Optimal. Leaf size=246 \[ -\frac{d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac{3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}-\frac{3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}-\frac{(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac{3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}-\frac{3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac{3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}-\frac{3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \]
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Rubi [A] time = 0.266411, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3729, 2176, 2194} \[ -\frac{d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac{3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}-\frac{3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}-\frac{(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac{3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}-\frac{3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac{(c+d x)^3}{24 a^3 d}-\frac{d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac{3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}-\frac{3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \]
Antiderivative was successfully verified.
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Rule 3729
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(a+a \tanh (e+f x))^3} \, dx &=\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*}
Mathematica [A] time = 1.42458, size = 371, normalized size = 1.51 \[ \frac{\text{sech}^3(e+f x) \left (-81 \left (24 c^2 f^2+4 c d f (12 f x+5)+d^2 \left (24 f^2 x^2+20 f x+9\right )\right ) \cosh (e+f x)+8 \left (18 c^2 f^2 (6 f x-1)+6 c d f \left (18 f^2 x^2-6 f x-1\right )+d^2 \left (36 f^3 x^3-18 f^2 x^2-6 f x-1\right )\right ) \cosh (3 (e+f x))+864 c^2 f^3 x \sinh (3 (e+f x))-648 c^2 f^2 \sinh (e+f x)+144 c^2 f^2 \sinh (3 (e+f x))+864 c d f^3 x^2 \sinh (3 (e+f x))-1296 c d f^2 x \sinh (e+f x)+288 c d f^2 x \sinh (3 (e+f x))-972 c d f \sinh (e+f x)+48 c d f \sinh (3 (e+f x))+288 d^2 f^3 x^3 \sinh (3 (e+f x))-648 d^2 f^2 x^2 \sinh (e+f x)+144 d^2 f^2 x^2 \sinh (3 (e+f x))-972 d^2 f x \sinh (e+f x)+48 d^2 f x \sinh (3 (e+f x))-567 d^2 \sinh (e+f x)+8 d^2 \sinh (3 (e+f x))\right )}{6912 a^3 f^3 (\tanh (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 2016, normalized size = 8.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 5.77989, size = 344, normalized size = 1.4 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20029, size = 1218, normalized size = 4.95 \begin{align*} \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 )}} \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*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22491, size = 447, normalized size = 1.82 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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