Integrand size = 21, antiderivative size = 53 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {\cosh (c+d x)}{(a+b) d} \]
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Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3745, 331, 214} \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\cosh (c+d x)}{d (a+b)}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{d (a+b)^{3/2}} \]
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Rule 214
Rule 331
Rule 3745
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{d} \\ & = \frac {\cosh (c+d x)}{(a+b) d}-\frac {b \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\text {sech}(c+d x)\right )}{(a+b) d} \\ & = -\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {\cosh (c+d x)}{(a+b) d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.02 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {-i \sqrt {b} \left (\arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )+\sqrt {a+b} \cosh (c+d x)}{(a+b)^{3/2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(45)=90\).
Time = 0.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.96
method | result | size |
derivativedivides | \(\frac {-\frac {b \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right ) \sqrt {a b +b^{2}}}+\frac {4}{\left (4 a +4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(104\) |
default | \(\frac {-\frac {b \,\operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right ) \sqrt {a b +b^{2}}}+\frac {4}{\left (4 a +4 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(104\) |
risch | \(\frac {{\mathrm e}^{d x +c}}{2 d \left (a +b \right )}+\frac {{\mathrm e}^{-d x -c}}{2 d \left (a +b \right )}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 \left (a +b \right )^{2} d}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{2 \left (a +b \right )^{2} d}\) | \(135\) |
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Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (45) = 90\).
Time = 0.31 (sec) , antiderivative size = 666, normalized size of antiderivative = 12.57 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\left [\frac {\sqrt {\frac {b}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \log \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a + b\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{a + b}} + a + b}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right ) + \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}{2 \, {\left ({\left (a + b\right )} d \cosh \left (d x + c\right ) + {\left (a + b\right )} d \sinh \left (d x + c\right )\right )}}, -\frac {2 \, \sqrt {-\frac {b}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (a - 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - 3 \, b\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a + b}}}{2 \, b}\right ) - 2 \, \sqrt {-\frac {b}{a + b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left ({\left (a + b\right )} \cosh \left (d x + c\right ) + {\left (a + b\right )} \sinh \left (d x + c\right )\right )} \sqrt {-\frac {b}{a + b}}}{2 \, b}\right ) - \cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) - \sinh \left (d x + c\right )^{2} - 1}{2 \, {\left ({\left (a + b\right )} d \cosh \left (d x + c\right ) + {\left (a + b\right )} d \sinh \left (d x + c\right )\right )}}\right ] \]
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\[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\sinh {\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
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\[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
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Time = 2.69 (sec) , antiderivative size = 520, normalized size of antiderivative = 9.81 \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{c+d\,x}}{2\,d\,\left (a+b\right )}+\frac {{\mathrm {e}}^{-c-d\,x}}{2\,d\,\left (a+b\right )}-\frac {\sqrt {b}\,\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2\,{\left (a+b\right )}^3}}{2\,\sqrt {b}\,d\,\left (a+b\right )}\right )-2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a\,\sqrt {b}}{d\,{\left (a+b\right )}^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {4\,\left (2\,a^2\,b^{3/2}\,d+2\,a\,b^{5/2}\,d\right )}{\left (a+b\right )\,\sqrt {-d^2\,{\left (a+b\right )}^3}\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )+\frac {2\,a\,\sqrt {b}\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{d\,{\left (a+b\right )}^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}\right )\,\left (a^4\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+b^4\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+4\,a\,b^3\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+4\,a^3\,b\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+6\,a^2\,b^2\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}\right )}{4\,a\,b}\right )\right )}{2\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}} \]
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