Integrand size = 27, antiderivative size = 235 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5924, 30, 5947, 4265, 2317, 2438} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {c^2 \sqrt {d-c^2 d x^2} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {c x-1} \sqrt {c x+1}} \]
[In]
[Out]
Rule 30
Rule 2317
Rule 2438
Rule 4265
Rule 5924
Rule 5947
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x^2} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {\left (c^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b c^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b c^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b c^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b c^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.89 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{2} \left (-\frac {a \sqrt {d-c^2 d x^2}}{x^2}-a c^2 \sqrt {d} \log (x)+a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b d (1+c x) \left (c x \sqrt {\frac {-1+c x}{1+c x}}-\text {arccosh}(c x)+c x \text {arccosh}(c x)+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{x^2 \sqrt {d-c^2 d x^2}}\right ) \]
[In]
[Out]
Time = 1.22 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.86
method | result | size |
default | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{2}}{2 \left (c x -1\right ) \left (c x +1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{2 x \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{2 x^{2} \left (c x -1\right ) \left (c x +1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(436\) |
parts | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) c^{2}}{2 \left (c x -1\right ) \left (c x +1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c}{2 x \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )}{2 x^{2} \left (c x -1\right ) \left (c x +1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(436\) |
[In]
[Out]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^3} \,d x \]
[In]
[Out]