Integrand size = 14, antiderivative size = 50 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\frac {x}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a \sqrt {a-b} d} \]
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Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4212, 3260, 211} \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\frac {x}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a d \sqrt {a-b}} \]
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Rule 211
Rule 3260
Rule 4212
Rubi steps \begin{align*} \text {integral}& = \frac {x}{a}+\frac {b \int \frac {1}{-b-a \sinh ^2(c+d x)} \, dx}{a} \\ & = \frac {x}{a}+\frac {b \text {Subst}\left (\int \frac {1}{-b-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{a d} \\ & = \frac {x}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{a \sqrt {a-b} d} \\ \end{align*}
Time = 2.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.04 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\frac {\frac {c}{d}+x-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a-b} d}}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(42)=84\).
Time = 0.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.48
method | result | size |
risch | \(\frac {x}{a}+\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-b \left (a -b \right )}-2 b}{a}\right )}{2 \left (a -b \right ) d a}-\frac {\sqrt {-b \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {-a +2 \sqrt {-b \left (a -b \right )}+2 b}{a}\right )}{2 \left (a -b \right ) d a}\) | \(124\) |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b^{2} \left (\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{a}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(208\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {2 b^{2} \left (\frac {\left (\sqrt {a \left (a -b \right )}+a \right ) \arctan \left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {\left (\sqrt {a \left (a -b \right )}-a \right ) \operatorname {arctanh}\left (\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 b \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{a}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(208\) |
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Time = 0.28 (sec) , antiderivative size = 457, normalized size of antiderivative = 9.14 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\left [\frac {2 \, d x + \sqrt {-\frac {b}{a - b}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} - a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} - {\left (a^{2} - 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left ({\left (a^{2} - a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} - a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} - a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 3 \, a b - 2 \, b^{2}\right )} \sqrt {-\frac {b}{a - b}}}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - {\left (a - 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right )}{2 \, a d}, \frac {d x - \sqrt {\frac {b}{a - b}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} - a + 2 \, b\right )} \sqrt {\frac {b}{a - b}}}{2 \, b}\right )}{a d}\right ] \]
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\[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\int \frac {1}{a + b \operatorname {csch}^{2}{\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\int { \frac {1}{b \operatorname {csch}\left (d x + c\right )^{2} + a} \,d x } \]
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Time = 2.79 (sec) , antiderivative size = 471, normalized size of antiderivative = 9.42 \[ \int \frac {1}{a+b \text {csch}^2(c+d x)} \, dx=\frac {x}{a}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\left (a^5\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-a^4\,b\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )\,\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {2\,\left (a^2-8\,a\,b+8\,b^2\right )\,\left (8\,b^{5/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-8\,a\,b^{3/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}+a^2\,\sqrt {b}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )}{a^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^3\,d^2-a^2\,b\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a-4\,b\right )\,\left (4\,d\,a^3\,b-12\,d\,a^2\,b^2+8\,d\,a\,b^3\right )}{a^7\,\left (a-b\right )\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}}\right )+\frac {2\,\left (2\,a\,b^{3/2}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}-a^2\,\sqrt {b}\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\right )\,\left (a^2-8\,a\,b+8\,b^2\right )}{a^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^3\,d^2-a^2\,b\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a^2\,b^2\,d-2\,a^3\,b\,d\right )\,\left (2\,a-4\,b\right )}{a^7\,\left (a-b\right )\,\sqrt {a^3\,d^2-a^2\,b\,d^2}\,\sqrt {a^2\,d^2\,\left (a-b\right )}}\right )}{4\,b}\right )}{\sqrt {a^3\,d^2-a^2\,b\,d^2}} \]
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