Optimal. Leaf size=49 \[ \frac{b}{2 a^2 d \left (a \cosh ^2(c+d x)+b\right )}+\frac{\log \left (a \cosh ^2(c+d x)+b\right )}{2 a^2 d} \]
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Rubi [A] time = 0.0624705, antiderivative size = 49, 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 = {4138, 266, 43} \[ \frac{b}{2 a^2 d \left (a \cosh ^2(c+d x)+b\right )}+\frac{\log \left (a \cosh ^2(c+d x)+b\right )}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 4138
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\tanh (c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{(b+a x)^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{b}{a (b+a x)^2}+\frac{1}{a (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{b}{2 a^2 d \left (b+a \cosh ^2(c+d x)\right )}+\frac{\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.571651, size = 79, normalized size = 1.61 \[ \frac{(a+2 b) \log (a \cosh (2 (c+d x))+a+2 b)+a \cosh (2 (c+d x)) \log (a \cosh (2 (c+d x))+a+2 b)+2 b}{2 a^2 d (a \cosh (2 (c+d x))+a+2 b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 60, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,da \left ( a+b \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) }}+{\frac{\ln \left ( a+b \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) }{2\,d{a}^{2}}}-{\frac{\ln \left ({\rm sech} \left (dx+c\right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26886, size = 143, normalized size = 2.92 \begin{align*} \frac{2 \, b e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} e^{\left (-4 \, d x - 4 \, c\right )} + a^{3} + 2 \,{\left (a^{3} + 2 \, a^{2} b\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac{d x + c}{a^{2} 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^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28445, size = 1243, normalized size = 25.37 \begin{align*} -\frac{2 \, a d x \cosh \left (d x + c\right )^{4} + 8 \, a d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, a d x \sinh \left (d x + c\right )^{4} + 2 \, a d x + 4 \,{\left ({\left (a + 2 \, b\right )} d x - b\right )} \cosh \left (d x + c\right )^{2} + 4 \,{\left (3 \, a d x \cosh \left (d x + c\right )^{2} +{\left (a + 2 \, b\right )} d x - b\right )} \sinh \left (d x + c\right )^{2} -{\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \,{\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (a \cosh \left (d x + c\right )^{3} +{\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \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 ) + 8 \,{\left (a d x \cosh \left (d x + c\right )^{3} +{\left ({\left (a + 2 \, b\right )} d x - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \,{\left (a^{3} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} d \sinh \left (d x + c\right )^{4} + a^{3} d + 2 \,{\left (a^{3} + 2 \, a^{2} b\right )} d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{3} d \cosh \left (d x + c\right )^{2} +{\left (a^{3} + 2 \, a^{2} b\right )} d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (a^{3} d \cosh \left (d x + c\right )^{3} +{\left (a^{3} + 2 \, a^{2} b\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \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 = 1.59107, size = 163, normalized size = 3.33 \begin{align*} -\frac{\frac{2 \, d x}{a^{2}} + \frac{e^{\left (4 \, d x + 4 \, c\right )} + 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}{{\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} - \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}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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