Integrand size = 15, antiderivative size = 38 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3859, 209} \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d} \]
[In]
[Out]
Rule 209
Rule 3859
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 i a) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {i a \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d} \\ & = \frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d} \\ \end{align*}
Time = 1.41 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.84 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\frac {\sqrt {1+e^{2 (c+d x)}} \left (\text {arcsinh}\left (e^{c+d x}\right )+\text {arctanh}\left (\sqrt {1+e^{2 (c+d x)}}\right )\right ) \sqrt {a-a \text {sech}(c+d x)}}{d \left (-1+e^{c+d x}\right )} \]
[In]
[Out]
\[\int \sqrt {a -\operatorname {sech}\left (d x +c \right ) a}d x\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 642, normalized size of antiderivative = 16.89 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\frac {\sqrt {a} \log \left (\frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} + 3 \, a \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) + 3 \, a\right )} \sinh \left (d x + c\right )^{3} + 5 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} + 9 \, a \cosh \left (d x + c\right ) + 5 \, a\right )} \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{5} + {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{4} + \sinh \left (d x + c\right )^{5} + 3 \, \cosh \left (d x + c\right )^{4} + {\left (10 \, \cosh \left (d x + c\right )^{2} + 12 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{3} + 5 \, \cosh \left (d x + c\right )^{3} + {\left (10 \, \cosh \left (d x + c\right )^{3} + 18 \, \cosh \left (d x + c\right )^{2} + 15 \, \cosh \left (d x + c\right ) + 7\right )} \sinh \left (d x + c\right )^{2} + 7 \, \cosh \left (d x + c\right )^{2} + {\left (5 \, \cosh \left (d x + c\right )^{4} + 12 \, \cosh \left (d x + c\right )^{3} + 15 \, \cosh \left (d x + c\right )^{2} + 14 \, \cosh \left (d x + c\right ) + 4\right )} \sinh \left (d x + c\right ) + 4 \, \cosh \left (d x + c\right ) + 4\right )} \sqrt {a} \sqrt {\frac {a}{\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} + 1}} + 4 \, a \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} + 9 \, a \cosh \left (d x + c\right )^{2} + 10 \, a \cosh \left (d x + c\right ) + 4 \, a\right )} \sinh \left (d x + c\right ) + 4 \, a}{\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}}\right ) + \sqrt {a} \log \left (-\frac {a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} - \cosh \left (d x + c\right )^{2} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right ) - 1\right )} \sqrt {a} \sqrt {\frac {a}{\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} + 1}} - a \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) - a\right )} \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}\right )}{2 \, d} \]
[In]
[Out]
\[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\int \sqrt {- a \operatorname {sech}{\left (c + d x \right )} + a}\, dx \]
[In]
[Out]
\[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\int { \sqrt {-a \operatorname {sech}\left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (32) = 64\).
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.66 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=-\frac {\frac {2 \, a \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (e^{\left (d x + c\right )} - 1\right )}{\sqrt {-a}} + \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) \mathrm {sgn}\left (e^{\left (d x + c\right )} - 1\right )}{d} \]
[In]
[Out]
Timed out. \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\int \sqrt {a-\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
[In]
[Out]