Integrand size = 27, antiderivative size = 197 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {b c^3 d x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}+\frac {3 c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \sqrt {d-c^2 d x^2} \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5928, 5895, 5893, 30, 74, 14} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {3 c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}+\frac {b c d \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 14
Rule 30
Rule 74
Rule 5893
Rule 5895
Rule 5928
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x}-\left (3 c^2 d\right ) \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x) (1+c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3}{2} c^2 d x \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))}{x}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {-1+c^2 x^2}{x} \, dx}{\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)}{\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 x \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \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))}{x}+\frac {3 c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {1}{x}+c^2 x\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c^3 d x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \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))}{x}+\frac {3 c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{4 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \sqrt {d-c^2 d x^2} \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{8} \left (-\frac {4 a d \left (2+c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{x}+12 a c d^{3/2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+4 b c d \sqrt {d-c^2 d x^2} \left (-\frac {2 \text {arccosh}(c x)}{c x}+\frac {\text {arccosh}(c x)^2+2 \log (c x)}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )+\frac {b c d \sqrt {d-c^2 d x^2} \left (2 \text {arccosh}(c x)^2+\cosh (2 \text {arccosh}(c x))-2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right ) \]
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Time = 1.02 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.24
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}-a \,c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 a \,c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a \,c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+2 c^{3} x^{3}+6 \operatorname {arccosh}\left (c x \right )^{2} x c -8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-8 c x \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -c x \right ) d}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\) | \(244\) |
parts | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}-a \,c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 a \,c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a \,c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+2 c^{3} x^{3}+6 \operatorname {arccosh}\left (c x \right )^{2} x c -8 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-8 c x \,\operatorname {arccosh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -c x \right ) d}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\) | \(244\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{2}}\, dx \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^2} \,d x \]
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