Optimal. Leaf size=215 \[ \frac{a^3 b x}{\left (a^2-b^2\right )^3}+\frac{a b x \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3}+\frac{a b x}{\left (a^2-b^2\right )^2}+\frac{a b^3 x}{\left (a^2-b^2\right )^3}+\frac{a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac{a^2 b}{\left (a^2-b^2\right )^2 (a \coth (x)+b)}-\frac{a b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^2}-\frac{a^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac{3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]
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Rubi [A] time = 0.536012, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {3111, 3109, 2564, 30, 2635, 8, 3097, 3133, 3099, 3085, 3483, 3531, 3530} \[ \frac{a^3 b x}{\left (a^2-b^2\right )^3}+\frac{a b x \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3}+\frac{a b x}{\left (a^2-b^2\right )^2}+\frac{a b^3 x}{\left (a^2-b^2\right )^3}+\frac{a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac{a^2 b}{\left (a^2-b^2\right )^2 (a \coth (x)+b)}-\frac{a b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^2}-\frac{a^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac{3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3111
Rule 3109
Rule 2564
Rule 30
Rule 2635
Rule 8
Rule 3097
Rule 3133
Rule 3099
Rule 3085
Rule 3483
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{\cosh (x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac{a \int \frac{\sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac{b \int \frac{\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac{(a b) \int \frac{\sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac{a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac{a^3 \int \frac{\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}-2 \frac{(a b) \int \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}+\frac{b^2 \int \cosh (x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}-\frac{\left (a b^2\right ) \int \frac{\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}-\frac{(a b) \int \frac{1}{(-i b-i a \coth (x))^2} \, dx}{a^2-b^2}\\ &=\frac{a^3 b x}{\left (a^2-b^2\right )^3}+\frac{a b^3 x}{\left (a^2-b^2\right )^3}-\frac{a^2 b}{\left (a^2-b^2\right )^2 (b+a \coth (x))}+\frac{a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac{\left (i a^4\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}-\frac{\left (i a^2 b^2\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}-2 \left (\frac{a b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}-\frac{(a b) \int 1 \, dx}{2 \left (a^2-b^2\right )^2}\right )-\frac{(a b) \int \frac{-i b+i a \coth (x)}{-i b-i a \coth (x)} \, dx}{\left (a^2-b^2\right )^2}-\frac{b^2 \operatorname{Subst}(\int x \, dx,x,i \sinh (x))}{\left (a^2-b^2\right )^2}\\ &=\frac{a^3 b x}{\left (a^2-b^2\right )^3}+\frac{a b^3 x}{\left (a^2-b^2\right )^3}+\frac{a b \left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^3}-\frac{a^2 b}{\left (a^2-b^2\right )^2 (b+a \coth (x))}-\frac{a^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac{a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac{a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-2 \left (-\frac{a b x}{2 \left (a^2-b^2\right )^2}+\frac{a b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}\right )-\frac{\left (2 i a^2 b^2\right ) \int \frac{-a-b \coth (x)}{-i b-i a \coth (x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac{a^3 b x}{\left (a^2-b^2\right )^3}+\frac{a b^3 x}{\left (a^2-b^2\right )^3}+\frac{a b \left (a^2+b^2\right ) x}{\left (a^2-b^2\right )^3}-\frac{a^2 b}{\left (a^2-b^2\right )^2 (b+a \coth (x))}-\frac{a^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac{3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac{a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-2 \left (-\frac{a b x}{2 \left (a^2-b^2\right )^2}+\frac{a b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^2}\right )\\ \end{align*}
Mathematica [A] time = 0.950532, size = 176, normalized size = 0.82 \[ \frac{a \left (a^2-b^2\right )^2 \cosh (3 x)+a \cosh (x) \left (-8 a^2 \left (a^2+3 b^2\right ) \log (a \cosh (x)+b \sinh (x))+2 a^2 b^2+24 a^3 b x+a^4+8 a b^3 x-3 b^4\right )-2 b \sinh (x) \left (\left (a^2-b^2\right )^2 \cosh (2 x)+2 a \left (2 a \left (a^2+3 b^2\right ) \log (a \cosh (x)+b \sinh (x))-6 a^2 b x+3 a^3-3 a b^2-2 b^3 x\right )\right )}{8 (a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 253, normalized size = 1.2 \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}{ \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-2\,{\frac{{a}^{4}b\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}^{2}{b}^{3}\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 ) }}-{\frac{{a}^{4}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }-3\,{\frac{{a}^{2}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ){b}^{2}}{ \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}{ \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 = 1.27537, size = 325, normalized size = 1.51 \begin{align*} -\frac{a x}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac{{\left (a^{4} + 3 \, a^{2} b^{2}\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} - 20 \, a^{3} b + 6 \, 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 = 2.68962, size = 3650, normalized size = 16.98 \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 = 1.17677, size = 321, normalized size = 1.49 \begin{align*} \frac{a x}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{{\left (a^{4} + 3 \, a^{2} b^{2}\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{2 \, a^{3} e^{\left (4 \, x\right )} - 4 \, a^{2} b e^{\left (4 \, x\right )} + 2 \, a b^{2} e^{\left (4 \, x\right )} + 3 \, a^{3} e^{\left (2 \, x\right )} + 11 \, a^{2} b e^{\left (2 \, x\right )} + a b^{2} e^{\left (2 \, x\right )} + b^{3} 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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