Integrand size = 27, antiderivative size = 329 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}} \]
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Time = 0.29 (sec) , antiderivative size = 329, 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 = {5932, 5936, 5946, 4265, 2317, 2438, 35, 213, 84} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {3 c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x \sqrt {d-c^2 d x^2}} \]
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Rule 35
Rule 84
Rule 213
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 \sqrt {d-c^2 d x^2}}+\frac {1}{2} \left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 (-1+c x) (1+c x)} \, dx}{2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx}{2 d}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (-\frac {1}{x^2}+\frac {c^2}{-1+c^2 x^2}\right ) \, dx}{2 d \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 x) (1+c x)} \, dx}{2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 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 \sqrt {d-c^2 d x^2}}+\frac {\left (b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d \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}{2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 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 \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 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 \sqrt {d-c^2 d x^2}}+\frac {\left (3 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 \sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 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 \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {\left (3 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 \sqrt {d-c^2 d x^2}}+\frac {\left (3 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 \sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 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 \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {3 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 \sqrt {d-c^2 d x^2}}+\frac {3 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 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 3.68 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.33 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (-\frac {a \left (-1+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{d^2 x^2 \left (-1+c^2 x^2\right )}+\frac {3 a c^2 \log (x)}{d^{3/2}}-\frac {3 a c^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{3/2}}-\frac {b c^2 \left (-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\left (-1+\frac {1}{c^2 x^2}\right ) \text {arccosh}(c x)-2 \text {arccosh}(c x) \cosh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \sinh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{d \sqrt {d-c^2 d x^2}}\right ) \]
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Time = 1.29 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.79
method | result | size |
default | \(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )\) | \(588\) |
parts | \(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )\) | \(588\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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