Integrand size = 27, antiderivative size = 226 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^4 d^2}-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c^5 d \sqrt {d-c^2 d x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 226, 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 = {5934, 5938, 5892, 30, 84, 266} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x^3 (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^4 d^2}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{4 c^3 d \sqrt {d-c^2 d x^2}} \]
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Rule 30
Rule 84
Rule 266
Rule 5892
Rule 5934
Rule 5938
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx}{c^2 d}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{(-1+c x) (1+c x)} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {x^3 (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^4 d^2}-\frac {3 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}} \, dx}{2 c^4 d}+\frac {\left (3 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x \, dx}{2 c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (\frac {x}{c^2}+\frac {x}{c^2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^4 d^2}-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{-1+c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}} \\ & = \frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 c^4 d^2}-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{4 b c^5 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c^5 d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-4 a c d x \left (-3+c^2 x^2\right )+12 a \sqrt {d} \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b d \left (8 c x \text {arccosh}(c x)-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (6 \text {arccosh}(c x)^2-\cosh (2 \text {arccosh}(c x))+8 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )+2 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))\right )\right )}{8 c^5 d^2 \sqrt {d-c^2 d x^2}} \]
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Time = 1.34 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.33
method | result | size |
default | \(-\frac {a \,x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-2 c^{4} x^{4}+6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -8 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+3 c^{2} x^{2}-6 \operatorname {arccosh}\left (c x \right )^{2}+8 \,\operatorname {arccosh}\left (c x \right )-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{8 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}\) | \(301\) |
parts | \(-\frac {a \,x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{3} x^{3}-2 c^{4} x^{4}+6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -8 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right ) x^{2} c^{2}+3 c^{2} x^{2}-6 \operatorname {arccosh}\left (c x \right )^{2}+8 \,\operatorname {arccosh}\left (c x \right )-8 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )-1\right )}{8 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{5}}\) | \(301\) |
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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