Integrand size = 25, antiderivative size = 179 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {b \text {arccosh}(c x)}{2 c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d^2} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5934, 5913, 3797, 2221, 2317, 2438, 91, 12, 79, 54} \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d^2}+\frac {\log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d^2}+\frac {b \text {arccosh}(c x)}{2 c^4 d^2}-\frac {b \sqrt {c x-1}}{2 c^4 d^2 \sqrt {c x+1}}-\frac {b}{2 c^4 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
[In]
[Out]
Rule 12
Rule 54
Rule 79
Rule 91
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5913
Rule 5934
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac {\int \frac {x (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx}{c^2 d} \\ & = -\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {\text {Subst}(\int (a+b x) \coth (x) \, dx,x,\text {arccosh}(c x))}{c^4 d^2}+\frac {b \int \frac {c^2 x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^4 d^2} \\ & = -\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d^2}-\frac {2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text {arccosh}(c x)\right )}{c^4 d^2}+\frac {b \int \frac {x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 c^2 d^2} \\ & = -\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d^2}-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^4 d^2}+\frac {b \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^3 d^2} \\ & = -\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {b \text {arccosh}(c x)}{2 c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d^2}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d^2} \\ & = -\frac {b}{2 c^4 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x}}{2 c^4 d^2 \sqrt {1+c x}}+\frac {b \text {arccosh}(c x)}{2 c^4 d^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d^2} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.17 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {-b \sqrt {\frac {-1+c x}{1+c x}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a}{-1+c^2 x^2}+\frac {b \text {arccosh}(c x)}{1-c x}+\frac {b \text {arccosh}(c x)}{1+c x}-2 b \text {arccosh}(c x)^2+4 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )+4 b \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+2 a \log \left (1-c^2 x^2\right )+4 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+4 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c^4 d^2} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}}{c^{4}}\) | \(202\) |
default | \(\frac {\frac {a \left (\frac {1}{4 c x +4}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}}{c^{4}}\) | \(202\) |
parts | \(\frac {a \left (\frac {1}{4 c^{4} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2 c^{4}}-\frac {1}{4 c^{4} \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2 c^{4}}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2} c^{4}}\) | \(213\) |
[In]
[Out]
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
[In]
[Out]
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
[In]
[Out]