Integrand size = 27, antiderivative size = 317 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
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Time = 0.31 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5936, 5946, 4265, 2317, 2438, 35, 213, 41, 205} \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {c x-1} \sqrt {c x+1}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
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Rule 35
Rule 41
Rule 205
Rule 213
Rule 2317
Rule 2438
Rule 4265
Rule 5936
Rule 5946
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{d}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^2 (1+c x)^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx}{d^2}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x) (1+c x)} \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {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 {7 b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b \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 )}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {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 {7 b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b \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 )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b \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 )}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {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 {7 b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 7.07 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.23 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {a}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {a \log (x)}{d^{5/2}}-\frac {a \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{d^{5/2}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (14 \text {arccosh}(c x) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {csch}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {1}{2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \text {csch}^4\left (\frac {1}{2} \text {arccosh}(c x)\right )-24 i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )+24 i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+28 \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-28 \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-24 i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+24 i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )-\text {sech}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {8 \text {arccosh}(c x) \sinh ^4\left (\frac {1}{2} \text {arccosh}(c x)\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}-14 \text {arccosh}(c x) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{24 d^2 \sqrt {-d (-1+c x) (1+c x)}} \]
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Time = 1.33 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.77
method | result | size |
default | \(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(560\) |
parts | \(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(560\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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