Integrand size = 27, antiderivative size = 298 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5914, 5889, 5901, 5903, 4267, 2317, 2438, 75} \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
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Rule 75
Rule 2317
Rule 2438
Rule 4267
Rule 5889
Rule 5901
Rule 5903
Rule 5914
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{(-1+c x)^2 (1+c x)^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{-1+c^2 x^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 2.50 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.30 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {4 a^2+b^2 \left (-2+4 \text {arccosh}(c x)^2+2 \cosh (2 \text {arccosh}(c x))-3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )+3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+4 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-4 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))+\text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-\text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))\right )+a b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (4 c x+3 \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )+\frac {8 \text {arccosh}(c x)+\left (-\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right ) \sinh (3 \text {arccosh}(c x))}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )}{12 c^2 d \left (d-c^2 d x^2\right )^{3/2}} \]
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Time = 1.19 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {a^{2}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(582\) |
parts | \(\frac {a^{2}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +2 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(582\) |
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\[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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