Integrand size = 25, antiderivative size = 136 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {-1+c x}}{4 c^4 d^3 \sqrt {1+c x}}-\frac {b \text {arccosh}(c x)}{4 c^4 d^3}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2} \]
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Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5917, 100, 21, 91, 12, 79, 54} \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \text {arccosh}(c x)}{4 c^4 d^3}+\frac {b \sqrt {c x-1}}{4 c^4 d^3 \sqrt {c x+1}}+\frac {b}{4 c^4 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^3}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rule 12
Rule 21
Rule 54
Rule 79
Rule 91
Rule 100
Rule 5917
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^4}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3} \\ & = \frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b \int \frac {x^2 (-3-3 c x)}{(-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{12 c d^3} \\ & = \frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{4 c d^3} \\ & = \frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {c^2 x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{4 c^4 d^3} \\ & = \frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{4 c^2 d^3} \\ & = \frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {-1+c x}}{4 c^4 d^3 \sqrt {1+c x}}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c^3 d^3} \\ & = \frac {b x^3}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^4 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \sqrt {-1+c x}}{4 c^4 d^3 \sqrt {1+c x}}-\frac {b \text {arccosh}(c x)}{4 c^4 d^3}+\frac {x^4 (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.61 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-3+4 c^2 x^2\right )+a \left (-3+6 c^2 x^2\right )+3 b \left (-1+2 c^2 x^2\right ) \text {arccosh}(c x)}{12 c^4 d^3 \left (-1+c^2 x^2\right )^2} \]
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Time = 0.51 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (-\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {3}{16 \left (c x -1\right )}\right )}{d^{3}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{16 \left (c x +1\right )^{2}}+\frac {3 \,\operatorname {arccosh}\left (c x \right )}{16 \left (c x +1\right )}-\frac {\operatorname {arccosh}\left (c x \right )}{16 \left (c x -1\right )^{2}}-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{16 \left (c x -1\right )}-\frac {c x \left (4 c^{2} x^{2}-3\right )}{12 \left (c x +1\right )^{\frac {3}{2}} \left (c x -1\right )^{\frac {3}{2}}}\right )}{d^{3}}}{c^{4}}\) | \(136\) |
default | \(\frac {-\frac {a \left (-\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {3}{16 \left (c x -1\right )}\right )}{d^{3}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{16 \left (c x +1\right )^{2}}+\frac {3 \,\operatorname {arccosh}\left (c x \right )}{16 \left (c x +1\right )}-\frac {\operatorname {arccosh}\left (c x \right )}{16 \left (c x -1\right )^{2}}-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{16 \left (c x -1\right )}-\frac {c x \left (4 c^{2} x^{2}-3\right )}{12 \left (c x +1\right )^{\frac {3}{2}} \left (c x -1\right )^{\frac {3}{2}}}\right )}{d^{3}}}{c^{4}}\) | \(136\) |
parts | \(-\frac {a \left (-\frac {1}{16 c^{4} \left (c x +1\right )^{2}}+\frac {3}{16 c^{4} \left (c x +1\right )}-\frac {1}{16 c^{4} \left (c x -1\right )^{2}}-\frac {3}{16 c^{4} \left (c x -1\right )}\right )}{d^{3}}-\frac {b \left (-\frac {\operatorname {arccosh}\left (c x \right )}{16 \left (c x +1\right )^{2}}+\frac {3 \,\operatorname {arccosh}\left (c x \right )}{16 \left (c x +1\right )}-\frac {\operatorname {arccosh}\left (c x \right )}{16 \left (c x -1\right )^{2}}-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{16 \left (c x -1\right )}-\frac {c x \left (4 c^{2} x^{2}-3\right )}{12 \left (c x +1\right )^{\frac {3}{2}} \left (c x -1\right )^{\frac {3}{2}}}\right )}{d^{3} c^{4}}\) | \(147\) |
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.74 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {3 \, a c^{4} x^{4} + 3 \, {\left (2 \, b c^{2} x^{2} - b\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (4 \, b c^{3} x^{3} - 3 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
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\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a x^{3}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b x^{3} \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
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\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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