Optimal. Leaf size=129 \[ \frac{2 a b}{d \left (a^2-b^2\right )^2 (a+b \tanh (c+d x))}+\frac{b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}-\frac{b \left (3 a^2+b^2\right ) \log (a \cosh (c+d x)+b \sinh (c+d x))}{d \left (a^2-b^2\right )^3}+\frac{a x \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3} \]
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Rubi [A] time = 0.177583, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3483, 3529, 3531, 3530} \[ \frac{2 a b}{d \left (a^2-b^2\right )^2 (a+b \tanh (c+d x))}+\frac{b}{2 d \left (a^2-b^2\right ) (a+b \tanh (c+d x))^2}-\frac{b \left (3 a^2+b^2\right ) \log (a \cosh (c+d x)+b \sinh (c+d x))}{d \left (a^2-b^2\right )^3}+\frac{a x \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3483
Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{1}{(a+b \tanh (c+d x))^3} \, dx &=\frac{b}{2 \left (a^2-b^2\right ) d (a+b \tanh (c+d x))^2}+\frac{\int \frac{a-b \tanh (c+d x)}{(a+b \tanh (c+d x))^2} \, dx}{a^2-b^2}\\ &=\frac{b}{2 \left (a^2-b^2\right ) d (a+b \tanh (c+d x))^2}+\frac{2 a b}{\left (a^2-b^2\right )^2 d (a+b \tanh (c+d x))}+\frac{\int \frac{a^2+b^2-2 a b \tanh (c+d x)}{a+b \tanh (c+d x)} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac{b}{2 \left (a^2-b^2\right ) d (a+b \tanh (c+d x))^2}+\frac{2 a b}{\left (a^2-b^2\right )^2 d (a+b \tanh (c+d x))}-\frac{\left (i b \left (3 a^2+b^2\right )\right ) \int \frac{-i b-i a \tanh (c+d x)}{a+b \tanh (c+d x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac{a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}-\frac{b \left (3 a^2+b^2\right ) \log (a \cosh (c+d x)+b \sinh (c+d x))}{\left (a^2-b^2\right )^3 d}+\frac{b}{2 \left (a^2-b^2\right ) d (a+b \tanh (c+d x))^2}+\frac{2 a b}{\left (a^2-b^2\right )^2 d (a+b \tanh (c+d x))}\\ \end{align*}
Mathematica [A] time = 2.29119, size = 122, normalized size = 0.95 \[ \frac{\frac{b \left (\frac{\left (a^2-b^2\right ) \left (5 a^2+4 a b \tanh (c+d x)-b^2\right )}{(a+b \tanh (c+d x))^2}-2 \left (3 a^2+b^2\right ) \log (a+b \tanh (c+d x))\right )}{\left (a^2-b^2\right )^3}-\frac{\log (1-\tanh (c+d x))}{(a+b)^3}+\frac{\log (\tanh (c+d x)+1)}{(a-b)^3}}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 166, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{2\,d \left ( a-b \right ) ^{3}}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d \left ( a+b \right ) ^{3}}}+{\frac{b}{2\,d \left ( a-b \right ) \left ( a+b \right ) \left ( a+b\tanh \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{ab}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( a+b\tanh \left ( dx+c \right ) \right ) }}-3\,{\frac{b\ln \left ( a+b\tanh \left ( dx+c \right ) \right ){a}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}}-{\frac{{b}^{3}\ln \left ( a+b\tanh \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.42882, size = 439, normalized size = 3.4 \begin{align*} -\frac{{\left (3 \, a^{2} b + b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a - b\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d} - \frac{2 \,{\left (3 \, a^{2} b^{2} + 3 \, a b^{3} +{\left (3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7} + 2 \,{\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{7} - 3 \, a^{6} b + a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac{d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.57486, size = 3182, normalized size = 24.67 \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 [A] time = 1.25006, size = 288, normalized size = 2.23 \begin{align*} -\frac{{\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | -a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{6} d - 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d - b^{6} d} + \frac{d x + c}{a^{3} d - 3 \, a^{2} b d + 3 \, a b^{2} d - b^{3} d} + \frac{2 \,{\left ({\left (3 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )} + \frac{3 \,{\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}}{a + b}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b\right )}^{2}{\left (a + b\right )}^{2}{\left (a - b\right )}^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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