Optimal. Leaf size=147 \[ -\frac {a (3 a+b) \log (1-\tanh (x))}{16 (a+b)^3}+\frac {a (3 a-b) \log (1+\tanh (x))}{16 (a-b)^3}-\frac {a^4 b \log (a+b \tanh (x))}{\left (a^2-b^2\right )^3}-\frac {\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac {\cosh ^2(x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \tanh (x)\right )}{8 \left (a^2-b^2\right )^2} \]
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Rubi [A]
time = 0.26, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3597, 1661,
815} \begin {gather*} -\frac {\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac {\cosh ^2(x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}-\frac {a^4 b \log (a+b \tanh (x))}{\left (a^2-b^2\right )^3}-\frac {a (3 a+b) \log (1-\tanh (x))}{16 (a+b)^3}+\frac {a (3 a-b) \log (\tanh (x)+1)}{16 (a-b)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 815
Rule 1661
Rule 3597
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{a+b \tanh (x)} \, dx &=-\left (b \text {Subst}\left (\int \frac {x^4}{(a+x) \left (-b^2+x^2\right )^3} \, dx,x,b \tanh (x)\right )\right )\\ &=-\frac {\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^4}{a^2-b^2}-\frac {3 a b^4 x}{a^2-b^2}+4 b^2 x^2}{(a+x) \left (-b^2+x^2\right )^2} \, dx,x,b \tanh (x)\right )}{4 b}\\ &=-\frac {\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac {\cosh ^2(x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}-\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^4 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^2}-\frac {a b^4 \left (5 a^2-b^2\right ) x}{\left (a^2-b^2\right )^2}}{(a+x) \left (-b^2+x^2\right )} \, dx,x,b \tanh (x)\right )}{8 b^3}\\ &=-\frac {\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac {\cosh ^2(x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}-\frac {\text {Subst}\left (\int \left (-\frac {a b^3 (3 a+b)}{2 (a+b)^3 (b-x)}+\frac {8 a^4 b^4}{(a-b)^3 (a+b)^3 (a+x)}-\frac {a (3 a-b) b^3}{2 (a-b)^3 (b+x)}\right ) \, dx,x,b \tanh (x)\right )}{8 b^3}\\ &=-\frac {a (3 a+b) \log (1-\tanh (x))}{16 (a+b)^3}+\frac {a (3 a-b) \log (1+\tanh (x))}{16 (a-b)^3}-\frac {a^4 b \log (a+b \tanh (x))}{\left (a^2-b^2\right )^3}-\frac {\cosh ^4(x) (b-a \tanh (x))}{4 \left (a^2-b^2\right )}+\frac {\cosh ^2(x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \tanh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 144, normalized size = 0.98 \begin {gather*} \frac {12 a^5 x+24 a^3 b^2 x-4 a b^4 x+4 b \left (3 a^4-4 a^2 b^2+b^4\right ) \cosh (2 x)-b \left (a^2-b^2\right )^2 \cosh (4 x)-32 a^4 b \log (a \cosh (x)+b \sinh (x))-8 a^3 \left (a^2-b^2\right ) \sinh (2 x)+a^5 \sinh (4 x)-2 a^3 b^2 \sinh (4 x)+a b^4 \sinh (4 x)}{32 (a-b)^3 (a+b)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 246, normalized size = 1.67
method | result | size |
risch | \(\frac {3 a^{2} x}{8 \left (a +b \right )^{3}}+\frac {a x b}{8 \left (a +b \right )^{3}}+\frac {{\mathrm e}^{4 x}}{64 a +64 b}-\frac {{\mathrm e}^{2 x} a}{8 \left (a +b \right )^{2}}-\frac {{\mathrm e}^{2 x} b}{16 \left (a +b \right )^{2}}-\frac {{\mathrm e}^{-2 x} b}{16 \left (a -b \right )^{2}}+\frac {{\mathrm e}^{-2 x} a}{8 \left (a -b \right )^{2}}-\frac {{\mathrm e}^{-4 x}}{64 \left (a -b \right )}+\frac {2 a^{4} b x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}-\frac {a^{4} b \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}\) | \(183\) |
default | \(-\frac {8}{\left (32 a -32 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {32}{\left (64 a -64 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a -b}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 a -b}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {a \left (3 a -b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 \left (a -b \right )^{3}}+\frac {8}{\left (32 a +32 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {32}{\left (64 a +64 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a -b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3 a +b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {a \left (3 a +b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {a^{4} b \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 b \tanh \left (\frac {x}{2}\right )+a \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 163, normalized size = 1.11 \begin {gather*} -\frac {a^{4} b \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 {{\left (3 \, a^{2} + a b\right )} x}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (4 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} - a - b\right )} e^{\left (4 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {4 \, {\left (2 \, a - b\right )} e^{\left (-2 \, x\right )} - {\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} \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. 1226 vs.
\(2 (139) = 278\).
time = 0.36, size = 1226, normalized size = 8.34 \begin {gather*} \frac {{\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{8} + 8 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{7} + {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \sinh \left (x\right )^{8} - 4 \, {\left (2 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{6} - 4 \, {\left (2 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - b^{5} - 7 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{6} + 8 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 6 \, a^{3} b^{2} - a b^{4}\right )} x \cosh \left (x\right )^{4} + 8 \, {\left (7 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (2 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - a^{5} - a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - a b^{4} - b^{5} + 2 \, {\left (35 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{4} - 30 \, {\left (2 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 6 \, a^{3} b^{2} - a b^{4}\right )} x\right )} \sinh \left (x\right )^{4} + 8 \, {\left (7 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{5} - 10 \, {\left (2 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{3} + 4 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 6 \, a^{3} b^{2} - a b^{4}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (2 \, a^{5} + 3 \, a^{4} b - 2 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (7 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{6} + 2 \, a^{5} + 3 \, a^{4} b - 2 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + b^{5} - 15 \, {\left (2 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{4} + 12 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 6 \, a^{3} b^{2} - a b^{4}\right )} x \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} - 64 \, {\left (a^{4} b \cosh \left (x\right )^{4} + 4 \, a^{4} b \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, a^{4} b \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, a^{4} b \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{4} b \sinh \left (x\right )^{4}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 8 \, {\left ({\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} \cosh \left (x\right )^{7} - 3 \, {\left (2 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - b^{5}\right )} \cosh \left (x\right )^{5} + 4 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 6 \, a^{3} b^{2} - a b^{4}\right )} x \cosh \left (x\right )^{3} + {\left (2 \, a^{5} + 3 \, a^{4} b - 2 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{64 \, {\left ({\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sinh \left (x\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{4}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 214, normalized size = 1.46 \begin {gather*} -\frac {a^{4} b \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 {{\left (3 \, a^{2} - a b\right )} x}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, x\right )} - 6 \, a b e^{\left (4 \, x\right )} - 8 \, a^{2} e^{\left (2 \, x\right )} + 12 \, a b 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^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 8 \, a e^{\left (2 \, x\right )} - 4 \, b e^{\left (2 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.67, size = 135, normalized size = 0.92 \begin {gather*} \frac {{\mathrm {e}}^{4\,x}}{64\,a+64\,b}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a-64\,b}+\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,a-b\right )}{16\,{\left (a-b\right )}^2}-\frac {{\mathrm {e}}^{2\,x}\,\left (2\,a+b\right )}{16\,{\left (a+b\right )}^2}-\frac {a^4\,b\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {a\,x\,\left (3\,a-b\right )}{8\,{\left (a-b\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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