Optimal. Leaf size=68 \[ -\frac{1}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log \left (a+b \tanh ^2(c+d x)\right )}{2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
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Rubi [A] time = 0.0857955, antiderivative size = 68, 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 = {3670, 444, 44} \[ -\frac{1}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log \left (a+b \tanh ^2(c+d x)\right )}{2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 444
Rule 44
Rubi steps
\begin{align*} \int \frac{\tanh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) (a+b x)^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{(a+b)^2 (-1+x)}+\frac{b}{(a+b) (a+b x)^2}+\frac{b}{(a+b)^2 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\log (\cosh (c+d x))}{(a+b)^2 d}+\frac{\log \left (a+b \tanh ^2(c+d x)\right )}{2 (a+b)^2 d}-\frac{1}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.371088, size = 55, normalized size = 0.81 \[ -\frac{\frac{a+b}{a+b \tanh ^2(c+d x)}-\log \left (a+b \tanh ^2(c+d x)\right )-2 \log (\cosh (c+d x))}{2 d (a+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 113, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{2\,d \left ( a+b \right ) ^{2}}}-{\frac{a}{2\,d \left ( a+b \right ) ^{2} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{b}{2\,d \left ( a+b \right ) ^{2} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}+{\frac{\ln \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d \left ( a+b \right ) ^{2}}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d \left ( a+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09235, size = 230, normalized size = 3.38 \begin{align*} -\frac{2 \, b e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} + 2 \,{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac{d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac{\log \left (2 \,{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95637, size = 1611, normalized size = 23.69 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22318, size = 208, normalized size = 3.06 \begin{align*} \frac{\log \left ({\left | a{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b \right |}\right )}{2 \,{\left (a^{2} d + 2 \, a b d + b^{2} d\right )}} - \frac{e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2}{2 \,{\left (a d + b d\right )}{\left (a{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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