Integrand size = 27, antiderivative size = 311 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.33 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5928, 5926, 5947, 4265, 2317, 2438, 8, 74, 14} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {3 c^2 d \sqrt {d-c^2 d x^2} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d \sqrt {d-c^2 d x^2}}{2 x \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 8
Rule 14
Rule 74
Rule 2317
Rule 2438
Rule 4265
Rule 5926
Rule 5928
Rule 5947
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {1}{2} \left (3 c^2 d\right ) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x} \, dx-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x) (1+c x)}{x^2} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {-1+c^2 x^2}{x^2} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c^2 d \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 (3 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3 b c^3 d x \sqrt {d-c^2 d x^2}}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (c^2-\frac {1}{x^2}\right ) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c^2 d \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 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {3 c^2 d \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 (3 i b c^2 d \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 (3 i b c^2 d \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 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {3 c^2 d \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 (3 i b c^2 d \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 (3 i b c^2 d \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 \sqrt {d-c^2 d x^2}}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{2 x^2}+\frac {3 c^2 d \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 {3 i b c^2 d \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 {3 i b c^2 d \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 = 1.47 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.61 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{2} \left (-\frac {a d \left (1+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{x^2}-3 a c^2 d^{3/2} \log (x)+3 a c^2 d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )-\frac {2 b c^2 d \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b d^2 (1+c x) \left (c x \sqrt {\frac {-1+c x}{1+c x}}-\text {arccosh}(c x)+c x \text {arccosh}(c x)+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{x^2 \sqrt {d-c^2 d x^2}}\right ) \]
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Time = 0.97 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.73
method | result | size |
default | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}-\frac {3 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d \,\operatorname {arccosh}\left (c x \right ) x^{2}}{\left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} d x}{\sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} d \,\operatorname {arccosh}\left (c x \right )}{2 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c}{2 x \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right )}{2 x^{2} \left (c x +1\right ) \left (c x -1\right )}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(538\) |
parts | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}-\frac {3 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d \,\operatorname {arccosh}\left (c x \right ) x^{2}}{\left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} d x}{\sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} d \,\operatorname {arccosh}\left (c x \right )}{2 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c}{2 x \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right )}{2 x^{2} \left (c x +1\right ) \left (c x -1\right )}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {3 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2} d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(538\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{3}}\, dx \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^3} \,d x \]
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