Optimal. Leaf size=205 \[ -\frac {a^2 x}{2 \left (a^2-b^2\right )^2}-\frac {4 a^2 b^2 x}{\left (a^2-b^2\right )^3}+\frac {b^2 x}{2 \left (a^2-b^2\right )^2}-\frac {a b \sinh ^2(x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}+\frac {a b^2 \sinh (x)}{\left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))}+\frac {b^2 \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )^2}+\frac {2 a b^3 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {2 a^3 b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]
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Rubi [A] time = 0.66, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 8
Rule 30
Rule 2564
Rule 2635
Rule 3075
Rule 3097
Rule 3098
Rule 3109
Rule 3111
Rule 3133
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*}
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Mathematica [A] time = 2.05, size = 174, normalized size = 0.85 \[ \frac {1}{8} \left (\frac {2 \left (a^2+b^2\right ) \sinh (2 x)}{(a-b)^2 (a+b)^2}+\frac {16 a b \left (a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {\sinh (x)}{a^2 \cosh (x)+a b \sinh (x)}-\frac {4 x \left (a^4+6 a^2 b^2+b^4\right )}{(a-b)^3 (a+b)^3}+\frac {\left (a^4+6 a^2 b^2+b^4\right ) \sinh (x)}{a (a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))}-\frac {4 a b \cosh (2 x)}{(a-b)^2 (a+b)^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 1726, normalized size = 8.42 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 232, normalized size = 1.13 \[ -\frac {{\left (a + b\right )} x}{2 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {2 \, {\left (a^{3} b + a b^{3}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {e^{\left (2 \, x\right )}}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {a^{3} e^{\left (4 \, x\right )} - 3 \, a^{2} b e^{\left (4 \, x\right )} + 3 \, a b^{2} e^{\left (4 \, x\right )} - b^{3} e^{\left (4 \, x\right )} - 8 \, a^{2} b e^{\left (2 \, x\right )} - 8 \, a b^{2} e^{\left (2 \, x\right )} - a^{3} - a^{2} b + a b^{2} + b^{3}}{8 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + a e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 286, normalized size = 1.40 \[ \frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) b}{2 \left (a +b \right )^{3}}+\frac {2 a^{3} b^{2} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}-\frac {2 a \,b^{4} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}+\frac {2 a^{3} b \ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}+\frac {2 a \,b^{3} \ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 \left (a -b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) b}{2 \left (a -b \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 244, normalized size = 1.19 \[ -\frac {{\left (a - b\right )} x}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {2 \, {\left (a^{3} b + a b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4} + {\left (a^{4} - 4 \, a^{3} b + 22 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, x\right )}}{8 \, {\left ({\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} e^{\left (-2 \, x\right )} + {\left (a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right )} e^{\left (-4 \, x\right )}\right )}} - \frac {e^{\left (-2 \, x\right )}}{8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.85, size = 132, normalized size = 0.64 \[ \frac {{\mathrm {e}}^{2\,x}}{8\,{\left (a+b\right )}^2}-\frac {{\mathrm {e}}^{-2\,x}}{8\,{\left (a-b\right )}^2}+\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (2\,a^3\,b+2\,a\,b^3\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}-\frac {x\,\left (a+b\right )}{2\,{\left (a-b\right )}^3}-\frac {2\,a^2\,b^2}{{\left (a+b\right )}^3\,{\left (a-b\right )}^2\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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