Integrand size = 27, antiderivative size = 201 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5926, 5939, 5893, 30} \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 30
Rule 5893
Rule 5926
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b x^2 \sqrt {d-c^2 d x^2}}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2}}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{16 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.85 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.75 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {-16 a c x \left (-1+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}+16 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {b \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}}{128 c^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs. \(2(169)=338\).
Time = 0.58 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.83
method | result | size |
default | \(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\) | \(367\) |
parts | \(-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+4 c x -8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c^{5} x^{5}+8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )\) | \(367\) |
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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