Optimal. Leaf size=95 \[ -\frac{b (2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 d (a+b)^2}+\frac{\log (\tanh (c+d x))}{a^2 d}+\frac{b}{2 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
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Rubi [A] time = 0.149079, antiderivative size = 95, 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, 446, 72} \[ -\frac{b (2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 d (a+b)^2}+\frac{\log (\tanh (c+d x))}{a^2 d}+\frac{b}{2 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{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) 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{1}{a^2 x}-\frac{b^2}{a (a+b) (a+b x)^2}-\frac{b^2 (2 a+b)}{a^2 (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 (\tanh (c+d x))}{a^2 d}-\frac{b (2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 (a+b)^2 d}+\frac{b}{2 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.8582, size = 83, normalized size = 0.87 \[ \frac{\frac{\frac{b \left (\frac{a (a+b)}{a+b \tanh ^2(c+d x)}-(2 a+b) \log \left (a+b \tanh ^2(c+d x)\right )\right )}{(a+b)^2}+2 \log (\tanh (c+d x))}{a^2}+\frac{2 \log (\cosh (c+d x))}{(a+b)^2}}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 325, normalized size = 3.4 \begin{align*} -{\frac{1}{d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-2\,{\frac{{b}^{2} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d \left ( a+b \right ) ^{2}a \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }}-2\,{\frac{{b}^{3} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d \left ( a+b \right ) ^{2}{a}^{2} \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }}-{\frac{b}{d \left ( a+b \right ) ^{2}a}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }-{\frac{{b}^{2}}{2\,d \left ( a+b \right ) ^{2}{a}^{2}}\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) }+{\frac{1}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{d \left ( a+b \right ) ^{2}}\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 = 1.12821, size = 317, normalized size = 3.34 \begin{align*} \frac{2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + 2 \,{\left (a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac{{\left (2 \, a b + b^{2}\right )} \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^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d} + \frac{d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.14217, size = 2678, normalized size = 28.19 \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 \frac{\coth{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21831, size = 279, normalized size = 2.94 \begin{align*} -\frac{{\left (2 \, a b + b^{2}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{2 \,{\left (a^{4} d + 2 \, a^{3} b d + a^{2} b^{2} d\right )}} - \frac{d x + c}{a^{2} d + 2 \, a b d + b^{2} d} + \frac{2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{\left (a + b\right )}^{2} a d} + \frac{\log \left ({\left | -e^{\left (2 \, d x + 2 \, c\right )} + 1 \right |}\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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