Optimal. Leaf size=83 \[ -\frac{a^2}{2 b^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 b^2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
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Rubi [A] time = 0.150087, antiderivative size = 83, 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, 88} \[ -\frac{a^2}{2 b^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 b^2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 88
Rubi steps
\begin{align*} \int \frac{\tanh ^5(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{\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{x^2}{(1-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{a^2}{b (a+b) (a+b x)^2}-\frac{a (a+2 b)}{b (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{a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 b^2 (a+b)^2 d}-\frac{a^2}{2 b^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.473557, size = 69, normalized size = 0.83 \[ -\frac{\frac{a^2 (a+b)}{b^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac{a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{b^2}-2 \log (\cosh (c+d x))}{2 d (a+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 156, normalized size = 1.9 \begin{align*} -{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{2\,d \left ( a+b \right ) ^{2}}}-{\frac{{a}^{3}}{2\,d \left ( a+b \right ) ^{2}{b}^{2} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{{a}^{2}}{2\,d \left ( a+b \right ) ^{2}b \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}-{\frac{{a}^{2}\ln \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d \left ( a+b \right ) ^{2}{b}^{2}}}-{\frac{a\ln \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }{d \left ( a+b \right ) ^{2}b}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d \left ( a+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63801, size = 293, normalized size = 3.53 \begin{align*} -\frac{2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4} + 2 \,{\left (a^{3} b + a^{2} b^{2} - a b^{3} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac{{\left (a^{2} + 2 \, a b\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^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} d} + \frac{d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.61627, size = 2678, normalized size = 32.27 \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(-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 = 1.26749, size = 275, normalized size = 3.31 \begin{align*} -\frac{{\left (a^{2} + 2 \, a b\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^{2} b^{2} d + 2 \, a b^{3} d + b^{4} d\right )}} - \frac{d x + c}{a^{2} d + 2 \, a b d + b^{2} d} - \frac{2 \, a^{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} b d} + \frac{\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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