Integrand size = 21, antiderivative size = 31 \[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3970, 65, 213} \[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
[In]
[Out]
Rule 65
Rule 213
Rule 3970
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \text {sech}(c+d x)\right )}{d} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{d \sqrt {a}}\) | \(26\) |
default | \(\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{d \sqrt {a}}\) | \(26\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (25) = 50\).
Time = 0.68 (sec) , antiderivative size = 558, normalized size of antiderivative = 18.00 \[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\left [\frac {\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 )}{2 \, \sqrt {a} 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 )}{a d}\right ] \]
[In]
[Out]
\[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {\tanh {\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\tanh \left (d x + c\right )}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\tanh \left (d x + c\right )}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
Time = 2.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\tanh (c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}}{\sqrt {a}}\right )}{\sqrt {a}\,d} \]
[In]
[Out]