Integrand size = 10, antiderivative size = 76 \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=-\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{b}+\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{2 b} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2320, 5882, 3799, 2221, 2317, 2438} \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (e^{2 \text {arccosh}\left (c e^{a+b x}\right )}+1\right )}{b} \]
[In]
[Out]
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 3799
Rule 5882
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arccosh}(c x)}{x} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int x \tanh (x) \, dx,x,\text {arccosh}\left (c e^{a+b x}\right )\right )}{b} \\ & = -\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {arccosh}\left (c e^{a+b x}\right )\right )}{b} \\ & = -\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}\left (c e^{a+b x}\right )\right )}{b} \\ & = -\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{2 b} \\ & = -\frac {\text {arccosh}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\text {arccosh}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{b}+\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (c e^{a+b x}\right )}\right )}{2 b} \\ \end{align*}
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \text {arccosh}\left (c e^{a+b x}\right ) \, dx \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.45
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )}{2}}{b}\) | \(110\) |
default | \(\frac {-\frac {\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2}+\operatorname {arccosh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )}{2}}{b}\) | \(110\) |
[In]
[Out]
Exception generated. \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \operatorname {acosh}{\left (c e^{a + b x} \right )}\, dx \]
[In]
[Out]
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int { \operatorname {arcosh}\left (c e^{\left (b x + a\right )}\right ) \,d x } \]
[In]
[Out]
\[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int { \operatorname {arcosh}\left (c e^{\left (b x + a\right )}\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int \text {arccosh}\left (c e^{a+b x}\right ) \, dx=\int \mathrm {acosh}\left (c\,{\mathrm {e}}^{a+b\,x}\right ) \,d x \]
[In]
[Out]