Optimal. Leaf size=195 \[ \frac {3 a^2 b \text {ArcTan}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{-a^2 b^2+b^4}+\frac {a \left (a^2+2 b^2\right ) \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}+\frac {2 a^2 \left (a+b \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac {x}{2}\right )+a \tanh ^2\left (\frac {x}{2}\right )\right )} \]
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Rubi [A]
time = 0.94, antiderivative size = 301, normalized size of antiderivative = 1.54, number of steps
used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4486, 2717,
2718, 6874, 652, 632, 210, 3179, 3153, 212} \begin {gather*} -\frac {a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {a^3}{b^3 (a-b)^2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {2 a^2 \left (3 a^2-b^2\right ) \text {ArcTan}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2}}+\frac {2 a^2 b \text {ArcTan}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {3 a^2 \text {ArcTan}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}-\frac {3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}+\frac {2 a^2 \left (a+b \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a \tanh ^2\left (\frac {x}{2}\right )+a+2 b \tanh \left (\frac {x}{2}\right )\right )}+\frac {3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac {2 a \sinh (x)}{b^3}+\frac {\cosh (x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 632
Rule 652
Rule 2717
Rule 2718
Rule 3153
Rule 3179
Rule 4486
Rule 6874
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=i \int \left (\frac {2 i a \cosh (x)}{b^3}-\frac {i \sinh (x)}{b^2}-\frac {i a^3 \cosh ^3(x)}{b^3 (i a \cosh (x)+i b \sinh (x))^2}-\frac {3 i a^2 \cosh ^2(x)}{b^3 (a \cosh (x)+b \sinh (x))}\right ) \, dx\\ &=-\frac {(2 a) \int \cosh (x) \, dx}{b^3}+\frac {\left (3 a^2\right ) \int \frac {\cosh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{b^3}+\frac {a^3 \int \frac {\cosh ^3(x)}{(i a \cosh (x)+i b \sinh (x))^2} \, dx}{b^3}+\frac {\int \sinh (x) \, dx}{b^2}\\ &=\frac {\cosh (x)}{b^2}-\frac {3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac {2 a \sinh (x)}{b^3}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {\left (-1-x^2\right )^3}{\left (1-x^2\right )^2 \left (a+2 b x+a x^2\right )^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^3}+\frac {\left (3 a^3\right ) \int \cosh (x) \, dx}{b^3 \left (a^2-b^2\right )}-\frac {\left (3 a^2\right ) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac {\cosh (x)}{b^2}-\frac {3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac {2 a \sinh (x)}{b^3}+\frac {3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \left (-\frac {1}{2 (a+b)^2 (-1+x)^2}-\frac {1}{2 (a-b)^2 (1+x)^2}+\frac {2 b^3 x}{a \left (-a^2+b^2\right ) \left (a+2 b x+a x^2\right )^2}+\frac {3 a^2 b^2-b^4}{a \left (a^2-b^2\right )^2 \left (a+2 b x+a x^2\right )}\right ) \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^3}-\frac {\left (3 i a^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 )}{b \left (a^2-b^2\right )}\\ &=-\frac {3 a^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac {\cosh (x)}{b^2}-\frac {3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac {2 a \sinh (x)}{b^3}+\frac {3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}-\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {x}{\left (a+2 b x+a x^2\right )^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2}+\frac {\left (2 a^2 \left (3 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b \left (a^2-b^2\right )^2}\\ &=-\frac {3 a^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac {\cosh (x)}{b^2}-\frac {3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac {2 a \sinh (x)}{b^3}+\frac {3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}+\frac {2 a^2 \left (a+b \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac {x}{2}\right )+a \tanh ^2\left (\frac {x}{2}\right )\right )}+\frac {\left (2 a^2 b\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2}-\frac {\left (4 a^2 \left (3 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tanh \left (\frac {x}{2}\right )\right )}{b \left (a^2-b^2\right )^2}\\ &=-\frac {3 a^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac {2 a^2 \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2}}+\frac {\cosh (x)}{b^2}-\frac {3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac {2 a \sinh (x)}{b^3}+\frac {3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}+\frac {2 a^2 \left (a+b \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac {x}{2}\right )+a \tanh ^2\left (\frac {x}{2}\right )\right )}-\frac {\left (4 a^2 b\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac {3 a^2 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{3/2}}+\frac {2 a^2 b \tan ^{-1}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {2 a^2 \left (3 a^2-b^2\right ) \tan ^{-1}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b \left (a^2-b^2\right )^{5/2}}+\frac {\cosh (x)}{b^2}-\frac {3 a^2 \cosh (x)}{b^2 \left (a^2-b^2\right )}-\frac {2 a \sinh (x)}{b^3}+\frac {3 a^3 \sinh (x)}{b^3 \left (a^2-b^2\right )}-\frac {a^3}{b^3 (a+b)^2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {a^3}{(a-b)^2 b^3 \left (1+\tanh \left (\frac {x}{2}\right )\right )}+\frac {2 a^2 \left (a+b \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2 \left (a+2 b \tanh \left (\frac {x}{2}\right )+a \tanh ^2\left (\frac {x}{2}\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 205, normalized size = 1.05 \begin {gather*} \frac {a \sqrt {a-b} \left (a^3+a^2 b+a b^2+b^3\right ) \cosh ^2(x)-b \cosh (x) \left (-6 a^3 \sqrt {a+b} \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )+(a-b)^{3/2} (a+b)^2 \sinh (x)\right )+a \left (a^2 \sqrt {a-b} (a+b)+6 a b^2 \sqrt {a+b} \text {ArcTan}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \sinh (x)-2 \sqrt {a-b} b^2 (a+b) \sinh ^2(x)\right )}{(a-b)^{5/2} (a+b)^3 (a \cosh (x)+b \sinh (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.39, size = 121, normalized size = 0.62
method | result | size |
default | \(\frac {1}{\left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {4 a^{2} \left (\frac {-\frac {b \tanh \left (\frac {x}{2}\right )}{2}-\frac {a}{2}}{a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}-\frac {3 b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}\) | \(121\) |
risch | \(\frac {{\mathrm e}^{x}}{2 a^{2}+4 a b +2 b^{2}}+\frac {{\mathrm e}^{-x}}{2 a^{2}-4 a b +2 b^{2}}+\frac {2 \,{\mathrm e}^{x} a^{3}}{\left (a -b \right )^{2} \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )}-\frac {3 b \,a^{2} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {3 b \,a^{2} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(185\) |
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. 789 vs.
\(2 (180) = 360\).
time = 0.39, size = 1633, normalized size = 8.37 \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 = 0.43, size = 174, normalized size = 0.89 \begin {gather*} \frac {6 \, a^{2} b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {e^{x}}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {5 \, a^{3} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{\left (2 \, x\right )} + 3 \, 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}}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a e^{\left (3 \, x\right )} + b e^{\left (3 \, x\right )} + a e^{x} - b e^{x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.79, size = 255, normalized size = 1.31 \begin {gather*} \frac {{\mathrm {e}}^{-x}}{2\,{\left (a-b\right )}^2}+\frac {{\mathrm {e}}^x}{2\,{\left (a+b\right )}^2}+\frac {6\,\mathrm {atan}\left (\frac {a^2\,b\,{\mathrm {e}}^x\,\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}{a^5\,\sqrt {a^4\,b^2}-b^5\,\sqrt {a^4\,b^2}+2\,a^2\,b^3\,\sqrt {a^4\,b^2}-2\,a^3\,b^2\,\sqrt {a^4\,b^2}+a\,b^4\,\sqrt {a^4\,b^2}-a^4\,b\,\sqrt {a^4\,b^2}}\right )\,\sqrt {a^4\,b^2}}{\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}+\frac {2\,a^3\,{\mathrm {e}}^x}{{\left (a+b\right )}^2\,{\left (a-b\right )}^2\,\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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