Integrand size = 23, antiderivative size = 75 \[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a+2 b) \tanh (c+d x)}{b^2 d}+\frac {\tanh ^3(c+d x)}{3 b d} \]
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Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3756, 398, 211} \[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a+2 b) \tanh (c+d x)}{b^2 d}+\frac {\tanh ^3(c+d x)}{3 b d} \]
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Rule 211
Rule 398
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a+2 b}{b^2}+\frac {x^2}{b}+\frac {a^2+2 a b+b^2}{b^2 \left (a+b x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {(a+2 b) \tanh (c+d x)}{b^2 d}+\frac {\tanh ^3(c+d x)}{3 b d}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{b^2 d} \\ & = \frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a+2 b) \tanh (c+d x)}{b^2 d}+\frac {\tanh ^3(c+d x)}{3 b d} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {(a+b)^2 \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {\left (3 a+5 b+b \text {sech}^2(c+d x)\right ) \tanh (c+d x)}{3 b^2 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(251\) vs. \(2(65)=130\).
Time = 115.36 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.36
method | result | size |
derivativedivides | \(\frac {\frac {2 a \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\left (a -\sqrt {\left (a +b \right ) b}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}+\frac {\left (-a -\sqrt {\left (a +b \right ) b}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{b^{2}}+\frac {2 \left (-a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-2 a -\frac {8 b}{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (-a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{3}}}{d}\) | \(252\) |
default | \(\frac {\frac {2 a \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\left (a -\sqrt {\left (a +b \right ) b}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}+\frac {\left (-a -\sqrt {\left (a +b \right ) b}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{b^{2}}+\frac {2 \left (-a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+2 \left (-2 a -\frac {8 b}{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2 \left (-a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )^{3}}}{d}\) | \(252\) |
risch | \(\frac {2 a \,{\mathrm e}^{4 d x +4 c}+2 b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} a +8 b \,{\mathrm e}^{2 d x +2 c}+2 a +\frac {10 b}{3}}{b^{2} d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{\sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{\sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}\) | \(433\) |
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Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (65) = 130\).
Time = 0.30 (sec) , antiderivative size = 2032, normalized size of antiderivative = 27.09 \[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\operatorname {sech}^{6}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (65) = 130\).
Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.87 \[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {2 \, {\left (6 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, a + 5 \, b\right )}}{3 \, {\left (3 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + b^{2} e^{\left (-6 \, d x - 6 \, c\right )} + b^{2}\right )} d} - \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} b^{2} d} \]
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\[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{6}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
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Time = 2.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.36 \[ \int \frac {\text {sech}^6(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {4}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {8}{3\,b\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {2\,\left (a+b\right )}{b^2\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\ln \left (-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{b^2}-\frac {2\,\left (a+b\right )\,\left (a+b+a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{5/2}}\right )\,{\left (a+b\right )}^2}{2\,\sqrt {-a}\,b^{5/2}\,d}-\frac {\ln \left (\frac {2\,\left (a+b\right )\,\left (a+b+a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{5/2}}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{b^2}\right )\,{\left (a+b\right )}^2}{2\,\sqrt {-a}\,b^{5/2}\,d} \]
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