Integrand size = 27, antiderivative size = 229 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5936, 5946, 4265, 2317, 2438, 35, 213} \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}} \]
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Rule 35
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)}{d \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}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x) (1+c x)} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {a+b \text {arccosh}(c x)}{d \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 \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 \sqrt {d-c^2 d x^2}} \\ & = \frac {a+b \text {arccosh}(c x)}{d \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 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \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 \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 \sqrt {d-c^2 d x^2}} \\ & = \frac {a+b \text {arccosh}(c x)}{d \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 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \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 \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 \sqrt {d-c^2 d x^2}} \\ & = \frac {a+b \text {arccosh}(c x)}{d \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 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \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 \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 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 2.14 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.47 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {\frac {a \sqrt {d-c^2 d x^2}}{-1+c^2 x^2}-a \sqrt {d} \log (x)+a \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {i b d \left (i \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}}{d^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (238 ) = 476\).
Time = 1.22 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.11
method | result | size |
default | \(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \left (c^{2} x^{2}-1\right )}\) | \(483\) |
parts | \(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \left (c^{2} x^{2}-1\right )}\) | \(483\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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