Optimal. Leaf size=46 \[ \frac{\log (\sinh (c+d x))}{d (a+b)}+\frac{b \log \left (a \cosh ^2(c+d x)+b\right )}{2 a d (a+b)} \]
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Rubi [A] time = 0.087361, antiderivative size = 46, 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, 446, 72} \[ \frac{\log (\sinh (c+d x))}{d (a+b)}+\frac{b \log \left (a \cosh ^2(c+d x)+b\right )}{2 a d (a+b)} \]
Antiderivative was successfully verified.
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Rule 4138
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{\coth (c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x}{(1-x) (b+a x)} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{(-a-b) (-1+x)}-\frac{b}{(a+b) (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac{b \log \left (b+a \cosh ^2(c+d x)\right )}{2 a (a+b) d}+\frac{\log (\sinh (c+d x))}{(a+b) d}\\ \end{align*}
Mathematica [A] time = 0.0890806, size = 42, normalized size = 0.91 \[ \frac{b \log \left (a \sinh ^2(c+d x)+a+b\right )+2 a \log (\sinh (c+d x))}{2 a^2 d+2 a b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 133, normalized size = 2.9 \begin{align*} -{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{b}{2\,da \left ( a+b \right ) }\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+b \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) }+{\frac{1}{d \left ( a+b \right ) }\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04251, size = 135, normalized size = 2.93 \begin{align*} \frac{b \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 \,{\left (a^{2} + a b\right )} d} + \frac{d x + c}{a d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a + b\right )} d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a + b\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.79042, size = 305, normalized size = 6.63 \begin{align*} -\frac{2 \,{\left (a + b\right )} d x - b \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 \log \left (\frac{2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{2 \,{\left (a^{2} + a b\right )} d} \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 \frac{\coth{\left (c + d x \right )}}{a + b \operatorname{sech}^{2}{\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 = 1.46974, size = 131, normalized size = 2.85 \begin{align*} -\frac{\frac{2 \, d x}{a} - \frac{b \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} + a b} - \frac{2 \, e^{\left (2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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