Optimal. Leaf size=51 \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b (2 a+b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0645143, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3664, 390, 207} \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b (2 a+b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 390
Rule 207
Rubi steps
\begin{align*} \int \text{csch}(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^2}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b (2 a+b)+b^2 x^2+\frac{a^2}{-1+x^2}\right ) \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{b (2 a+b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{b (2 a+b) \text{sech}(c+d x)}{d}+\frac{b^2 \text{sech}^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.155003, size = 50, normalized size = 0.98 \[ \frac{3 a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-3 b (2 a+b) \text{sech}(c+d x)+b^2 \text{sech}^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 97, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( -2\,{a}^{2}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +2\,ab \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}-\cosh \left ( dx+c \right ) \right ) +{b}^{2} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3\,\cosh \left ( dx+c \right ) }}-{\frac{2\,\cosh \left ( dx+c \right ) }{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04153, size = 265, normalized size = 5.2 \begin{align*} -\frac{2}{3} \, b^{2}{\left (\frac{3 \, e^{\left (-d x - c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac{2 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac{a^{2} \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} - \frac{4 \, a b}{d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00274, size = 2331, normalized size = 45.71 \begin{align*} \text{result too large to display} \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*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname{csch}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33456, size = 170, normalized size = 3.33 \begin{align*} -\frac{3 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac{2 \,{\left (6 \, a b e^{\left (5 \, d x + 5 \, c\right )} + 3 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 12 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 2 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a b e^{\left (d x + c\right )} + 3 \, b^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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