Integrand size = 27, antiderivative size = 236 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {8 b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4 b x^3 \sqrt {-1+c x} \sqrt {1+c x}}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^5 \sqrt {-1+c x} \sqrt {1+c x}}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2 d} \]
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Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5938, 5914, 8, 30} \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2 d}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^4 d}-\frac {b x^5 \sqrt {c x-1} \sqrt {c x+1}}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 b x \sqrt {c x-1} \sqrt {c x+1}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4 b x^3 \sqrt {c x-1} \sqrt {c x+1}}{45 c^3 \sqrt {d-c^2 d x^2}} \]
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Rule 8
Rule 30
Rule 5914
Rule 5938
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2 d}+\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx}{5 c^2}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^4 \, dx}{5 c \sqrt {d-c^2 d x^2}} \\ & = -\frac {b x^5 \sqrt {-1+c x} \sqrt {1+c x}}{25 c \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2 d}+\frac {8 \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx}{15 c^4}-\frac {\left (4 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 \, dx}{15 c^3 \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 b x^3 \sqrt {-1+c x} \sqrt {1+c x}}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^5 \sqrt {-1+c x} \sqrt {1+c x}}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2 d}-\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int 1 \, dx}{15 c^5 \sqrt {d-c^2 d x^2}} \\ & = -\frac {8 b x \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5 \sqrt {d-c^2 d x^2}}-\frac {4 b x^3 \sqrt {-1+c x} \sqrt {1+c x}}{45 c^3 \sqrt {d-c^2 d x^2}}-\frac {b x^5 \sqrt {-1+c x} \sqrt {1+c x}}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 c^2 d} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.59 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (120+20 c^2 x^2+9 c^4 x^4\right )-15 a \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )-15 b \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right ) \text {arccosh}(c x)\right )}{225 c^6 d (-1+c x) (1+c x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(669\) vs. \(2(200)=400\).
Time = 0.79 (sec) , antiderivative size = 670, normalized size of antiderivative = 2.84
method | result | size |
default | \(a \left (-\frac {x^{4} \sqrt {-c^{2} d \,x^{2}+d}}{5 c^{2} d}+\frac {-\frac {4 x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{15 c^{2} d}-\frac {8 \sqrt {-c^{2} d \,x^{2}+d}}{15 d \,c^{4}}}{c^{2}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+13 c^{2} x^{2}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+16 c^{6} x^{6}+20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}-1\right )}\right )\) | \(670\) |
parts | \(a \left (-\frac {x^{4} \sqrt {-c^{2} d \,x^{2}+d}}{5 c^{2} d}+\frac {-\frac {4 x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{15 c^{2} d}-\frac {8 \sqrt {-c^{2} d \,x^{2}+d}}{15 d \,c^{4}}}{c^{2}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+13 c^{2} x^{2}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{288 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+16 c^{6} x^{6}+20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}-1\right )}\right )\) | \(670\) |
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Time = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.75 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {15 \, {\left (3 \, b c^{6} x^{6} + b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{5} x^{5} + 20 \, b c^{3} x^{3} + 120 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 15 \, {\left (3 \, a c^{6} x^{6} + a c^{4} x^{4} + 4 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{225 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \]
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.83 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a + \frac {{\left (9 \, c^{4} \sqrt {-d} x^{5} + 20 \, c^{2} \sqrt {-d} x^{3} + 120 \, \sqrt {-d} x\right )} b}{225 \, c^{5} d} \]
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Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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