Integrand size = 21, antiderivative size = 90 \[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 a^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}-\frac {a \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]
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Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2806, 3852, 8, 2686, 2739, 632, 210} \[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 a^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {a \tanh (c+d x)}{d \left (a^2+b^2\right )}-\frac {b \text {sech}(c+d x)}{d \left (a^2+b^2\right )} \]
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Rule 8
Rule 210
Rule 632
Rule 2686
Rule 2739
Rule 2806
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {a \int \text {sech}^2(c+d x) \, dx}{a^2+b^2}+\frac {a^2 \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {b \int \text {sech}(c+d x) \tanh (c+d x) \, dx}{a^2+b^2} \\ & = -\frac {(i a) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {\left (2 i a^2\right ) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2+b^2\right ) d}-\frac {b \text {Subst}(\int 1 \, dx,x,\text {sech}(c+d x))}{\left (a^2+b^2\right ) d} \\ & = -\frac {b \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}-\frac {a \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\left (4 i a^2\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {2 a^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}-\frac {a \tanh (c+d x)}{\left (a^2+b^2\right ) d} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.18 \[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {-b \sqrt {-a^2-b^2} \text {sech}(c+d x)+a \left (2 a \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )-\sqrt {-a^2-b^2} \tanh (c+d x)\right )}{\left (-a^2-b^2\right )^{3/2} d} \]
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Time = 0.91 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {-2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {8 a^{2} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{2}+4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}}{d}\) | \(103\) |
default | \(\frac {\frac {-2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {8 a^{2} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{2}+4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}}{d}\) | \(103\) |
risch | \(\frac {-2 b \,{\mathrm e}^{d x +c}+2 a}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (87) = 174\).
Time = 0.26 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.90 \[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a^{3} + 2 \, a b^{2} + {\left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d \sinh \left (d x + c\right )^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} \]
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\[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.28 \[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^{2} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} d} - \frac {2 \, {\left (b e^{\left (-d x - c\right )} + a\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} \]
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Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.20 \[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {a^{2} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b e^{\left (d x + c\right )} - a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \]
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Time = 1.51 (sec) , antiderivative size = 422, normalized size of antiderivative = 4.69 \[ \int \frac {\tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2\,a}{d\,\left (a^2+b^2\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^2}{b^2\,d\,\sqrt {a^4}\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (a^3\,d\,\sqrt {a^4}+a\,b^2\,d\,\sqrt {a^4}\right )}{a\,b^2\,\sqrt {-d^2\,{\left (a^2+b^2\right )}^3}\,\left (a^2+b^2\right )\,\sqrt {-a^6\,d^2-3\,a^4\,b^2\,d^2-3\,a^2\,b^4\,d^2-b^6\,d^2}}\right )-\frac {2\,\left (b^3\,d\,\sqrt {a^4}+a^2\,b\,d\,\sqrt {a^4}\right )}{a\,b^2\,\sqrt {-d^2\,{\left (a^2+b^2\right )}^3}\,\left (a^2+b^2\right )\,\sqrt {-a^6\,d^2-3\,a^4\,b^2\,d^2-3\,a^2\,b^4\,d^2-b^6\,d^2}}\right )\,\left (\frac {b^3\,\sqrt {-a^6\,d^2-3\,a^4\,b^2\,d^2-3\,a^2\,b^4\,d^2-b^6\,d^2}}{2}+\frac {a^2\,b\,\sqrt {-a^6\,d^2-3\,a^4\,b^2\,d^2-3\,a^2\,b^4\,d^2-b^6\,d^2}}{2}\right )\right )\,\sqrt {a^4}}{\sqrt {-a^6\,d^2-3\,a^4\,b^2\,d^2-3\,a^2\,b^4\,d^2-b^6\,d^2}} \]
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