c d x+d^2 x^2\right ) (a+b \tanh (e))\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(289\) vs. \(2(97)=194\).
Time = 0.24 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.82
method | result | size |
risch | \(\frac {a \,d^{2} x^{3}}{3}-\frac {d^{2} b \,x^{3}}{3}+a d c \,x^{2}-d b c \,x^{2}+a \,c^{2} x +b \,c^{2} x +\frac {a \,c^{3}}{3 d}+\frac {b \,c^{3}}{3 d}+\frac {b \,c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 b \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {4 b \,d^{2} e^{3}}{3 f^{3}}+\frac {b \,d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f}-\frac {b \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}-\frac {4 b d c e x}{f}+\frac {4 b d c e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {2 b \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {2 b \,d^{2} e^{2} x}{f^{2}}+\frac {b \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}-\frac {2 b d c \,e^{2}}{f^{2}}+\frac {2 b d c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {b d c \operatorname {polylog}\left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}\) | \(290\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.53 \[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\frac {{\left (a - b\right )} d^{2} f^{3} x^{3} + 3 \, {\left (a - b\right )} c d f^{3} x^{2} + 3 \, {\left (a - b\right )} c^{2} f^{3} x - 6 \, b d^{2} {\rm polylog}\left (3, i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) - 6 \, b d^{2} {\rm polylog}\left (3, -i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right ) + 6 \, {\left (b d^{2} f x + b c d f\right )} {\rm Li}_2\left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) + 6 \, {\left (b d^{2} f x + b c d f\right )} {\rm Li}_2\left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right ) + 3 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + i\right ) + 3 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - i\right ) + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right ) + 1\right ) + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right ) + 1\right )}{3 \, f^{3}} \]
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\[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\int \left (a + b \tanh {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.74 \[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\frac {1}{3} \, a d^{2} x^{3} + \frac {1}{3} \, b d^{2} x^{3} + a c d x^{2} + b c d x^{2} + a c^{2} x + \frac {b c^{2} \log \left (\cosh \left (f x + e\right )\right )}{f} + \frac {{\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )} b c d}{f^{2}} + \frac {{\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} b d^{2}}{2 \, f^{3}} - \frac {2 \, {\left (b d^{2} f^{3} x^{3} + 3 \, b c d f^{3} x^{2}\right )}}{3 \, f^{3}} \]
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\[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\int { {\left (d x + c\right )}^{2} {\left (b \tanh \left (f x + e\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (c+d x)^2 (a+b \tanh (e+f x)) \, dx=\int \left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2 \,d x \]
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