Integrand size = 20, antiderivative size = 246 \[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=-\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}-\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}-\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}-\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}-\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}-\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d} \]
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Time = 0.17 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3810, 2207, 2225} \[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=-\frac {d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac {3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}-\frac {3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}-\frac {(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}-\frac {3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}-\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \]
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Rule 2207
Rule 2225
Rule 3810
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(c+d x)^2}{8 a^3}+\frac {e^{-6 e-6 f x} (c+d x)^2}{8 a^3}+\frac {3 e^{-4 e-4 f x} (c+d x)^2}{8 a^3}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{8 a^3}\right ) \, dx \\ & = \frac {(c+d x)^3}{24 a^3 d}+\frac {\int e^{-6 e-6 f x} (c+d x)^2 \, dx}{8 a^3}+\frac {3 \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{8 a^3}+\frac {3 \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{8 a^3} \\ & = -\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}-\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}+\frac {d \int e^{-6 e-6 f x} (c+d x) \, dx}{24 a^3 f}+\frac {(3 d) \int e^{-4 e-4 f x} (c+d x) \, dx}{16 a^3 f}+\frac {(3 d) \int e^{-2 e-2 f x} (c+d x) \, dx}{8 a^3 f} \\ & = -\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}-\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}-\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}-\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}+\frac {d^2 \int e^{-6 e-6 f x} \, dx}{144 a^3 f^2}+\frac {\left (3 d^2\right ) \int e^{-4 e-4 f x} \, dx}{64 a^3 f^2}+\frac {\left (3 d^2\right ) \int e^{-2 e-2 f x} \, dx}{16 a^3 f^2} \\ & = -\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}-\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}-\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}-\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}-\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}-\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d} \\ \end{align*}
Time = 2.12 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.51 \[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=\frac {\text {sech}^3(e+f x) \left (-81 \left (24 c^2 f^2+4 c d f (5+12 f x)+d^2 \left (9+20 f x+24 f^2 x^2\right )\right ) \cosh (e+f x)+8 \left (18 c^2 f^2 (-1+6 f x)+6 c d f \left (-1-6 f x+18 f^2 x^2\right )+d^2 \left (-1-6 f x-18 f^2 x^2+36 f^3 x^3\right )\right ) \cosh (3 (e+f x))-567 d^2 \sinh (e+f x)-972 c d f \sinh (e+f x)-648 c^2 f^2 \sinh (e+f x)-972 d^2 f x \sinh (e+f x)-1296 c d f^2 x \sinh (e+f x)-648 d^2 f^2 x^2 \sinh (e+f x)+8 d^2 \sinh (3 (e+f x))+48 c d f \sinh (3 (e+f x))+144 c^2 f^2 \sinh (3 (e+f x))+48 d^2 f x \sinh (3 (e+f x))+288 c d f^2 x \sinh (3 (e+f x))+864 c^2 f^3 x \sinh (3 (e+f x))+144 d^2 f^2 x^2 \sinh (3 (e+f x))+864 c d f^3 x^2 \sinh (3 (e+f x))+288 d^2 f^3 x^3 \sinh (3 (e+f x))\right )}{6912 a^3 f^3 (1+\tanh (e+f x))^3} \]
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Time = 0.30 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {d^{2} x^{3}}{24 a^{3}}+\frac {d c \,x^{2}}{8 a^{3}}+\frac {c^{2} x}{8 a^{3}}+\frac {c^{3}}{24 a^{3} d}-\frac {3 \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}+2 d^{2} f x +2 c d f +d^{2}\right ) {\mathrm e}^{-2 f x -2 e}}{32 a^{3} f^{3}}-\frac {3 \left (8 d^{2} x^{2} f^{2}+16 c d \,f^{2} x +8 c^{2} f^{2}+4 d^{2} f x +4 c d f +d^{2}\right ) {\mathrm e}^{-4 f x -4 e}}{256 a^{3} f^{3}}-\frac {\left (18 d^{2} x^{2} f^{2}+36 c d \,f^{2} x +18 c^{2} f^{2}+6 d^{2} f x +6 c d f +d^{2}\right ) {\mathrm e}^{-6 f x -6 e}}{864 a^{3} f^{3}}\) | \(223\) |
parallelrisch | \(\frac {-328 d^{2}-417 d^{2} f x +255 x \tanh \left (f x +e \right )^{3} d^{2} f +198 x^{2} \tanh \left (f x +e \right )^{3} d^{2} f^{2}-396 \tanh \left (f x +e \right )^{2} c d f +72 d^{2} \tanh \left (f x +e \right )^{3} x^{3} f^{3}-567 \tanh \left (f x +e \right ) d^{2}-720 c^{2} f^{2}+216 c d \,x^{2} f^{3}+216 x \tanh \left (f x +e \right )^{3} c^{2} f^{3}-1044 c d \,f^{2} x -522 d^{2} x^{2} f^{2}-672 c d f -648 \tanh \left (f x +e \right ) c^{2} f^{2}-255 \tanh \left (f x +e \right )^{2} d^{2}+216 x \,c^{2} f^{3}+72 d^{2} x^{3} f^{3}+216 d^{2} \tanh \left (f x +e \right ) x^{3} f^{3}+648 x \tanh \left (f x +e \right ) c^{2} f^{3}-207 x \tanh \left (f x +e \right ) d^{2} f -54 x^{2} \tanh \left (f x +e \right ) d^{2} f^{2}+756 x \tanh \left (f x +e \right )^{2} c d \,f^{2}+648 x^{2} \tanh \left (f x +e \right )^{2} c d \,f^{3}+396 x \tanh \left (f x +e \right )^{3} c d \,f^{2}+216 x^{2} \tanh \left (f x +e \right )^{3} c d \,f^{3}-108 x \tanh \left (f x +e \right ) c d \,f^{2}-216 \tanh \left (f x +e \right )^{2} c^{2} f^{2}+216 d^{2} \tanh \left (f x +e \right )^{2} x^{3} f^{3}-972 \tanh \left (f x +e \right ) c d f +648 x \tanh \left (f x +e \right )^{2} c^{2} f^{3}+369 x \tanh \left (f x +e \right )^{2} d^{2} f +378 x^{2} \tanh \left (f x +e \right )^{2} d^{2} f^{2}+648 x^{2} \tanh \left (f x +e \right ) c d \,f^{3}}{1728 f^{3} a^{3} \left (1+\tanh \left (f x +e \right )\right )^{3}}\) | \(472\) |
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Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (217) = 434\).
Time = 0.25 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.16 \[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=\frac {8 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{3} + 24 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + 8 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \sinh \left (f x + e\right )^{3} - 81 \, {\left (24 \, d^{2} f^{2} x^{2} + 24 \, c^{2} f^{2} + 20 \, c d f + 9 \, d^{2} + 4 \, {\left (12 \, c d f^{2} + 5 \, d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) - 3 \, {\left (216 \, d^{2} f^{2} x^{2} + 216 \, c^{2} f^{2} + 324 \, c d f - 8 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + e\right )^{2} + 189 \, d^{2} + 108 \, {\left (4 \, c d f^{2} + 3 \, d^{2} f\right )} x\right )} \sinh \left (f x + e\right )}{6912 \, {\left (a^{3} f^{3} \cosh \left (f x + e\right )^{3} + 3 \, a^{3} f^{3} \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right ) + 3 \, a^{3} f^{3} \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{2} + a^{3} f^{3} \sinh \left (f x + e\right )^{3}\right )}} \]
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\[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=\frac {\int \frac {c^{2}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
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Time = 1.19 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.04 \[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=\frac {1}{96} \, c^{2} {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} - \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} + 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} - 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d e^{\left (-6 \, e\right )}}{576 \, a^{3} f^{2}} + \frac {{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} - 648 \, {\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 8 \, {\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{2} e^{\left (-6 \, e\right )}}{6912 \, a^{3} f^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=\frac {{\left (288 \, d^{2} f^{3} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c d f^{3} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c^{2} f^{3} x e^{\left (6 \, f x + 6 \, e\right )} - 1296 \, d^{2} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 648 \, d^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 144 \, d^{2} f^{2} x^{2} - 2592 \, c d f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 1296 \, c d f^{2} x e^{\left (2 \, f x + 2 \, e\right )} - 288 \, c d f^{2} x - 1296 \, c^{2} f^{2} e^{\left (4 \, f x + 4 \, e\right )} - 1296 \, d^{2} f x e^{\left (4 \, f x + 4 \, e\right )} - 648 \, c^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 324 \, d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} - 144 \, c^{2} f^{2} - 48 \, d^{2} f x - 1296 \, c d f e^{\left (4 \, f x + 4 \, e\right )} - 324 \, c d f e^{\left (2 \, f x + 2 \, e\right )} - 48 \, c d f - 648 \, d^{2} e^{\left (4 \, f x + 4 \, e\right )} - 81 \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} - 8 \, d^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{6912 \, a^{3} f^{3}} \]
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Time = 1.93 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx=\frac {c^2\,x}{8\,a^3}-{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {6\,c^2\,f^2+6\,c\,d\,f+3\,d^2}{32\,a^3\,f^3}+\frac {3\,d^2\,x^2}{16\,a^3\,f}+\frac {3\,d\,x\,\left (d+2\,c\,f\right )}{16\,a^3\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {24\,c^2\,f^2+12\,c\,d\,f+3\,d^2}{256\,a^3\,f^3}+\frac {3\,d^2\,x^2}{32\,a^3\,f}+\frac {3\,d\,x\,\left (d+4\,c\,f\right )}{64\,a^3\,f^2}\right )-{\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {18\,c^2\,f^2+6\,c\,d\,f+d^2}{864\,a^3\,f^3}+\frac {d^2\,x^2}{48\,a^3\,f}+\frac {d\,x\,\left (d+6\,c\,f\right )}{144\,a^3\,f^2}\right )+\frac {d^2\,x^3}{24\,a^3}+\frac {c\,d\,x^2}{8\,a^3} \]
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