Integrand size = 21, antiderivative size = 51 \[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{d} \]
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Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3970, 52, 65, 213} \[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{d} \]
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Rule 52
Rule 65
Rule 213
Rule 3970
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sqrt {a+x}}{x} \, dx,x,b \text {sech}(c+d x)\right )}{d} \\ & = -\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{d}-\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \text {sech}(c+d x)\right )}{d} \\ & = -\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{d}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d} \\ & = \frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {2 \sqrt {a+b \text {sech}(c+d x)}}{d} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.98 \[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=-\frac {-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \text {sech}(c+d x)}}{d} \]
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Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {2 \sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{d}\) | \(43\) |
default | \(-\frac {2 \sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{d}\) | \(43\) |
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (43) = 86\).
Time = 0.65 (sec) , antiderivative size = 605, normalized size of antiderivative = 11.86 \[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=\left [\frac {\sqrt {a} \log \left (-\frac {2 \, a^{2} \cosh \left (d x + c\right )^{4} + 2 \, a^{2} \sinh \left (d x + c\right )^{4} + 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (2 \, a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )^{3} + 4 \, a b \cosh \left (d x + c\right ) + {\left (4 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (12 \, a^{2} \cosh \left (d x + c\right )^{2} + 12 \, a b \cosh \left (d x + c\right ) + 4 \, a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, {\left (a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} + b \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right )^{3} + 2 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} + 3 \, b \cosh \left (d x + c\right ) + 2 \, a\right )} \sinh \left (d x + c\right )^{2} + b \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {a} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}} + 2 \, {\left (4 \, a^{2} \cosh \left (d x + c\right )^{3} + 6 \, a b \cosh \left (d x + c\right )^{2} + 2 \, a b + {\left (4 \, a^{2} + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 4 \, \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}}}{2 \, d}, -\frac {\sqrt {-a} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + b \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {-a} \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}}}{a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )}\right ) + 2 \, \sqrt {\frac {a \cosh \left (d x + c\right ) + b}{\cosh \left (d x + c\right )}}}{d}\right ] \]
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\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=\int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}} \tanh {\left (c + d x \right )}\, dx \]
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\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right ) \,d x } \]
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\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right ) \,d x } \]
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Time = 2.36 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x) \, dx=\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}}{\sqrt {a}}\right )}{d}-\frac {2\,\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}}{d} \]
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