Integrand size = 29, antiderivative size = 292 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {4 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {40 b^2 (1-c x) (1+c x)}{27 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^2 \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d} \]
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Time = 0.24 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5938, 5914, 5879, 75, 5883, 102, 12} \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}-\frac {2 b x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d}-\frac {4 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^2 (1-c x) (c x+1)}{27 c^2 \sqrt {d-c^2 d x^2}}-\frac {40 b^2 (1-c x) (c x+1)}{27 c^4 \sqrt {d-c^2 d x^2}} \]
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Rule 12
Rule 75
Rule 102
Rule 5879
Rule 5883
Rule 5914
Rule 5938
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^2}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 (a+b \text {arccosh}(c x)) \, dx}{3 c \sqrt {d-c^2 d x^2}} \\ & = -\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int (a+b \text {arccosh}(c x)) \, dx}{3 c^3 \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^2 \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}-\frac {\left (4 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \text {arccosh}(c x) \, dx}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 c^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^2 \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {\left (4 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 c^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {40 b^2 (1-c x) (1+c x)}{27 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^2 \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^3 \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (6 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (6+c^2 x^2\right )-9 a^2 \left (-2+c^2 x^2+c^4 x^4\right )-2 b^2 \left (-20+19 c^2 x^2+c^4 x^4\right )+6 b \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (6+c^2 x^2\right )-3 a \left (-2+c^2 x^2+c^4 x^4\right )\right ) \text {arccosh}(c x)-9 b^2 \left (-2+c^2 x^2+c^4 x^4\right ) \text {arccosh}(c x)^2\right )}{27 c^4 d (-1+c x) (1+c x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(751\) vs. \(2(252)=504\).
Time = 0.94 (sec) , antiderivative size = 752, normalized size of antiderivative = 2.58
method | result | size |
default | \(a^{2} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}-6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}+6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(752\) |
parts | \(a^{2} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}-6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}+6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(752\) |
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Time = 0.27 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {9 \, {\left (b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a b c^{3} x^{3} + 6 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left ({\left (b^{2} c^{3} x^{3} + 6 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 3 \, {\left (a b c^{4} x^{4} + a b c^{2} x^{2} - 2 \, a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} + {\left (9 \, a^{2} + 38 \, b^{2}\right )} c^{2} x^{2} - 18 \, a^{2} - 40 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{6} d x^{2} - c^{4} d\right )}} \]
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\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{3} \, b^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{3} \, a b {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} - \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} + \frac {20 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{2} d} - \frac {3 \, {\left (c^{2} \sqrt {-d} x^{3} + 6 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3} d}\right )} + \frac {2 \, {\left (c^{2} \sqrt {-d} x^{3} + 6 \, \sqrt {-d} x\right )} a b}{9 \, c^{3} d} \]
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Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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