Integrand size = 14, antiderivative size = 106 \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a d} \]
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Time = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3869} \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d} \]
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Rule 3869
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a d} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 b \sqrt {b+a \cosh (c+d x)} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a+b} \sqrt {a \cosh (c+d x)}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1+\text {sech}(c+d x))}{-a+b}} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a} \sqrt {a+b} d \sqrt {a \cosh (c+d x)} \sqrt {-\frac {b (-1+\text {sech}(c+d x))}{a+b}} \sqrt {a+b \text {sech}(c+d x)}} \]
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\[\int \frac {1}{\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}d x\]
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\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]
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