Optimal. Leaf size=314 \[ \frac {a b x}{4 \left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac {2 a^2 b^2 \sinh ^2(x)}{\left (a^2-b^2\right )^3}+\frac {b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac {a b \sinh (x) \cosh ^3(x)}{2 \left (a^2-b^2\right )^2}+\frac {a b \sinh (x) \cosh (x)}{4 \left (a^2-b^2\right )^2}+\frac {3 a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac {a b^3 \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {3 a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}-\frac {a^3 b x}{\left (a^2-b^2\right )^3}+\frac {a^3 b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {6 a^3 b^3 x}{\left (a^2-b^2\right )^4} \]
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Rubi [A] time = 1.73, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3111, 3109, 2565, 30, 2568, 2635, 8, 2564, 3098, 3133, 3097, 3075} \[ -\frac {a^3 b x}{\left (a^2-b^2\right )^3}-\frac {6 a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac {a b x}{4 \left (a^2-b^2\right )^2}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac {2 a^2 b^2 \sinh ^2(x)}{\left (a^2-b^2\right )^3}+\frac {b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac {a b \sinh (x) \cosh ^3(x)}{2 \left (a^2-b^2\right )^2}+\frac {a^3 b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac {a b \sinh (x) \cosh (x)}{4 \left (a^2-b^2\right )^2}+\frac {a b^3 \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {3 a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac {3 a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2564
Rule 2565
Rule 2568
Rule 2635
Rule 3075
Rule 3097
Rule 3098
Rule 3109
Rule 3111
Rule 3133
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac {a \int \frac {\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac {b \int \frac {\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac {(a b) \int \frac {\cosh ^2(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac {a^2 \int \cosh (x) \sinh ^3(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac {(a b) \int \cosh ^2(x) \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}+2 \frac {\left (a^2 b\right ) \int \frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac {b^2 \int \cosh ^3(x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac {\left (a b^2\right ) \int \frac {\cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac {\left (a^2 b^2\right ) \int \frac {\cosh (x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac {\left (a^3 b^2\right ) \int \frac {\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}+2 \left (\frac {\left (a^3 b\right ) \int \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^3}-\frac {\left (a^2 b^2\right ) \int \cosh (x) \sinh (x) \, dx}{\left (a^2-b^2\right )^3}+\frac {\left (a^3 b^2\right ) \int \frac {\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )-\frac {\left (a^2 b^3\right ) \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}-2 \left (\frac {\left (a^2 b^2\right ) \int \cosh (x) \sinh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac {\left (a b^3\right ) \int \cosh ^2(x) \, dx}{\left (a^2-b^2\right )^3}+\frac {\left (a^2 b^3\right ) \int \frac {\cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )+\frac {\left (a^3 b^3\right ) \int \frac {1}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^3}+\frac {a^2 \operatorname {Subst}\left (\int x^3 \, dx,x,i \sinh (x)\right )}{\left (a^2-b^2\right )^2}-2 \left (\frac {a b \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )^2}-\frac {(a b) \int \cosh ^2(x) \, dx}{4 \left (a^2-b^2\right )^2}\right )+\frac {b^2 \operatorname {Subst}\left (\int x^3 \, dx,x,\cosh (x)\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac {2 a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac {b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac {a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac {\left (i a^4 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 )^4}+\frac {\left (i a^2 b^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 )^4}+2 \left (-\frac {a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac {a^3 b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}+\frac {\left (i a^4 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 )^4}-\frac {\left (a^3 b\right ) \int 1 \, dx}{2 \left (a^2-b^2\right )^3}+\frac {\left (a^2 b^2\right ) \operatorname {Subst}(\int x \, dx,x,i \sinh (x))}{\left (a^2-b^2\right )^3}\right )-2 \left (\frac {a^3 b^3 x}{\left (a^2-b^2\right )^4}-\frac {a b^3 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}-\frac {\left (i a^2 b^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 )^4}-\frac {\left (a^2 b^2\right ) \operatorname {Subst}(\int x \, dx,x,i \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {\left (a b^3\right ) \int 1 \, dx}{2 \left (a^2-b^2\right )^3}\right )-2 \left (-\frac {a b \cosh (x) \sinh (x)}{8 \left (a^2-b^2\right )^2}+\frac {a b \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )^2}-\frac {(a b) \int 1 \, dx}{8 \left (a^2-b^2\right )^2}\right )\\ &=-\frac {2 a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac {b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac {a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac {a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac {a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac {a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}-2 \left (-\frac {a b x}{8 \left (a^2-b^2\right )^2}-\frac {a b \cosh (x) \sinh (x)}{8 \left (a^2-b^2\right )^2}+\frac {a b \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )^2}\right )+2 \left (-\frac {a^3 b^3 x}{\left (a^2-b^2\right )^4}-\frac {a^3 b x}{2 \left (a^2-b^2\right )^3}+\frac {a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac {a^3 b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}-\frac {a^2 b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^3}\right )-2 \left (\frac {a^3 b^3 x}{\left (a^2-b^2\right )^4}-\frac {a b^3 x}{2 \left (a^2-b^2\right )^3}-\frac {a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}-\frac {a b^3 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}+\frac {a^2 b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^3}\right )\\ \end {align*}
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Mathematica [A] time = 1.49, size = 366, normalized size = 1.17 \[ \frac {a^7 \cosh (5 x)+20 a^6 b \sinh (x)+9 a^6 b \sinh (3 x)-a^6 b \sinh (5 x)-48 a^5 b^2 x \sinh (x)-3 a^5 b^2 \cosh (5 x)+84 a^4 b^3 \sinh (x)-15 a^4 b^3 \sinh (3 x)+3 a^4 b^3 \sinh (5 x)+192 a^4 b^3 \sinh (x) \log (a \cosh (x)+b \sinh (x))-288 a^3 b^4 x \sinh (x)+3 a^3 b^4 \cosh (5 x)-100 a^2 b^5 \sinh (x)+3 a^2 b^5 \sinh (3 x)-3 a^2 b^5 \sinh (5 x)+192 a^2 b^5 \sinh (x) \log (a \cosh (x)+b \sinh (x))-3 a \left (a^2-b^2\right )^2 \left (a^2+3 b^2\right ) \cosh (3 x)-4 a \cosh (x) \left (a^6+12 a^5 b x+9 a^4 b^2+72 a^3 b^3 x-5 a^2 b^4-48 a^2 b^2 \left (a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))+12 a b^5 x-5 b^6\right )-48 a b^6 x \sinh (x)-a b^6 \cosh (5 x)-4 b^7 \sinh (x)+3 b^7 \sinh (3 x)+b^7 \sinh (5 x)}{64 (a-b)^4 (a+b)^4 (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 4001, normalized size = 12.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 384, normalized size = 1.22 \[ -\frac {3 \, a b x}{4 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )}} + \frac {{\left (36 \, a b e^{\left (4 \, x\right )} - 4 \, a^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} e^{\left (2 \, x\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )}} + \frac {3 \, {\left (a^{4} b^{2} + a^{2} b^{4}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {a^{2} e^{\left (4 \, x\right )} + 2 \, a b e^{\left (4 \, x\right )} + b^{2} e^{\left (4 \, x\right )} - 4 \, a^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} e^{\left (2 \, x\right )}}{64 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}} - \frac {3 \, a^{5} b^{2} e^{\left (2 \, x\right )} + 3 \, a^{4} b^{3} e^{\left (2 \, x\right )} + 3 \, a^{3} b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{5} e^{\left (2 \, x\right )} + 3 \, a^{5} b^{2} - a^{4} b^{3} + a^{3} b^{4} - 3 \, a^{2} b^{5}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 398, normalized size = 1.27 \[ \frac {1}{4 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {a}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {5 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {a}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 b}{8 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 a b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{4 \left (a +b \right )^{4}}+\frac {2 a^{4} b^{3} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{4} \left (a +b \right )^{4} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}-\frac {2 a^{2} b^{5} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{4} \left (a +b \right )^{4} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}+\frac {3 a^{4} b^{2} \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 )^{4} \left (a +b \right )^{4}}+\frac {3 a^{2} b^{4} \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 )^{4} \left (a +b \right )^{4}}+\frac {1}{4 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {a}{8 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {3 b}{8 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {a}{8 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5 b}{8 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 a b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{4 \left (a -b \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 384, normalized size = 1.22 \[ -\frac {3 \, a b x}{4 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}} + \frac {3 \, {\left (a^{4} b^{2} + a^{2} b^{4}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {4 \, {\left (a + b\right )} e^{\left (-2 \, x\right )} - {\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - 3 \, {\left (a^{6} - 4 \, a^{5} b + 5 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 4 \, a b^{5} - b^{6}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 52 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} e^{\left (-4 \, x\right )}}{64 \, {\left ({\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} e^{\left (-4 \, x\right )} + {\left (a^{8} - 2 \, a^{7} b - 2 \, a^{6} b^{2} + 6 \, a^{5} b^{3} - 6 \, a^{3} b^{5} + 2 \, a^{2} b^{6} + 2 \, a b^{7} - b^{8}\right )} e^{\left (-6 \, x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.92, size = 173, normalized size = 0.55 \[ \frac {{\mathrm {e}}^{4\,x}}{64\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{-4\,x}}{64\,{\left (a-b\right )}^2}+\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (3\,a^4\,b^2+3\,a^2\,b^4\right )}{a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8}-\frac {{\mathrm {e}}^{-2\,x}\,\left (a+b\right )}{16\,{\left (a-b\right )}^3}-\frac {{\mathrm {e}}^{2\,x}\,\left (a-b\right )}{16\,{\left (a+b\right )}^3}-\frac {3\,a\,b\,x}{4\,{\left (a-b\right )}^4}-\frac {2\,a^3\,b^3}{{\left (a+b\right )}^4\,{\left (a-b\right )}^3\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
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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