Optimal. Leaf size=205 \[ -\frac{a^2 x}{2 \left (a^2-b^2\right )^2}-\frac{4 a^2 b^2 x}{\left (a^2-b^2\right )^3}+\frac{b^2 x}{2 \left (a^2-b^2\right )^2}-\frac{a b \sinh ^2(x)}{\left (a^2-b^2\right )^2}+\frac{a^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}+\frac{a b^2 \sinh (x)}{\left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))}+\frac{b^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}+\frac{2 a^3 b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac{2 a b^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]
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Rubi [A] time = 0.659398, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3111, 3109, 2635, 8, 2564, 30, 3098, 3133, 3097, 3075} \[ -\frac{a^2 x}{2 \left (a^2-b^2\right )^2}-\frac{4 a^2 b^2 x}{\left (a^2-b^2\right )^3}+\frac{b^2 x}{2 \left (a^2-b^2\right )^2}-\frac{a b \sinh ^2(x)}{\left (a^2-b^2\right )^2}+\frac{a^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}+\frac{a b^2 \sinh (x)}{\left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))}+\frac{b^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}+\frac{2 a^3 b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac{2 a b^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3111
Rule 3109
Rule 2635
Rule 8
Rule 2564
Rule 30
Rule 3098
Rule 3133
Rule 3097
Rule 3075
Rubi steps
\begin{align*} \int \frac{\cosh ^2(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac{a \int \frac{\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac{b \int \frac{\cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac{(a b) \int \frac{\cosh (x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac{a^2 \int \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac{(a b) \int \cosh (x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}+2 \frac{\left (a^2 b\right ) \int \frac{\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac{b^2 \int \cosh ^2(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac{\left (a b^2\right ) \int \frac{\cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{1}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{a^2 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}+\frac{b^2 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}+\frac{a b^2 \sinh (x)}{\left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))}+2 \left (-\frac{a^2 b^2 x}{\left (a^2-b^2\right )^3}+\frac{\left (i a^3 b\right ) \int \frac{-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )-2 \left (\frac{a^2 b^2 x}{\left (a^2-b^2\right )^3}-\frac{\left (i a b^3\right ) \int \frac{-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )-\frac{a^2 \int 1 \, dx}{2 \left (a^2-b^2\right )^2}+2 \frac{(a b) \operatorname{Subst}(\int x \, dx,x,i \sinh (x))}{\left (a^2-b^2\right )^2}+\frac{b^2 \int 1 \, dx}{2 \left (a^2-b^2\right )^2}\\ &=-\frac{a^2 x}{2 \left (a^2-b^2\right )^2}+\frac{b^2 x}{2 \left (a^2-b^2\right )^2}+2 \left (-\frac{a^2 b^2 x}{\left (a^2-b^2\right )^3}+\frac{a^3 b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}\right )-2 \left (\frac{a^2 b^2 x}{\left (a^2-b^2\right )^3}-\frac{a b^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}\right )+\frac{a^2 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}+\frac{b^2 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}-\frac{a b \sinh ^2(x)}{\left (a^2-b^2\right )^2}+\frac{a b^2 \sinh (x)}{\left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 1.86382, size = 174, normalized size = 0.85 \[ \frac{1}{8} \left (-\frac{4 x \left (6 a^2 b^2+a^4+b^4\right )}{(a-b)^3 (a+b)^3}+\frac{2 \left (a^2+b^2\right ) \sinh (2 x)}{(a-b)^2 (a+b)^2}+\frac{\left (6 a^2 b^2+a^4+b^4\right ) \sinh (x)}{a (a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))}+\frac{16 a b \left (a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac{\sinh (x)}{a^2 \cosh (x)+a b \sinh (x)}-\frac{4 a b \cosh (2 x)}{(a-b)^2 (a+b)^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 286, normalized size = 1.4 \begin{align*} -{\frac{1}{2\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{1}{2\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{a}{2\, \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{b}{2\, \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+2\,{\frac{{a}^{3}{b}^{2}\tanh \left ( x/2 \right ) }{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{{b}^{4}a\tanh \left ( x/2 \right ) }{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{{a}^{3}b\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}}+2\,{\frac{a{b}^{3}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}}+{\frac{1}{2\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{2\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{b}{2\, \left ( a+b \right ) ^{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 [A] time = 1.21942, size = 329, normalized size = 1.6 \begin{align*} -\frac{{\left (a - b\right )} x}{2 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac{2 \,{\left (a^{3} b + a b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4} +{\left (a^{4} - 4 \, a^{3} b + 22 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, x\right )}}{8 \,{\left ({\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} e^{\left (-2 \, x\right )} +{\left (a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right )} e^{\left (-4 \, x\right )}\right )}} - \frac{e^{\left (-2 \, x\right )}}{8 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.69527, size = 3779, normalized size = 18.43 \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 = 1.17331, size = 313, normalized size = 1.53 \begin{align*} -\frac{{\left (a + b\right )} x}{2 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{2 \,{\left (a^{3} b + a b^{3}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{e^{\left (2 \, x\right )}}{8 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac{a^{3} e^{\left (4 \, x\right )} - 3 \, a^{2} b e^{\left (4 \, x\right )} + 3 \, a b^{2} e^{\left (4 \, x\right )} - b^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (2 \, x\right )} - 8 \, a b^{2} e^{\left (2 \, x\right )} - a^{3} - a^{2} b + a b^{2} + b^{3}}{8 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + a e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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