Optimal. Leaf size=23 \[ \frac{\log \left (a \cosh ^2(c+d x)+b\right )}{2 a d} \]
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Rubi [A] time = 0.0347621, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4138, 260} \[ \frac{\log \left (a \cosh ^2(c+d x)+b\right )}{2 a d} \]
Antiderivative was successfully verified.
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Rule 4138
Rule 260
Rubi steps
\begin{align*} \int \frac{\tanh (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\log \left (b+a \cosh ^2(c+d x)\right )}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.175158, size = 26, normalized size = 1.13 \[ \frac{\log (a \cosh (2 (c+d x))+a+2 b)}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 38, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( a+b \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) }{2\,da}}-{\frac{\ln \left ({\rm sech} \left (dx+c\right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13346, size = 69, normalized size = 3. \begin{align*} \frac{d x + c}{a d} + \frac{\log \left (2 \,{\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18883, size = 198, normalized size = 8.61 \begin{align*} -\frac{2 \, d x - \log \left (\frac{2 \,{\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.33816, size = 124, normalized size = 5.39 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \tanh{\left (c \right )}}{\operatorname{sech}^{2}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{1}{2 b d \operatorname{sech}^{2}{\left (c + d x \right )}} & \text{for}\: a = 0 \\\frac{x \tanh{\left (c \right )}}{a + b \operatorname{sech}^{2}{\left (c \right )}} & \text{for}\: d = 0 \\\frac{x - \frac{\log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d}}{a} & \text{for}\: b = 0 \\\frac{x}{a} + \frac{\log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \operatorname{sech}{\left (c + d x \right )} \right )}}{2 a d} + \frac{\log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \operatorname{sech}{\left (c + d x \right )} \right )}}{2 a d} - \frac{\log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{a d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39835, size = 76, normalized size = 3.3 \begin{align*} -\frac{\frac{2 \, d x}{a} - \frac{\log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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