Optimal. Leaf size=259 \[ \frac {3 a^3 b^2 \text {ArcTan}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {2 a b^4 \text {ArcTan}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {2 a^3 b \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {2 a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {4 a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {b^2 \sinh (x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))} \]
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Rubi [A]
time = 0.62, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps
used = 33, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3190, 3188,
2713, 2645, 30, 3179, 2717, 3153, 212, 2644, 2718, 3234} \begin {gather*} \frac {2 a b^4 \text {ArcTan}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh (x)}{\left (a^2-b^2\right )^2}-\frac {4 a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}-\frac {2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {2 a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac {3 a^3 b^2 \text {ArcTan}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {2 a^3 b \cosh (x)}{\left (a^2-b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 212
Rule 2644
Rule 2645
Rule 2713
Rule 2717
Rule 2718
Rule 3153
Rule 3179
Rule 3188
Rule 3190
Rule 3234
Rubi steps
\begin {align*} \int \frac {\cosh ^3(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac {a \int \frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac {b \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac {(a b) \int \frac {\cosh ^2(x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac {a^2 \int \cosh (x) \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac {(a b) \int \cosh ^2(x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}+2 \frac {\left (a^2 b\right ) \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac {b^2 \int \cosh ^3(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac {\left (a b^2\right ) \int \frac {\cosh ^2(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)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac {a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac {\left (a^3 b^2\right ) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}+2 \left (\frac {\left (a^3 b\right ) \int \sinh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac {\left (a^2 b^2\right ) \int \cosh (x) \, dx}{\left (a^2-b^2\right )^3}+\frac {\left (a^3 b^2\right ) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )-2 \left (-\frac {a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {\left (a^2 b^2\right ) \int \cosh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac {\left (a b^4\right ) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )+\frac {\left (i a^2\right ) \text {Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )}{\left (a^2-b^2\right )^2}-2 \frac {(a b) \text {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )}{\left (a^2-b^2\right )^2}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac {2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh (x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac {\left (i a^3 b^2\right ) \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}+2 \left (\frac {a^3 b \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {\left (i a^3 b^2\right ) \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}\right )-2 \left (-\frac {a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}-\frac {\left (i a b^4\right ) \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}\right )\\ &=\frac {a^3 b^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh (x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+2 \left (\frac {a^3 b^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {a^3 b \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}\right )-2 \left (-\frac {a b^4 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}\right )\\ \end {align*}
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Mathematica [A]
time = 1.49, size = 481, normalized size = 1.86 \begin {gather*} -\frac {\sqrt {a-b} b (a+b)+2 a^2 \sqrt {a+b} \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \cosh (x)+2 a b \sqrt {a+b} \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \sinh (x)}{8 (a-b)^{3/2} (a+b)^2 (a \cosh (x)+b \sinh (x))}+\frac {1}{16} \left (-\frac {6 a \left (a^2+3 b^2\right ) \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}-\frac {8 a b \cosh (x)}{(a-b)^2 (a+b)^2}+\frac {4 \left (a^2+b^2\right ) \sinh (x)}{(a-b)^2 (a+b)^2}-\frac {b \left (3 a^2+b^2\right )}{(a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))}\right )+\frac {1}{16} \left (\frac {10 a \left (a^4+10 a^2 b^2+5 b^4\right ) \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}+\frac {32 a b \left (a^2+b^2\right ) \cosh (x)}{(a-b)^3 (a+b)^3}-\frac {8 a b \cosh (3 x)}{3 (a-b)^2 (a+b)^2}-\frac {8 \left (a^4+6 a^2 b^2+b^4\right ) \sinh (x)}{(a-b)^3 (a+b)^3}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right )}{(a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))}+\frac {4 \left (a^2+b^2\right ) \sinh (3 x)}{3 (a-b)^2 (a+b)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.57, size = 198, normalized size = 0.76
method | result | size |
default | \(-\frac {1}{3 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {b}{\left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {2 a \,b^{2} \left (\frac {b^{2} \tanh \left (\frac {x}{2}\right )+a b}{a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}+\frac {\left (3 a^{2}+2 b^{2}\right ) \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {b}{\left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(198\) |
risch | \(\frac {{\mathrm e}^{3 x}}{24 a^{2}+48 a b +24 b^{2}}-\frac {{\mathrm e}^{x} a}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {3 \,{\mathrm e}^{x} b}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {a \,{\mathrm e}^{-x}}{8 a^{3}-24 a^{2} b +24 a \,b^{2}-8 b^{3}}+\frac {3 \,{\mathrm e}^{-x} b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {{\mathrm e}^{-3 x}}{24 \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 \,{\mathrm e}^{x} a^{2} b^{3}}{\left (a -b \right )^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {3 b^{2} a^{3} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {2 b^{4} a \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {3 b^{2} a^{3} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {2 b^{4} a \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(409\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2488 vs.
\(2 (245) = 490\).
time = 0.39, size = 5031, normalized size = 19.42 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 310, normalized size = 1.20 \begin {gather*} \frac {2 \, a^{2} b^{3} e^{x}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} + \frac {{\left (3 \, a e^{\left (2 \, x\right )} + 9 \, b e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {2 \, {\left (3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {a^{4} e^{\left (3 \, x\right )} + 4 \, a^{3} b e^{\left (3 \, x\right )} + 6 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 4 \, a b^{3} e^{\left (3 \, x\right )} + b^{4} e^{\left (3 \, x\right )} - 3 \, a^{4} e^{x} + 18 \, a^{2} b^{2} e^{x} + 24 \, a b^{3} e^{x} + 9 \, b^{4} e^{x}}{24 \, {\left (a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.97, size = 590, normalized size = 2.28 \begin {gather*} \frac {{\mathrm {e}}^{3\,x}}{24\,{\left (a+b\right )}^2}-\frac {{\mathrm {e}}^{-3\,x}}{24\,{\left (a-b\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a-3\,b\right )}{8\,{\left (a+b\right )}^3}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (2\,a\,b^4\,\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}+3\,a^3\,b^2\,\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}\right )}{a^7\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}+b^7\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}-3\,a^2\,b^5\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}+3\,a^3\,b^4\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}+3\,a^4\,b^3\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}-3\,a^5\,b^2\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}-a\,b^6\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}-a^6\,b\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}}\right )\,\sqrt {9\,a^6\,b^4+12\,a^4\,b^6+4\,a^2\,b^8}}{\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}}+\frac {{\mathrm {e}}^{-x}\,\left (a+3\,b\right )}{8\,{\left (a-b\right )}^3}+\frac {2\,a^2\,b^3\,{\mathrm {e}}^x}{{\left (a+b\right )}^3\,{\left (a-b\right )}^3\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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