Optimal. Leaf size=94 \[ -\frac {b x}{a^2-b^2}+\frac {a \log (\cosh (x))}{a^2-b^2}-\frac {\left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {a^5 \log (a+b \tanh (x))}{b^4 \left (a^2-b^2\right )}+\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b} \]
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Rubi [A] time = 0.37, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3566, 3647, 3648, 3626, 3617, 31, 3475} \[ -\frac {b x}{a^2-b^2}-\frac {\left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {a^5 \log (a+b \tanh (x))}{b^4 \left (a^2-b^2\right )}+\frac {a \log (\cosh (x))}{a^2-b^2}+\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3566
Rule 3617
Rule 3626
Rule 3647
Rule 3648
Rubi steps
\begin {align*} \int \frac {\tanh ^5(x)}{a+b \tanh (x)} \, dx &=-\frac {\tanh ^3(x)}{3 b}-\frac {\int \frac {\tanh ^2(x) \left (-3 a-3 b \tanh (x)+3 a \tanh ^2(x)\right )}{a+b \tanh (x)} \, dx}{3 b}\\ &=\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b}-\frac {\int \frac {\tanh (x) \left (6 a^2-6 \left (a^2+b^2\right ) \tanh ^2(x)\right )}{a+b \tanh (x)} \, dx}{6 b^2}\\ &=-\frac {\left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b}-\frac {\int \frac {-6 a \left (a^2+b^2\right )-6 b^3 \tanh (x)+6 a \left (a^2+b^2\right ) \tanh ^2(x)}{a+b \tanh (x)} \, dx}{6 b^3}\\ &=-\frac {b x}{a^2-b^2}-\frac {\left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b}+\frac {a \int \tanh (x) \, dx}{a^2-b^2}+\frac {a^5 \int \frac {1-\tanh ^2(x)}{a+b \tanh (x)} \, dx}{b^3 \left (a^2-b^2\right )}\\ &=-\frac {b x}{a^2-b^2}+\frac {a \log (\cosh (x))}{a^2-b^2}-\frac {\left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b}+\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tanh (x)\right )}{b^4 \left (a^2-b^2\right )}\\ &=-\frac {b x}{a^2-b^2}+\frac {a \log (\cosh (x))}{a^2-b^2}+\frac {a^5 \log (a+b \tanh (x))}{b^4 \left (a^2-b^2\right )}-\frac {\left (a^2+b^2\right ) \tanh (x)}{b^3}+\frac {a \tanh ^2(x)}{2 b^2}-\frac {\tanh ^3(x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 105, normalized size = 1.12 \[ -\frac {-6 a^5 \log (a \cosh (x)+b \sinh (x))+6 a \left (a^4-b^4\right ) \log (\cosh (x))+b^2 \left (b^2-a^2\right ) \text {sech}^2(x) (2 b \tanh (x)-3 a)+2 b \left (3 a^4+a^2 b^2-4 b^4\right ) \tanh (x)+6 b^5 x}{6 b^4 (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 1296, normalized size = 13.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 142, normalized size = 1.51 \[ \frac {a^{5} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{2} b^{4} - b^{6}} - \frac {x}{a - b} - \frac {{\left (a^{3} + a b^{2}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{4}} + \frac {2 \, {\left (3 \, a^{2} b + 4 \, b^{3} + 3 \, {\left (a^{2} b - a b^{2} + 2 \, b^{3}\right )} e^{\left (4 \, x\right )} + 3 \, {\left (2 \, a^{2} b - a b^{2} + 2 \, b^{3}\right )} e^{\left (2 \, x\right )}\right )}}{3 \, b^{4} {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 96, normalized size = 1.02 \[ -\frac {\tanh ^{3}\relax (x )}{3 b}+\frac {a \left (\tanh ^{2}\relax (x )\right )}{2 b^{2}}-\frac {a^{2} \tanh \relax (x )}{b^{3}}-\frac {\tanh \relax (x )}{b}-\frac {\ln \left (\tanh \relax (x )-1\right )}{2 b +2 a}-\frac {\ln \left (1+\tanh \relax (x )\right )}{2 a -2 b}+\frac {a^{5} \ln \left (a +b \tanh \relax (x )\right )}{b^{4} \left (a +b \right ) \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 150, normalized size = 1.60 \[ \frac {a^{5} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{2} b^{4} - b^{6}} - \frac {2 \, {\left (3 \, a^{2} + 4 \, b^{2} + 3 \, {\left (2 \, a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \, {\left (3 \, b^{3} e^{\left (-2 \, x\right )} + 3 \, b^{3} e^{\left (-4 \, x\right )} + b^{3} e^{\left (-6 \, x\right )} + b^{3}\right )}} + \frac {x}{a + b} - \frac {{\left (a^{3} + a b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 85, normalized size = 0.90 \[ \frac {x}{a+b}-\frac {{\mathrm {tanh}\relax (x)}^3}{3\,b}-\frac {a\,\ln \left (\mathrm {tanh}\relax (x)+1\right )}{a^2-b^2}+\frac {a\,{\mathrm {tanh}\relax (x)}^2}{2\,b^2}-\frac {\mathrm {tanh}\relax (x)\,\left (a^2+b^2\right )}{b^3}+\frac {a^5\,\ln \left (a+b\,\mathrm {tanh}\relax (x)\right )}{b^4\,\left (a^2-b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.57, size = 546, normalized size = 5.81 \[ \begin {cases} \tilde {\infty } \left (x - \frac {\tanh ^{3}{\relax (x )}}{3} - \tanh {\relax (x )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {27 x \tanh {\relax (x )}}{6 b \tanh {\relax (x )} - 6 b} - \frac {27 x}{6 b \tanh {\relax (x )} - 6 b} - \frac {12 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{6 b \tanh {\relax (x )} - 6 b} + \frac {12 \log {\left (\tanh {\relax (x )} + 1 \right )}}{6 b \tanh {\relax (x )} - 6 b} - \frac {2 \tanh ^{4}{\relax (x )}}{6 b \tanh {\relax (x )} - 6 b} - \frac {\tanh ^{3}{\relax (x )}}{6 b \tanh {\relax (x )} - 6 b} - \frac {9 \tanh ^{2}{\relax (x )}}{6 b \tanh {\relax (x )} - 6 b} + \frac {15}{6 b \tanh {\relax (x )} - 6 b} & \text {for}\: a = - b \\\frac {3 x \tanh {\relax (x )}}{6 b \tanh {\relax (x )} + 6 b} + \frac {3 x}{6 b \tanh {\relax (x )} + 6 b} + \frac {12 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{6 b \tanh {\relax (x )} + 6 b} + \frac {12 \log {\left (\tanh {\relax (x )} + 1 \right )}}{6 b \tanh {\relax (x )} + 6 b} - \frac {2 \tanh ^{4}{\relax (x )}}{6 b \tanh {\relax (x )} + 6 b} + \frac {\tanh ^{3}{\relax (x )}}{6 b \tanh {\relax (x )} + 6 b} - \frac {9 \tanh ^{2}{\relax (x )}}{6 b \tanh {\relax (x )} + 6 b} + \frac {15}{6 b \tanh {\relax (x )} + 6 b} & \text {for}\: a = b \\\frac {x - \log {\left (\tanh {\relax (x )} + 1 \right )} - \frac {\tanh ^{4}{\relax (x )}}{4} - \frac {\tanh ^{2}{\relax (x )}}{2}}{a} & \text {for}\: b = 0 \\\frac {6 a^{5} \log {\left (\frac {a}{b} + \tanh {\relax (x )} \right )}}{6 a^{2} b^{4} - 6 b^{6}} - \frac {6 a^{4} b \tanh {\relax (x )}}{6 a^{2} b^{4} - 6 b^{6}} + \frac {3 a^{3} b^{2} \tanh ^{2}{\relax (x )}}{6 a^{2} b^{4} - 6 b^{6}} - \frac {2 a^{2} b^{3} \tanh ^{3}{\relax (x )}}{6 a^{2} b^{4} - 6 b^{6}} + \frac {6 a b^{4} x}{6 a^{2} b^{4} - 6 b^{6}} - \frac {6 a b^{4} \log {\left (\tanh {\relax (x )} + 1 \right )}}{6 a^{2} b^{4} - 6 b^{6}} - \frac {3 a b^{4} \tanh ^{2}{\relax (x )}}{6 a^{2} b^{4} - 6 b^{6}} - \frac {6 b^{5} x}{6 a^{2} b^{4} - 6 b^{6}} + \frac {2 b^{5} \tanh ^{3}{\relax (x )}}{6 a^{2} b^{4} - 6 b^{6}} + \frac {6 b^{5} \tanh {\relax (x )}}{6 a^{2} b^{4} - 6 b^{6}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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