Integrand size = 29, antiderivative size = 144 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 b^4 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {\coth (c+d x)}{a d}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac {\tanh (c+d x)}{a d} \]
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Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2977, 2702, 327, 213, 2700, 14, 2775, 12, 2739, 632, 210} \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 b^4 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}+\frac {b \text {arctanh}(\cosh (c+d x))}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a^2 d \left (a^2+b^2\right )}-\frac {b \text {sech}(c+d x)}{a^2 d}-\frac {\tanh (c+d x)}{a d}-\frac {\coth (c+d x)}{a d} \]
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Rule 12
Rule 14
Rule 210
Rule 213
Rule 327
Rule 632
Rule 2700
Rule 2702
Rule 2739
Rule 2775
Rule 2977
Rubi steps \begin{align*} \text {integral}& = -\int \left (\frac {b \text {csch}(c+d x) \text {sech}^2(c+d x)}{a^2}-\frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a}-\frac {b^2 \text {sech}^2(c+d x)}{a^2 (a+b \sinh (c+d x))}\right ) \, dx \\ & = \frac {\int \text {csch}^2(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {\text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2} \\ & = \frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \int \frac {b^2}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {i \text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,i \tanh (c+d x)\right )}{a d}-\frac {b \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{a^2 d} \\ & = -\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}+\frac {b^4 \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {i \text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,i \tanh (c+d x)\right )}{a d}-\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{a^2 d} \\ & = \frac {b \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {\coth (c+d x)}{a d}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac {\tanh (c+d x)}{a d}-\frac {\left (2 i b^4\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 )}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {b \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {\coth (c+d x)}{a d}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac {\tanh (c+d x)}{a d}+\frac {\left (4 i b^4\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 )}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {b \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 b^4 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {\coth (c+d x)}{a d}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac {\tanh (c+d x)}{a d} \\ \end{align*}
Time = 1.70 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4 b^4 \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{a^2 \left (-a^2-b^2\right )^{3/2}}+\frac {\coth \left (\frac {1}{2} (c+d x)\right )}{a}-\frac {2 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {2 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2+b^2}+\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{a}}{2 d} \]
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Time = 10.36 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {2 b^{4} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {-2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\) | \(139\) |
default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {2 b^{4} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {-2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\) | \(139\) |
risch | \(-\frac {2 \left ({\mathrm e}^{3 d x +3 c} a b +b^{2} {\mathrm e}^{2 d x +2 c}-{\mathrm e}^{d x +c} a b +2 a^{2}+b^{2}\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )}-\frac {b \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+\frac {b^{4} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}-\frac {b^{4} \ln \left ({\mathrm e}^{d x +c}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}+\frac {b \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}\) | \(261\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1040 vs. \(2 (141) = 282\).
Time = 0.33 (sec) , antiderivative size = 1040, normalized size of antiderivative = 7.22 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.44 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{4} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {2 \, {\left (a b e^{\left (-d x - c\right )} + b^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a b e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, a^{2} + b^{2}\right )}}{{\left (a^{3} + a b^{2} - {\left (a^{3} + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \]
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Time = 0.64 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {b^{4} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {b \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} - \frac {b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (a b e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b e^{\left (d x + c\right )} + 2 \, a^{2} + b^{2}\right )}}{{\left (a^{3} + a b^{2}\right )} {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}}{d} \]
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Time = 6.55 (sec) , antiderivative size = 768, normalized size of antiderivative = 5.33 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^4\,\ln \left (\frac {64\,b^8\,\sqrt {{\left (a^2+b^2\right )}^3}-96\,a\,b^{10}-384\,a^3\,b^8-512\,a^5\,b^6-288\,a^7\,b^4-64\,a^9\,b^2+288\,a^2\,b^9\,{\mathrm {e}}^{c+d\,x}+960\,a^4\,b^7\,{\mathrm {e}}^{c+d\,x}+1152\,a^6\,b^5\,{\mathrm {e}}^{c+d\,x}+608\,a^8\,b^3\,{\mathrm {e}}^{c+d\,x}+128\,a^{10}\,b\,{\mathrm {e}}^{c+d\,x}-64\,a\,b^7\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+32\,a^3\,b^5\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}}{a^3\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}\,\left (a^2+b^2\right )}-\frac {32\,b\,\left (-4\,{\mathrm {e}}^{c+d\,x}\,a^3+2\,a^2\,b-5\,{\mathrm {e}}^{c+d\,x}\,a\,b^2+2\,b^3\right )}{a^3\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^8+3\,d\,a^6\,b^2+3\,d\,a^4\,b^4+d\,a^2\,b^6}-\frac {\frac {2\,b^4\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d\,\left (a^2\,b^3+b^5\right )}-\frac {2\,b^4\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {2\,b^3\,\left (2\,a^2+b^2\right )}{a\,d\,\left (a^2\,b^3+b^5\right )}+\frac {2\,b^5\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a\,d\,\left (a^2\,b^3+b^5\right )}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {b^4\,\ln \left (\frac {96\,a\,b^{10}+64\,b^8\,\sqrt {{\left (a^2+b^2\right )}^3}+384\,a^3\,b^8+512\,a^5\,b^6+288\,a^7\,b^4+64\,a^9\,b^2-288\,a^2\,b^9\,{\mathrm {e}}^{c+d\,x}-960\,a^4\,b^7\,{\mathrm {e}}^{c+d\,x}-1152\,a^6\,b^5\,{\mathrm {e}}^{c+d\,x}-608\,a^8\,b^3\,{\mathrm {e}}^{c+d\,x}-128\,a^{10}\,b\,{\mathrm {e}}^{c+d\,x}-64\,a\,b^7\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+32\,a^3\,b^5\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}}{a^3\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}\,\left (a^2+b^2\right )}-\frac {32\,b\,\left (-4\,{\mathrm {e}}^{c+d\,x}\,a^3+2\,a^2\,b-5\,{\mathrm {e}}^{c+d\,x}\,a\,b^2+2\,b^3\right )}{a^3\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^8+3\,d\,a^6\,b^2+3\,d\,a^4\,b^4+d\,a^2\,b^6}-\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )}{a^2\,d}+\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )}{a^2\,d} \]
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