Integrand size = 29, antiderivative size = 319 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}-\frac {b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{24 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.48 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {5926, 5939, 5893, 5883, 92, 54, 102, 12} \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{24 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{64 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2} \]
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 5883
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))^2-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\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 (a+b \text {arccosh}(c x)) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \text {arccosh}(c x))^2}{\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 (a+b \text {arccosh}(c x)) \, dx}{4 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{24 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b^2 x \sqrt {d-c^2 d x^2}}{16 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{24 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}-\frac {b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{16 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{24 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b^2 x \sqrt {d-c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d-c^2 d x^2}-\frac {b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{24 b c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 2.13 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.76 \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {-96 a^2 c x \left (-1+2 c^2 x^2\right ) \sqrt {d-c^2 d x^2}+96 a^2 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {12 a b \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b^2 \sqrt {d-c^2 d x^2} \left (32 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))-3 \left (1+8 \text {arccosh}(c x)^2\right ) \sinh (4 \text {arccosh}(c x))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}}{768 c^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(677\) vs. \(2(271)=542\).
Time = 0.65 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.13
method | result | size |
default | \(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} 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^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3}}{24 \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 (8 \operatorname {arccosh}\left (c x \right )^{2}-4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \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 (8 \operatorname {arccosh}\left (c x \right )^{2}+4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )+2 a 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 )\) | \(678\) |
parts | \(-\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}+\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{2}}+\frac {a^{2} 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^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{3}}{24 \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 (8 \operatorname {arccosh}\left (c x \right )^{2}-4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \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 (8 \operatorname {arccosh}\left (c x \right )^{2}+4 \,\operatorname {arccosh}\left (c x \right )+1\right )}{512 \left (c x +1\right ) c^{3} \left (c x -1\right )}\right )+2 a 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 )\) | \(678\) |
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^{2} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}\, dx \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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