Integrand size = 19, antiderivative size = 61 \[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\text {sech}^{-1}(a+b x)^2}{2 d}-\frac {\text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 d} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6454, 12, 6416, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 d}+\frac {\text {sech}^{-1}(a+b x)^2}{2 d}-\frac {\text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )}{d} \]
[In]
[Out]
Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 6416
Rule 6454
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b \text {sech}^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\text {sech}^{-1}(x)}{x} \, dx,x,a+b x\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {\text {arccosh}(x)}{x} \, dx,x,\frac {1}{a+b x}\right )}{d} \\ & = -\frac {\text {Subst}\left (\int x \tanh (x) \, dx,x,\text {arccosh}\left (\frac {1}{a+b x}\right )\right )}{d} \\ & = \frac {\text {arccosh}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {2 \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {arccosh}\left (\frac {1}{a+b x}\right )\right )}{d} \\ & = \frac {\text {arccosh}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {\text {arccosh}\left (\frac {1}{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{a+b x}\right )}\right )}{d}+\frac {\text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}\left (\frac {1}{a+b x}\right )\right )}{d} \\ & = \frac {\text {arccosh}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {\text {arccosh}\left (\frac {1}{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{a+b x}\right )}\right )}{d}+\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}\left (\frac {1}{a+b x}\right )}\right )}{2 d} \\ & = \frac {\text {arccosh}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {\text {arccosh}\left (\frac {1}{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{a+b x}\right )}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (\frac {1}{a+b x}\right )}\right )}{2 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.85 \[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {-\text {sech}^{-1}(a+b x) \left (\text {sech}^{-1}(a+b x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(a+b x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a+b x)}\right )}{2 d} \]
[In]
[Out]
Time = 0.78 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(\frac {\frac {b \operatorname {arcsech}\left (b x +a \right )^{2}}{2 d}-\frac {b \,\operatorname {arcsech}\left (b x +a \right ) \ln \left (1+\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{d}-\frac {b \operatorname {polylog}\left (2, -\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{2 d}}{b}\) | \(111\) |
default | \(\frac {\frac {b \operatorname {arcsech}\left (b x +a \right )^{2}}{2 d}-\frac {b \,\operatorname {arcsech}\left (b x +a \right ) \ln \left (1+\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{d}-\frac {b \operatorname {polylog}\left (2, -\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{2 d}}{b}\) | \(111\) |
[In]
[Out]
\[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \]
[In]
[Out]
\[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {b \int \frac {\operatorname {asech}{\left (a + b x \right )}}{a + b x}\, dx}{d} \]
[In]
[Out]
\[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \]
[In]
[Out]
\[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arsech}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}{d\,x+\frac {a\,d}{b}} \,d x \]
[In]
[Out]