Integrand size = 8, antiderivative size = 43 \[ \int \tanh ^6(a+b x) \, dx=x-\frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh ^5(a+b x)}{5 b} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8} \[ \int \tanh ^6(a+b x) \, dx=-\frac {\tanh ^5(a+b x)}{5 b}-\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh (a+b x)}{b}+x \]
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Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = -\frac {\tanh ^5(a+b x)}{5 b}+\int \tanh ^4(a+b x) \, dx \\ & = -\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh ^5(a+b x)}{5 b}+\int \tanh ^2(a+b x) \, dx \\ & = -\frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh ^5(a+b x)}{5 b}+\int 1 \, dx \\ & = x-\frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh ^5(a+b x)}{5 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23 \[ \int \tanh ^6(a+b x) \, dx=\frac {\text {arctanh}(\tanh (a+b x))}{b}-\frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}-\frac {\tanh ^5(a+b x)}{5 b} \]
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Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91
method | result | size |
parallelrisch | \(-\frac {3 \tanh \left (b x +a \right )^{5}+5 \tanh \left (b x +a \right )^{3}-15 b x +15 \tanh \left (b x +a \right )}{15 b}\) | \(39\) |
derivativedivides | \(\frac {-\frac {\tanh \left (b x +a \right )^{5}}{5}-\frac {\tanh \left (b x +a \right )^{3}}{3}-\tanh \left (b x +a \right )-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (b x +a \right )\right )}{2}}{b}\) | \(56\) |
default | \(\frac {-\frac {\tanh \left (b x +a \right )^{5}}{5}-\frac {\tanh \left (b x +a \right )^{3}}{3}-\tanh \left (b x +a \right )-\frac {\ln \left (-1+\tanh \left (b x +a \right )\right )}{2}+\frac {\ln \left (1+\tanh \left (b x +a \right )\right )}{2}}{b}\) | \(56\) |
risch | \(x +\frac {6 \,{\mathrm e}^{8 b x +8 a}+12 \,{\mathrm e}^{6 b x +6 a}+\frac {56 \,{\mathrm e}^{4 b x +4 a}}{3}+\frac {28 \,{\mathrm e}^{2 b x +2 a}}{3}+\frac {46}{15}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{5}}\) | \(67\) |
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (39) = 78\).
Time = 0.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 5.91 \[ \int \tanh ^6(a+b x) \, dx=\frac {{\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{5} + 5 \, {\left (15 \, b x + 23\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 23 \, \sinh \left (b x + a\right )^{5} + 5 \, {\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{3} - 5 \, {\left (46 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right )^{3} + 5 \, {\left (2 \, {\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (15 \, b x + 23\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 10 \, {\left (15 \, b x + 23\right )} \cosh \left (b x + a\right ) - 5 \, {\left (23 \, \cosh \left (b x + a\right )^{4} + 15 \, \cosh \left (b x + a\right )^{2} + 10\right )} \sinh \left (b x + a\right )}{15 \, {\left (b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 5 \, b \cosh \left (b x + a\right )^{3} + 5 \, {\left (2 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 10 \, b \cosh \left (b x + a\right )\right )}} \]
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Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \tanh ^6(a+b x) \, dx=\begin {cases} x - \frac {\tanh ^{5}{\left (a + b x \right )}}{5 b} - \frac {\tanh ^{3}{\left (a + b x \right )}}{3 b} - \frac {\tanh {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \tanh ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (39) = 78\).
Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.67 \[ \int \tanh ^6(a+b x) \, dx=x + \frac {a}{b} - \frac {2 \, {\left (70 \, e^{\left (-2 \, b x - 2 \, a\right )} + 140 \, e^{\left (-4 \, b x - 4 \, a\right )} + 90 \, e^{\left (-6 \, b x - 6 \, a\right )} + 45 \, e^{\left (-8 \, b x - 8 \, a\right )} + 23\right )}}{15 \, b {\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} + 10 \, e^{\left (-6 \, b x - 6 \, a\right )} + 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-10 \, b x - 10 \, a\right )} + 1\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.72 \[ \int \tanh ^6(a+b x) \, dx=\frac {15 \, b x + 15 \, a + \frac {2 \, {\left (45 \, e^{\left (8 \, b x + 8 \, a\right )} + 90 \, e^{\left (6 \, b x + 6 \, a\right )} + 140 \, e^{\left (4 \, b x + 4 \, a\right )} + 70 \, e^{\left (2 \, b x + 2 \, a\right )} + 23\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{5}}}{15 \, b} \]
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Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \tanh ^6(a+b x) \, dx=x-\frac {\frac {{\mathrm {tanh}\left (a+b\,x\right )}^5}{5}+\frac {{\mathrm {tanh}\left (a+b\,x\right )}^3}{3}+\mathrm {tanh}\left (a+b\,x\right )}{b} \]
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