Integrand size = 27, antiderivative size = 479 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \]
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Time = 0.41 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {5932, 5936, 5946, 4265, 2317, 2438, 35, 213, 41, 205, 74, 296, 331} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {5 c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {3 b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {c x-1} \sqrt {c x+1}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
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Rule 35
Rule 41
Rule 74
Rule 205
Rule 213
Rule 296
Rule 331
Rule 2317
Rule 2438
Rule 4265
Rule 5932
Rule 5936
Rule 5946
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {1}{2} \left (5 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 (-1+c x)^2 (1+c x)^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (5 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{2 d}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \left (-1+c^2 x^2\right )^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^2 (1+c x)^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx}{2 d^2}+\frac {\left (3 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \left (-1+c^2 x^2\right )} \, dx}{4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\left (-1+c^2 x^2\right )^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x) (1+c x)} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{12 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 7.21 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a}{2 d^3 x^2}+\frac {a c^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {2 a c^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {5 a c^2 \log (x)}{2 d^{5/2}}-\frac {5 a c^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{2 d^{5/2}}+\frac {b c^2 \left (\frac {6 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\frac {6 (-1+c x) (1+c x) \text {arccosh}(c x)}{c^2 x^2}+26 \text {arccosh}(c x) \cosh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {arccosh}(c x) \coth ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )+30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+26 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-26 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )-26 \text {arccosh}(c x) \sinh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {arccosh}(c x) \tanh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{12 d^2 \sqrt {-d (-1+c x) (1+c x)}} \]
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Time = 1.22 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.37
method | result | size |
default | \(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-20 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +3 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}-\frac {13 \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 ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \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 ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \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-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \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+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(655\) |
parts | \(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-20 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +3 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}-\frac {13 \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 ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \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 ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \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-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \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+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(655\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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