Optimal. Leaf size=122 \[ -\frac{d (c+d x)}{2 f^2 (a \tanh (e+f x)+a)}-\frac{(c+d x)^2}{2 f (a \tanh (e+f x)+a)}+\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a \tanh (e+f x)+a)}+\frac{d^2 x}{4 a f^2} \]
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Rubi [A] time = 0.118215, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3723, 3479, 8} \[ -\frac{d (c+d x)}{2 f^2 (a \tanh (e+f x)+a)}-\frac{(c+d x)^2}{2 f (a \tanh (e+f x)+a)}+\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a \tanh (e+f x)+a)}+\frac{d^2 x}{4 a f^2} \]
Antiderivative was successfully verified.
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Rule 3723
Rule 3479
Rule 8
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{a+a \tanh (e+f x)} \, dx &=\frac{(c+d x)^3}{6 a d}-\frac{(c+d x)^2}{2 f (a+a \tanh (e+f x))}+\frac{d \int \frac{c+d x}{a+a \tanh (e+f x)} \, dx}{f}\\ &=\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d (c+d x)}{2 f^2 (a+a \tanh (e+f x))}-\frac{(c+d x)^2}{2 f (a+a \tanh (e+f x))}+\frac{d^2 \int \frac{1}{a+a \tanh (e+f x)} \, dx}{2 f^2}\\ &=\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a+a \tanh (e+f x))}-\frac{d (c+d x)}{2 f^2 (a+a \tanh (e+f x))}-\frac{(c+d x)^2}{2 f (a+a \tanh (e+f x))}+\frac{d^2 \int 1 \, dx}{4 a f^2}\\ &=\frac{d^2 x}{4 a f^2}+\frac{(c+d x)^2}{4 a f}+\frac{(c+d x)^3}{6 a d}-\frac{d^2}{4 f^3 (a+a \tanh (e+f x))}-\frac{d (c+d x)}{2 f^2 (a+a \tanh (e+f x))}-\frac{(c+d x)^2}{2 f (a+a \tanh (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.260144, size = 169, normalized size = 1.39 \[ \frac{\text{sech}(e+f x) (\sinh (f x)+\cosh (f x)) \left (\frac{4}{3} f^3 x \left (3 c^2+3 c d x+d^2 x^2\right ) (\sinh (e)+\cosh (e))+(\sinh (e)-\cosh (e)) \cosh (2 f x) \left (2 c^2 f^2+2 c d f (2 f x+1)+d^2 \left (2 f^2 x^2+2 f x+1\right )\right )+(\cosh (e)-\sinh (e)) \sinh (2 f x) \left (2 c^2 f^2+2 c d f (2 f x+1)+d^2 \left (2 f^2 x^2+2 f x+1\right )\right )\right )}{8 a f^3 (\tanh (e+f x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 446, normalized size = 3.7 \begin{align*}{\frac{1}{{f}^{3}a} \left ( -{d}^{2} \left ({\frac{ \left ( fx+e \right ) ^{2} \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{2}}-{\frac{ \left ( fx+e \right ) \cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}-{\frac{ \left ( fx+e \right ) ^{2}}{4}}+{\frac{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{4}} \right ) -2\,cdf \left ( 1/2\, \left ( fx+e \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}-1/4\,\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) -1/4\,fx-e/4 \right ) +2\,{d}^{2}e \left ( 1/2\, \left ( fx+e \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}-1/4\,\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) -1/4\,fx-e/4 \right ) -{\frac{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}{c}^{2}{f}^{2}}{2}}+ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}cdfe-{\frac{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}{d}^{2}{e}^{2}}{2}}+{d}^{2} \left ({\frac{ \left ( fx+e \right ) ^{2}\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}+{\frac{ \left ( fx+e \right ) ^{3}}{6}}-{\frac{ \left ( fx+e \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{2}}+{\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{4}}+{\frac{fx}{4}}+{\frac{e}{4}} \right ) +2\,cdf \left ( 1/2\, \left ( fx+e \right ) \cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) +1/4\, \left ( fx+e \right ) ^{2}-1/4\, \left ( \cosh \left ( fx+e \right ) \right ) ^{2} \right ) -2\,{d}^{2}e \left ( 1/2\, \left ( fx+e \right ) \cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) +1/4\, \left ( fx+e \right ) ^{2}-1/4\, \left ( \cosh \left ( fx+e \right ) \right ) ^{2} \right ) +{c}^{2}{f}^{2} \left ({\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -2\,cdfe \left ( 1/2\,\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +{d}^{2}{e}^{2} \left ({\frac{\cosh \left ( fx+e \right ) \sinh \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22346, size = 170, normalized size = 1.39 \begin{align*} \frac{1}{4} \, c^{2}{\left (\frac{2 \,{\left (f x + e\right )}}{a f} - \frac{e^{\left (-2 \, f x - 2 \, e\right )}}{a f}\right )} + \frac{{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} -{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} c d e^{\left (-2 \, e\right )}}{4 \, a f^{2}} + \frac{{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} - 3 \,{\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x\right )}\right )} d^{2} e^{\left (-2 \, e\right )}}{24 \, a f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14139, size = 419, normalized size = 3.43 \begin{align*} \frac{{\left (4 \, d^{2} f^{3} x^{3} - 6 \, c^{2} f^{2} - 6 \, c d f + 6 \,{\left (2 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - 3 \, d^{2} + 6 \,{\left (2 \, c^{2} f^{3} - 2 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + e\right ) +{\left (4 \, d^{2} f^{3} x^{3} + 6 \, c^{2} f^{2} + 6 \, c d f + 6 \,{\left (2 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + 3 \, d^{2} + 6 \,{\left (2 \, c^{2} f^{3} + 2 \, c d f^{2} + d^{2} f\right )} x\right )} \sinh \left (f x + e\right )}{24 \,{\left (a f^{3} \cosh \left (f x + e\right ) + a f^{3} \sinh \left (f x + e\right )\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{\left (e + f x \right )} + 1}\, dx + \int \frac{d^{2} x^{2}}{\tanh{\left (e + f x \right )} + 1}\, dx + \int \frac{2 c d x}{\tanh{\left (e + f x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18131, size = 166, normalized size = 1.36 \begin{align*} \frac{{\left (4 \, d^{2} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c d f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 12 \, c^{2} f^{3} x e^{\left (2 \, f x + 2 \, e\right )} - 6 \, d^{2} f^{2} x^{2} - 12 \, c d f^{2} x - 6 \, c^{2} f^{2} - 6 \, d^{2} f x - 6 \, c d f - 3 \, d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{24 \, a f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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