Optimal. Leaf size=230 \[ -\frac{3 d^2 (c+d x) e^{-4 e-4 f x}}{128 a^2 f^3}-\frac{3 d^2 (c+d x) e^{-2 e-2 f x}}{8 a^2 f^3}-\frac{3 d (c+d x)^2 e^{-4 e-4 f x}}{64 a^2 f^2}-\frac{3 d (c+d x)^2 e^{-2 e-2 f x}}{8 a^2 f^2}-\frac{(c+d x)^3 e^{-4 e-4 f x}}{16 a^2 f}-\frac{(c+d x)^3 e^{-2 e-2 f x}}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}-\frac{3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4} \]
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Rubi [A] time = 0.279114, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3729, 2176, 2194} \[ -\frac{3 d^2 (c+d x) e^{-4 e-4 f x}}{128 a^2 f^3}-\frac{3 d^2 (c+d x) e^{-2 e-2 f x}}{8 a^2 f^3}-\frac{3 d (c+d x)^2 e^{-4 e-4 f x}}{64 a^2 f^2}-\frac{3 d (c+d x)^2 e^{-2 e-2 f x}}{8 a^2 f^2}-\frac{(c+d x)^3 e^{-4 e-4 f x}}{16 a^2 f}-\frac{(c+d x)^3 e^{-2 e-2 f x}}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}-\frac{3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4} \]
Antiderivative was successfully verified.
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Rule 3729
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{(a+a \tanh (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^3}{4 a^2}+\frac{e^{-4 e-4 f x} (c+d x)^3}{4 a^2}+\frac{e^{-2 e-2 f x} (c+d x)^3}{2 a^2}\right ) \, dx\\ &=\frac{(c+d x)^4}{16 a^2 d}+\frac{\int e^{-4 e-4 f x} (c+d x)^3 \, dx}{4 a^2}+\frac{\int e^{-2 e-2 f x} (c+d x)^3 \, dx}{2 a^2}\\ &=-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}-\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{(3 d) \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{16 a^2 f}+\frac{(3 d) \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{4 a^2 f}\\ &=-\frac{3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}-\frac{3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}-\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{\left (3 d^2\right ) \int e^{-4 e-4 f x} (c+d x) \, dx}{32 a^2 f^2}+\frac{\left (3 d^2\right ) \int e^{-2 e-2 f x} (c+d x) \, dx}{4 a^2 f^2}\\ &=-\frac{3 d^2 e^{-4 e-4 f x} (c+d x)}{128 a^2 f^3}-\frac{3 d^2 e^{-2 e-2 f x} (c+d x)}{8 a^2 f^3}-\frac{3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}-\frac{3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}-\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{\left (3 d^3\right ) \int e^{-4 e-4 f x} \, dx}{128 a^2 f^3}+\frac{\left (3 d^3\right ) \int e^{-2 e-2 f x} \, dx}{8 a^2 f^3}\\ &=-\frac{3 d^3 e^{-4 e-4 f x}}{512 a^2 f^4}-\frac{3 d^3 e^{-2 e-2 f x}}{16 a^2 f^4}-\frac{3 d^2 e^{-4 e-4 f x} (c+d x)}{128 a^2 f^3}-\frac{3 d^2 e^{-2 e-2 f x} (c+d x)}{8 a^2 f^3}-\frac{3 d e^{-4 e-4 f x} (c+d x)^2}{64 a^2 f^2}-\frac{3 d e^{-2 e-2 f x} (c+d x)^2}{8 a^2 f^2}-\frac{e^{-4 e-4 f x} (c+d x)^3}{16 a^2 f}-\frac{e^{-2 e-2 f x} (c+d x)^3}{4 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.05263, size = 420, normalized size = 1.83 \[ \frac{\text{sech}^2(e+f x) (\sinh (f x)+\cosh (f x))^2 \left (f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) (\sinh (2 e)+\cosh (2 e))+\frac{1}{32} (\sinh (2 e)-\cosh (2 e)) \cosh (4 f x) \left (24 c^2 d f^2 (4 f x+1)+32 c^3 f^3+12 c d^2 f \left (8 f^2 x^2+4 f x+1\right )+d^3 \left (32 f^3 x^3+24 f^2 x^2+12 f x+3\right )\right )+\frac{1}{32} (\cosh (2 e)-\sinh (2 e)) \sinh (4 f x) \left (24 c^2 d f^2 (4 f x+1)+32 c^3 f^3+12 c d^2 f \left (8 f^2 x^2+4 f x+1\right )+d^3 \left (32 f^3 x^3+24 f^2 x^2+12 f x+3\right )\right )+\sinh (2 f x) \left (6 c^2 d f^2 (2 f x+1)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 f x+1\right )+d^3 \left (4 f^3 x^3+6 f^2 x^2+6 f x+3\right )\right )-\cosh (2 f x) \left (6 c^2 d f^2 (2 f x+1)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 f x+1\right )+d^3 \left (4 f^3 x^3+6 f^2 x^2+6 f x+3\right )\right )\right )}{16 a^2 f^4 (\tanh (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 2226, normalized size = 9.7 \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 = 2.48075, size = 400, normalized size = 1.74 \begin{align*} \frac{1}{16} \, c^{3}{\left (\frac{4 \,{\left (f x + e\right )}}{a^{2} f} - \frac{4 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )}}{a^{2} f}\right )} + \frac{3 \,{\left (8 \, f^{2} x^{2} e^{\left (4 \, e\right )} - 8 \,{\left (2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} -{\left (4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} c^{2} d e^{\left (-4 \, e\right )}}{64 \, a^{2} f^{2}} + \frac{{\left (32 \, f^{3} x^{3} e^{\left (4 \, e\right )} - 48 \,{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 3 \,{\left (8 \, f^{2} x^{2} + 4 \, f x + 1\right )} e^{\left (-4 \, f x\right )}\right )} c d^{2} e^{\left (-4 \, e\right )}}{128 \, a^{2} f^{3}} + \frac{{\left (32 \, f^{4} x^{4} e^{\left (4 \, e\right )} - 32 \,{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} + 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (-2 \, f x\right )} -{\left (32 \, f^{3} x^{3} + 24 \, f^{2} x^{2} + 12 \, f x + 3\right )} e^{\left (-4 \, f x\right )}\right )} d^{3} e^{\left (-4 \, e\right )}}{512 \, a^{2} f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19901, size = 1258, normalized size = 5.47 \begin{align*} -\frac{128 \, d^{3} f^{3} x^{3} + 128 \, c^{3} f^{3} + 192 \, c^{2} d f^{2} + 192 \, c d^{2} f + 96 \, d^{3} + 192 \,{\left (2 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} -{\left (32 \, d^{3} f^{4} x^{4} - 32 \, c^{3} f^{3} - 24 \, c^{2} d f^{2} - 12 \, c d^{2} f + 32 \,{\left (4 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 24 \,{\left (8 \, c^{2} d f^{4} - 4 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 4 \,{\left (32 \, c^{3} f^{4} - 24 \, c^{2} d f^{3} - 12 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right )^{2} - 2 \,{\left (32 \, d^{3} f^{4} x^{4} + 32 \, c^{3} f^{3} + 24 \, c^{2} d f^{2} + 12 \, c d^{2} f + 32 \,{\left (4 \, c d^{2} f^{4} + d^{3} f^{3}\right )} x^{3} + 3 \, d^{3} + 24 \,{\left (8 \, c^{2} d f^{4} + 4 \, c d^{2} f^{3} + d^{3} f^{2}\right )} x^{2} + 4 \,{\left (32 \, c^{3} f^{4} + 24 \, c^{2} d f^{3} + 12 \, c d^{2} f^{2} + 3 \, d^{3} f\right )} x\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) -{\left (32 \, d^{3} f^{4} x^{4} - 32 \, c^{3} f^{3} - 24 \, c^{2} d f^{2} - 12 \, c d^{2} f + 32 \,{\left (4 \, c d^{2} f^{4} - d^{3} f^{3}\right )} x^{3} - 3 \, d^{3} + 24 \,{\left (8 \, c^{2} d f^{4} - 4 \, c d^{2} f^{3} - d^{3} f^{2}\right )} x^{2} + 4 \,{\left (32 \, c^{3} f^{4} - 24 \, c^{2} d f^{3} - 12 \, c d^{2} f^{2} - 3 \, d^{3} f\right )} x\right )} \sinh \left (f x + e\right )^{2} + 192 \,{\left (2 \, c^{2} d f^{3} + 2 \, c d^{2} f^{2} + d^{3} f\right )} x}{512 \,{\left (a^{2} f^{4} \cosh \left (f x + e\right )^{2} + 2 \, a^{2} f^{4} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + a^{2} f^{4} \sinh \left (f x + e\right )^{2}\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^{3}}{\tanh ^{2}{\left (e + f x \right )} + 2 \tanh{\left (e + f x \right )} + 1}\, dx + \int \frac{d^{3} x^{3}}{\tanh ^{2}{\left (e + f x \right )} + 2 \tanh{\left (e + f x \right )} + 1}\, dx + \int \frac{3 c d^{2} x^{2}}{\tanh ^{2}{\left (e + f x \right )} + 2 \tanh{\left (e + f x \right )} + 1}\, dx + \int \frac{3 c^{2} d x}{\tanh ^{2}{\left (e + f x \right )} + 2 \tanh{\left (e + f x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22322, size = 517, normalized size = 2.25 \begin{align*} \frac{{\left (32 \, d^{3} f^{4} x^{4} e^{\left (4 \, f x + 4 \, e\right )} + 128 \, c d^{2} f^{4} x^{3} e^{\left (4 \, f x + 4 \, e\right )} + 192 \, c^{2} d f^{4} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 128 \, d^{3} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} - 32 \, d^{3} f^{3} x^{3} + 128 \, c^{3} f^{4} x e^{\left (4 \, f x + 4 \, e\right )} - 384 \, c d^{2} f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c d^{2} f^{3} x^{2} - 384 \, c^{2} d f^{3} x e^{\left (2 \, f x + 2 \, e\right )} - 192 \, d^{3} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c^{2} d f^{3} x - 24 \, d^{3} f^{2} x^{2} - 128 \, c^{3} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 384 \, c d^{2} f^{2} x e^{\left (2 \, f x + 2 \, e\right )} - 32 \, c^{3} f^{3} - 48 \, c d^{2} f^{2} x - 192 \, c^{2} d f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 192 \, d^{3} f x e^{\left (2 \, f x + 2 \, e\right )} - 24 \, c^{2} d f^{2} - 12 \, d^{3} f x - 192 \, c d^{2} f e^{\left (2 \, f x + 2 \, e\right )} - 12 \, c d^{2} f - 96 \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, d^{3}\right )} e^{\left (-4 \, f x - 4 \, e\right )}}{512 \, a^{2} f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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