Optimal. Leaf size=48 \[ -\frac {a^2+b^2}{2 b^3 (a+b \sinh (x))^2}+\frac {2 a}{b^3 (a+b \sinh (x))}+\frac {\log (a+b \sinh (x))}{b^3} \]
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Rubi [A] time = 0.08, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4391, 2668, 697} \[ -\frac {a^2+b^2}{2 b^3 (a+b \sinh (x))^2}+\frac {2 a}{b^3 (a+b \sinh (x))}+\frac {\log (a+b \sinh (x))}{b^3} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rule 4391
Rubi steps
\begin {align*} \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^3} \, dx &=\int \frac {\cosh ^3(x)}{(a+b \sinh (x))^3} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {-b^2-x^2}{(a+x)^3} \, dx,x,b \sinh (x)\right )}{b^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{-a-x}+\frac {-a^2-b^2}{(a+x)^3}+\frac {2 a}{(a+x)^2}\right ) \, dx,x,b \sinh (x)\right )}{b^3}\\ &=\frac {\log (a+b \sinh (x))}{b^3}-\frac {a^2+b^2}{2 b^3 (a+b \sinh (x))^2}+\frac {2 a}{b^3 (a+b \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 42, normalized size = 0.88 \[ -\frac {\frac {-3 a^2-4 a b \sinh (x)+b^2}{2 (a+b \sinh (x))^2}-\log (a+b \sinh (x))}{b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 543, normalized size = 11.31 \[ -\frac {b^{2} x \cosh \relax (x)^{4} + b^{2} x \sinh \relax (x)^{4} + 4 \, {\left (a b x - a b\right )} \cosh \relax (x)^{3} + 4 \, {\left (b^{2} x \cosh \relax (x) + a b x - a b\right )} \sinh \relax (x)^{3} + b^{2} x - 2 \, {\left (3 \, a^{2} - b^{2} - {\left (2 \, a^{2} - b^{2}\right )} x\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} x \cosh \relax (x)^{2} - 3 \, a^{2} + b^{2} + {\left (2 \, a^{2} - b^{2}\right )} x + 6 \, {\left (a b x - a b\right )} \cosh \relax (x)\right )} \sinh \relax (x)^{2} - 4 \, {\left (a b x - a b\right )} \cosh \relax (x) - {\left (b^{2} \cosh \relax (x)^{4} + b^{2} \sinh \relax (x)^{4} + 4 \, a b \cosh \relax (x)^{3} + 4 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x)^{3} - 4 \, a b \cosh \relax (x) + 2 \, {\left (2 \, a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} + 6 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2}\right )} \sinh \relax (x)^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \relax (x)^{3} + 3 \, a b \cosh \relax (x)^{2} - a b + {\left (2 \, a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 4 \, {\left (b^{2} x \cosh \relax (x)^{3} - a b x + 3 \, {\left (a b x - a b\right )} \cosh \relax (x)^{2} + a b - {\left (3 \, a^{2} - b^{2} - {\left (2 \, a^{2} - b^{2}\right )} x\right )} \cosh \relax (x)\right )} \sinh \relax (x)}{b^{5} \cosh \relax (x)^{4} + b^{5} \sinh \relax (x)^{4} + 4 \, a b^{4} \cosh \relax (x)^{3} - 4 \, a b^{4} \cosh \relax (x) + b^{5} + 4 \, {\left (b^{5} \cosh \relax (x) + a b^{4}\right )} \sinh \relax (x)^{3} + 2 \, {\left (2 \, a^{2} b^{3} - b^{5}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{5} \cosh \relax (x)^{2} + 6 \, a b^{4} \cosh \relax (x) + 2 \, a^{2} b^{3} - b^{5}\right )} \sinh \relax (x)^{2} + 4 \, {\left (b^{5} \cosh \relax (x)^{3} + 3 \, a b^{4} \cosh \relax (x)^{2} - a b^{4} + {\left (2 \, a^{2} b^{3} - b^{5}\right )} \cosh \relax (x)\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 75, normalized size = 1.56 \[ \frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{3}} - \frac {3 \, b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 4 \, a {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, b}{2 \, {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 241, normalized size = 5.02 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{3}}+\frac {2 a \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{b^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )^{2}}-\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )^{2} a}-\frac {6 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{b \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )^{2}}+\frac {2 b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )^{2} a^{2}}-\frac {2 a \tanh \left (\frac {x}{2}\right )}{b^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )^{2}}+\frac {2 \tanh \left (\frac {x}{2}\right )}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )^{2} a}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 117, normalized size = 2.44 \[ \frac {2 \, {\left (2 \, a b e^{\left (-x\right )} - 2 \, a b e^{\left (-3 \, x\right )} + {\left (3 \, a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )}\right )}}{4 \, a b^{4} e^{\left (-x\right )} - 4 \, a b^{4} e^{\left (-3 \, x\right )} + b^{5} e^{\left (-4 \, x\right )} + b^{5} + 2 \, {\left (2 \, a^{2} b^{3} - b^{5}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{b^{3}} + \frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (b\,\mathrm {tanh}\relax (x)+\frac {a}{\mathrm {cosh}\relax (x)}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.89, size = 651, normalized size = 13.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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