Integrand size = 12, antiderivative size = 44 \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=-\frac {2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d} \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2739, 632, 210} \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=-\frac {2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{d \sqrt {a^2+b^2}} \]
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Rule 210
Rule 632
Rule 2739
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 i) \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 )}{d} \\ & = \frac {(4 i) \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 )}{d} \\ & = -\frac {2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.18 \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=\frac {2 \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d} \]
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Time = 0.78 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {2 \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \sqrt {a^{2}+b^{2}}}\) | \(43\) |
default | \(\frac {2 \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \sqrt {a^{2}+b^{2}}}\) | \(43\) |
risch | \(\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, d}-\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, d}\) | \(111\) |
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (41) = 82\).
Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.68 \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=\frac {\log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right )}{\sqrt {a^{2} + b^{2}} d} \]
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Result contains complex when optimal does not.
Time = 3.95 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.52 \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=\begin {cases} \frac {\tilde {\infty } x}{\sinh {\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )}}{b d} & \text {for}\: a = 0 \\\frac {x}{a + b \sinh {\left (c \right )}} & \text {for}\: d = 0 \\\frac {2 i}{b d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - i b d} & \text {for}\: a = - i b \\- \frac {2 i}{b d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + i b d} & \text {for}\: a = i b \\- \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{d \sqrt {a^{2} + b^{2}}} + \frac {\log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{d \sqrt {a^{2} + b^{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.52 \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=\frac {\log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} d} \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.52 \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=\frac {\log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} d} \]
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Time = 1.51 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.25 \[ \int \frac {1}{a+b \sinh (c+d x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {a\,d+b\,d\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{\sqrt {-a^2\,d^2-b^2\,d^2}}\right )}{\sqrt {-a^2\,d^2-b^2\,d^2}} \]
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