Optimal. Leaf size=85 \[ \frac{b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 d (a+b)}+\frac{(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac{\log (\cosh (c+d x))}{d (a+b)}-\frac{\coth ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.137222, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 446, 72} \[ \frac{b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 d (a+b)}+\frac{(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac{\log (\cosh (c+d x))}{d (a+b)}-\frac{\coth ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) x^2 (a+b x)} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{(a+b) (-1+x)}+\frac{1}{a x^2}+\frac{a-b}{a^2 x}+\frac{b^3}{a^2 (a+b) (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\coth ^2(c+d x)}{2 a d}+\frac{\log (\cosh (c+d x))}{(a+b) d}+\frac{(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac{b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 (a+b) d}\\ \end{align*}
Mathematica [A] time = 0.163663, size = 60, normalized size = 0.71 \[ -\frac{-\frac{b^2 \log \left (a \coth ^2(c+d x)+b\right )}{a^2 (a+b)}-\frac{2 \log (\sinh (c+d x))}{a+b}+\frac{\coth ^2(c+d x)}{a}}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.086, size = 180, normalized size = 2.1 \begin{align*} -{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{d \left ( a+b \right ) }\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{{b}^{2}}{2\,d{a}^{2} \left ( a+b \right ) }\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}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{b}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{d \left ( a+b \right ) }\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 [A] time = 1.08124, size = 215, normalized size = 2.53 \begin{align*} \frac{b^{2} \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^{3} + a^{2} b\right )} d} + \frac{d x + c}{{\left (a + b\right )} d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )} - a\right )} d} + \frac{{\left (a - b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} + \frac{{\left (a - b\right )} \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 = 2.48711, size = 1817, normalized size = 21.38 \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 ^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19926, size = 190, normalized size = 2.24 \begin{align*} \frac{b^{2} \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^{3} d + a^{2} b d\right )}} - \frac{d x + c}{a d + b d} + \frac{{\left (a - b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{2} d} - \frac{2 \, e^{\left (2 \, d x + 2 \, c\right )}}{a d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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