Integrand size = 17, antiderivative size = 69 \[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=\frac {b d \text {arctanh}(\cosh (x))}{c^2}-\frac {2 \left (a c^2+b d^2\right ) \text {arctanh}\left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {c^2+d^2}}\right )}{c^2 \sqrt {c^2+d^2}}-\frac {b \coth (x)}{c} \]
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Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4318, 3135, 3080, 3855, 2739, 632, 212} \[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=-\frac {2 \left (a c^2+b d^2\right ) \text {arctanh}\left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {c^2+d^2}}\right )}{c^2 \sqrt {c^2+d^2}}+\frac {b d \text {arctanh}(\cosh (x))}{c^2}-\frac {b \coth (x)}{c} \]
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Rule 212
Rule 632
Rule 2739
Rule 3080
Rule 3135
Rule 3855
Rule 4318
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\text {csch}^2(x) \left (-b-a \sinh ^2(x)\right )}{c+d \sinh (x)} \, dx \\ & = -\frac {b \coth (x)}{c}-\frac {i \int \frac {\text {csch}(x) (-i b d+i a c \sinh (x))}{c+d \sinh (x)} \, dx}{c} \\ & = -\frac {b \coth (x)}{c}-\frac {(b d) \int \text {csch}(x) \, dx}{c^2}+\left (a+\frac {b d^2}{c^2}\right ) \int \frac {1}{c+d \sinh (x)} \, dx \\ & = \frac {b d \text {arctanh}(\cosh (x))}{c^2}-\frac {b \coth (x)}{c}+\left (2 \left (a+\frac {b d^2}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x-c x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right ) \\ & = \frac {b d \text {arctanh}(\cosh (x))}{c^2}-\frac {b \coth (x)}{c}-\left (4 \left (a+\frac {b d^2}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (c^2+d^2\right )-x^2} \, dx,x,2 d-2 c \tanh \left (\frac {x}{2}\right )\right ) \\ & = \frac {b d \text {arctanh}(\cosh (x))}{c^2}-\frac {2 \left (a+\frac {b d^2}{c^2}\right ) \text {arctanh}\left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {c^2+d^2}}\right )}{\sqrt {c^2+d^2}}-\frac {b \coth (x)}{c} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=\frac {\text {csch}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \left (-b c \cosh (x)+\left (\frac {2 \left (a c^2+b d^2\right ) \arctan \left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {-c^2-d^2}}\right )}{\sqrt {-c^2-d^2}}+b d \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )\right ) \sinh (x)\right )}{2 c^2} \]
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Time = 0.72 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.25
method | result | size |
default | \(-\frac {b \tanh \left (\frac {x}{2}\right )}{2 c}-\frac {\left (-4 a \,c^{2}-4 b \,d^{2}\right ) \operatorname {arctanh}\left (\frac {2 c \tanh \left (\frac {x}{2}\right )-2 d}{2 \sqrt {c^{2}+d^{2}}}\right )}{2 c^{2} \sqrt {c^{2}+d^{2}}}-\frac {b}{2 c \tanh \left (\frac {x}{2}\right )}-\frac {b d \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{c^{2}}\) | \(86\) |
parts | \(\frac {2 a \,\operatorname {arctanh}\left (\frac {2 c \tanh \left (\frac {x}{2}\right )-2 d}{2 \sqrt {c^{2}+d^{2}}}\right )}{\sqrt {c^{2}+d^{2}}}+b \left (-\frac {\tanh \left (\frac {x}{2}\right )}{2 c}-\frac {1}{2 c \tanh \left (\frac {x}{2}\right )}-\frac {d \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{c^{2}}+\frac {2 d^{2} \operatorname {arctanh}\left (\frac {2 c \tanh \left (\frac {x}{2}\right )-2 d}{2 \sqrt {c^{2}+d^{2}}}\right )}{c^{2} \sqrt {c^{2}+d^{2}}}\right )\) | \(111\) |
risch | \(-\frac {2 b}{c \left ({\mathrm e}^{2 x}-1\right )}+\frac {b d \ln \left ({\mathrm e}^{x}+1\right )}{c^{2}}-\frac {b d \ln \left ({\mathrm e}^{x}-1\right )}{c^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}+d^{2}}\, c -c^{2}-d^{2}}{\sqrt {c^{2}+d^{2}}\, d}\right ) a}{\sqrt {c^{2}+d^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}+d^{2}}\, c -c^{2}-d^{2}}{\sqrt {c^{2}+d^{2}}\, d}\right ) b \,d^{2}}{\sqrt {c^{2}+d^{2}}\, c^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}+d^{2}}\, c +c^{2}+d^{2}}{\sqrt {c^{2}+d^{2}}\, d}\right ) a}{\sqrt {c^{2}+d^{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}+d^{2}}\, c +c^{2}+d^{2}}{\sqrt {c^{2}+d^{2}}\, d}\right ) b \,d^{2}}{\sqrt {c^{2}+d^{2}}\, c^{2}}\) | \(245\) |
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Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (65) = 130\).
Time = 0.48 (sec) , antiderivative size = 401, normalized size of antiderivative = 5.81 \[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=\frac {2 \, b c^{3} + 2 \, b c d^{2} + {\left (a c^{2} + b d^{2} - {\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a c^{2} + b d^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {c^{2} + d^{2}} \log \left (\frac {d^{2} \cosh \left (x\right )^{2} + d^{2} \sinh \left (x\right )^{2} + 2 \, c d \cosh \left (x\right ) + 2 \, c^{2} + d^{2} + 2 \, {\left (d^{2} \cosh \left (x\right ) + c d\right )} \sinh \left (x\right ) - 2 \, \sqrt {c^{2} + d^{2}} {\left (d \cosh \left (x\right ) + d \sinh \left (x\right ) + c\right )}}{d \cosh \left (x\right )^{2} + d \sinh \left (x\right )^{2} + 2 \, c \cosh \left (x\right ) + 2 \, {\left (d \cosh \left (x\right ) + c\right )} \sinh \left (x\right ) - d}\right ) + {\left (b c^{2} d + b d^{3} - {\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (b c^{2} d + b d^{3}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (b c^{2} d + b d^{3} - {\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (b c^{2} d + b d^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (b c^{2} d + b d^{3}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{c^{4} + c^{2} d^{2} - {\left (c^{4} + c^{2} d^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (c^{4} + c^{2} d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (c^{4} + c^{2} d^{2}\right )} \sinh \left (x\right )^{2}} \]
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\[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=\int \frac {a + b \operatorname {csch}^{2}{\left (x \right )}}{c + d \sinh {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (65) = 130\).
Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.29 \[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=b {\left (\frac {d^{2} \log \left (\frac {d e^{\left (-x\right )} - c - \sqrt {c^{2} + d^{2}}}{d e^{\left (-x\right )} - c + \sqrt {c^{2} + d^{2}}}\right )}{\sqrt {c^{2} + d^{2}} c^{2}} + \frac {d \log \left (e^{\left (-x\right )} + 1\right )}{c^{2}} - \frac {d \log \left (e^{\left (-x\right )} - 1\right )}{c^{2}} + \frac {2}{c e^{\left (-2 \, x\right )} - c}\right )} + \frac {a \log \left (\frac {d e^{\left (-x\right )} - c - \sqrt {c^{2} + d^{2}}}{d e^{\left (-x\right )} - c + \sqrt {c^{2} + d^{2}}}\right )}{\sqrt {c^{2} + d^{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.58 \[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=\frac {b d \log \left (e^{x} + 1\right )}{c^{2}} - \frac {b d \log \left ({\left | e^{x} - 1 \right |}\right )}{c^{2}} + \frac {{\left (a c^{2} + b d^{2}\right )} \log \left (\frac {{\left | 2 \, d e^{x} + 2 \, c - 2 \, \sqrt {c^{2} + d^{2}} \right |}}{{\left | 2 \, d e^{x} + 2 \, c + 2 \, \sqrt {c^{2} + d^{2}} \right |}}\right )}{\sqrt {c^{2} + d^{2}} c^{2}} - \frac {2 \, b}{c {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
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Time = 4.58 (sec) , antiderivative size = 613, normalized size of antiderivative = 8.88 \[ \int \frac {a+b \text {csch}^2(x)}{c+d \sinh (x)} \, dx=\frac {b\,d\,\ln \left ({\mathrm {e}}^x+1\right )}{c^2}-\frac {b\,d\,\ln \left ({\mathrm {e}}^x-1\right )}{c^2}-\frac {2\,b}{c\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {\ln \left (\frac {\left (a\,c^2+b\,d^2\right )\,\left (\frac {32\,\left (a^2\,c^4+2\,a\,b\,c^2\,d^2-4\,{\mathrm {e}}^x\,b^2\,c^3\,d+2\,b^2\,c^2\,d^2-3\,{\mathrm {e}}^x\,b^2\,c\,d^3+2\,b^2\,d^4\right )}{c^2\,d^4}-\frac {\left (a\,c^2+b\,d^2\right )\,\left (\frac {32\,c\,\left (4\,a\,c^3\,{\mathrm {e}}^x-2\,b\,d^3-2\,a\,c^2\,d+a\,c\,d^2\,{\mathrm {e}}^x+3\,b\,c\,d^2\,{\mathrm {e}}^x\right )}{d^5}+\frac {32\,\left (a\,c^2+b\,d^2\right )\,\left (-4\,{\mathrm {e}}^x\,c^3+3\,c^2\,d-3\,{\mathrm {e}}^x\,c\,d^2+2\,d^3\right )}{d^5\,\sqrt {c^2+d^2}}\right )}{c^2\,\sqrt {c^2+d^2}}\right )}{c^2\,\sqrt {c^2+d^2}}-\frac {32\,b\,\left (a\,c^2+b\,d^2\right )\,\left (2\,b\,d+a\,c\,{\mathrm {e}}^x-4\,b\,c\,{\mathrm {e}}^x\right )}{c^3\,d^3}\right )\,\left (a\,c^2+b\,d^2\right )\,\sqrt {c^2+d^2}}{c^4+c^2\,d^2}+\frac {\ln \left (-\frac {\left (a\,c^2+b\,d^2\right )\,\left (\frac {32\,\left (a^2\,c^4+2\,a\,b\,c^2\,d^2-4\,{\mathrm {e}}^x\,b^2\,c^3\,d+2\,b^2\,c^2\,d^2-3\,{\mathrm {e}}^x\,b^2\,c\,d^3+2\,b^2\,d^4\right )}{c^2\,d^4}+\frac {\left (a\,c^2+b\,d^2\right )\,\left (\frac {32\,c\,\left (4\,a\,c^3\,{\mathrm {e}}^x-2\,b\,d^3-2\,a\,c^2\,d+a\,c\,d^2\,{\mathrm {e}}^x+3\,b\,c\,d^2\,{\mathrm {e}}^x\right )}{d^5}-\frac {32\,\left (a\,c^2+b\,d^2\right )\,\left (-4\,{\mathrm {e}}^x\,c^3+3\,c^2\,d-3\,{\mathrm {e}}^x\,c\,d^2+2\,d^3\right )}{d^5\,\sqrt {c^2+d^2}}\right )}{c^2\,\sqrt {c^2+d^2}}\right )}{c^2\,\sqrt {c^2+d^2}}-\frac {32\,b\,\left (a\,c^2+b\,d^2\right )\,\left (2\,b\,d+a\,c\,{\mathrm {e}}^x-4\,b\,c\,{\mathrm {e}}^x\right )}{c^3\,d^3}\right )\,\left (a\,c^2+b\,d^2\right )\,\sqrt {c^2+d^2}}{c^4+c^2\,d^2} \]
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