Integrand size = 27, antiderivative size = 404 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.44 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {5928, 5930, 5926, 5947, 4265, 2317, 2438, 8, 41, 74, 276} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 8
Rule 41
Rule 74
Rule 276
Rule 2317
Rule 2438
Rule 4265
Rule 5926
Rule 5928
Rule 5930
Rule 5947
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} \left (5 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^2 (1+c x)^2}{x^2} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} \left (5 c^2 d^2\right ) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x} \, dx+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2}{x^2} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x) (1+c x) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 3.71 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.48 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{36} d^2 \left (\frac {6 a \sqrt {d-c^2 d x^2} \left (-3-14 c^2 x^2+2 c^4 x^4\right )}{x^2}+\frac {b c^2 \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-90 a c^2 \sqrt {d} \log (x)+90 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )-\frac {72 b c^2 \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {18 b d (1+c x) \left (c x \sqrt {\frac {-1+c x}{1+c x}}-\text {arccosh}(c x)+c x \text {arccosh}(c x)+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{x^2 \sqrt {d-c^2 d x^2}}\right ) \]
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Time = 1.99 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.97
method | result | size |
default | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}-\frac {5 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-6 i \operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+2 i x^{5} c^{5}+42 i \operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-42 i x^{3} c^{3}+45 \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}-45 \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}-45 \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+45 \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+9 i \operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}+9 i c x \right ) d^{2}}{18 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}\) | \(390\) |
parts | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}-\frac {5 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-6 i \operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+2 i x^{5} c^{5}+42 i \operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-42 i x^{3} c^{3}+45 \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}-45 \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) \operatorname {arccosh}\left (c x \right ) x^{2} c^{2}-45 \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+45 \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+9 i \operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}+9 i c x \right ) d^{2}}{18 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}\) | \(390\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \]
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