Optimal. Leaf size=48 \[ -\frac{a^2+b^2}{2 b^3 (a+b \sinh (x))^2}+\frac{2 a}{b^3 (a+b \sinh (x))}+\frac{\log (a+b \sinh (x))}{b^3} \]
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Rubi [A] time = 0.0798414, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4391, 2668, 697} \[ -\frac{a^2+b^2}{2 b^3 (a+b \sinh (x))^2}+\frac{2 a}{b^3 (a+b \sinh (x))}+\frac{\log (a+b \sinh (x))}{b^3} \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{1}{(a \text{sech}(x)+b \tanh (x))^3} \, dx &=\int \frac{\cosh ^3(x)}{(a+b \sinh (x))^3} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{-b^2-x^2}{(a+x)^3} \, dx,x,b \sinh (x)\right )}{b^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{-a-x}+\frac{-a^2-b^2}{(a+x)^3}+\frac{2 a}{(a+x)^2}\right ) \, dx,x,b \sinh (x)\right )}{b^3}\\ &=\frac{\log (a+b \sinh (x))}{b^3}-\frac{a^2+b^2}{2 b^3 (a+b \sinh (x))^2}+\frac{2 a}{b^3 (a+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.133074, size = 42, normalized size = 0.88 \[ -\frac{\frac{-3 a^2-4 a b \sinh (x)+b^2}{2 (a+b \sinh (x))^2}-\log (a+b \sinh (x))}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.088, size = 241, normalized size = 5. \begin{align*} -{\frac{1}{{b}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+2\,{\frac{a \left ( \tanh \left ( x/2 \right ) \right ) ^{3}}{{b}^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) ^{2}}}-2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{3}}{ \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) ^{2}a}}-6\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}}{b \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) ^{2}}}+2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b}{ \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) ^{2}{a}^{2}}}-2\,{\frac{a\tanh \left ( x/2 \right ) }{{b}^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) ^{2}}}+2\,{\frac{\tanh \left ( x/2 \right ) }{ \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) ^{2}a}}+{\frac{1}{{b}^{3}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }-{\frac{1}{{b}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08794, size = 158, normalized size = 3.29 \begin{align*} \frac{2 \,{\left (2 \, a b e^{\left (-x\right )} - 2 \, a b e^{\left (-3 \, x\right )} +{\left (3 \, a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )}\right )}}{4 \, a b^{4} e^{\left (-x\right )} - 4 \, a b^{4} e^{\left (-3 \, x\right )} + b^{5} e^{\left (-4 \, x\right )} + b^{5} + 2 \,{\left (2 \, a^{2} b^{3} - b^{5}\right )} e^{\left (-2 \, x\right )}} + \frac{x}{b^{3}} + \frac{\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.56839, size = 1362, normalized size = 28.38 \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(-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 = 1.15932, size = 101, normalized size = 2.1 \begin{align*} \frac{\log \left ({\left | -b{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{3}} - \frac{3 \, b{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 4 \, a{\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, b}{2 \,{\left (b{\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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