Optimal. Leaf size=50 \[ \frac {2 b}{a^3 (a \cosh (x)+b)}+\frac {\log (a \cosh (x)+b)}{a^3}+\frac {a^2-b^2}{2 a^3 (a \cosh (x)+b)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 50, 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 = {4392, 2668, 697} \[ \frac {a^2-b^2}{2 a^3 (a \cosh (x)+b)^2}+\frac {2 b}{a^3 (a \cosh (x)+b)}+\frac {\log (a \cosh (x)+b)}{a^3} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rule 4392
Rubi steps
\begin {align*} \int \frac {1}{(a \coth (x)+b \text {csch}(x))^3} \, dx &=-\left (i \int \frac {\sinh ^3(x)}{(i b+i a \cosh (x))^3} \, dx\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {-a^2-x^2}{(i b+x)^3} \, dx,x,i a \cosh (x)\right )}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {-a^2+b^2}{(i b+x)^3}+\frac {2 i b}{(i b+x)^2}-\frac {1}{i b+x}\right ) \, dx,x,i a \cosh (x)\right )}{a^3}\\ &=\frac {a^2-b^2}{2 a^3 (b+a \cosh (x))^2}+\frac {2 b}{a^3 (b+a \cosh (x))}+\frac {\log (b+a \cosh (x))}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 77, normalized size = 1.54 \[ \frac {a^2 \cosh (2 x) \log (a \cosh (x)+b)+a^2 \log (a \cosh (x)+b)+a^2+2 b^2 \log (a \cosh (x)+b)+4 a b \cosh (x) (\log (a \cosh (x)+b)+1)+3 b^2}{2 a^3 (a \cosh (x)+b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 521, normalized size = 10.42 \[ -\frac {a^{2} x \cosh \relax (x)^{4} + a^{2} x \sinh \relax (x)^{4} + 4 \, {\left (a b x - a b\right )} \cosh \relax (x)^{3} + 4 \, {\left (a^{2} x \cosh \relax (x) + a b x - a b\right )} \sinh \relax (x)^{3} + a^{2} x - 2 \, {\left (a^{2} + 3 \, b^{2} - {\left (a^{2} + 2 \, b^{2}\right )} x\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, a^{2} x \cosh \relax (x)^{2} - a^{2} - 3 \, b^{2} + {\left (a^{2} + 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 (a^{2} \cosh \relax (x)^{4} + a^{2} \sinh \relax (x)^{4} + 4 \, a b \cosh \relax (x)^{3} + 4 \, {\left (a^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x)^{3} + 4 \, a b \cosh \relax (x) + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, a^{2} \cosh \relax (x)^{2} + 6 \, a b \cosh \relax (x) + a^{2} + 2 \, b^{2}\right )} \sinh \relax (x)^{2} + a^{2} + 4 \, {\left (a^{2} \cosh \relax (x)^{3} + 3 \, a b \cosh \relax (x)^{2} + a b + {\left (a^{2} + 2 \, b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (a \cosh \relax (x) + b\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 4 \, {\left (a^{2} x \cosh \relax (x)^{3} + a b x + 3 \, {\left (a b x - a b\right )} \cosh \relax (x)^{2} - a b - {\left (a^{2} + 3 \, b^{2} - {\left (a^{2} + 2 \, b^{2}\right )} x\right )} \cosh \relax (x)\right )} \sinh \relax (x)}{a^{5} \cosh \relax (x)^{4} + a^{5} \sinh \relax (x)^{4} + 4 \, a^{4} b \cosh \relax (x)^{3} + 4 \, a^{4} b \cosh \relax (x) + a^{5} + 4 \, {\left (a^{5} \cosh \relax (x) + a^{4} b\right )} \sinh \relax (x)^{3} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, a^{5} \cosh \relax (x)^{2} + 6 \, a^{4} b \cosh \relax (x) + a^{5} + 2 \, a^{3} b^{2}\right )} \sinh \relax (x)^{2} + 4 \, {\left (a^{5} \cosh \relax (x)^{3} + 3 \, a^{4} b \cosh \relax (x)^{2} + a^{4} b + {\left (a^{5} + 2 \, a^{3} b^{2}\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 = 0.15, size = 66, normalized size = 1.32 \[ \frac {\log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{3}} - \frac {3 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, b {\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a}{2 \, {\left (a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 144, normalized size = 2.88 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{3}}-\frac {2}{a^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )}{a^{3}}+\frac {2}{\left (a -b \right ) \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )^{2}}+\frac {2 b}{a \left (a -b \right ) \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 111, normalized size = 2.22 \[ \frac {2 \, {\left (2 \, a b e^{\left (-x\right )} + 2 \, a b e^{\left (-3 \, x\right )} + {\left (a^{2} + 3 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )}}{4 \, a^{4} b e^{\left (-x\right )} + 4 \, a^{4} b e^{\left (-3 \, x\right )} + a^{5} e^{\left (-4 \, x\right )} + a^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{a^{3}} + \frac {\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{3}} \]
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 \[ \int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\relax (x)}+a\,\mathrm {coth}\relax (x)\right )}^3} \,d x \]
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 \[ \int \frac {1}{\left (a \coth {\relax (x )} + b \operatorname {csch}{\relax (x )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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