Integrand size = 27, antiderivative size = 272 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3} \]
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Time = 0.15 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {272, 45, 5922, 12} \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 12
Rule 45
Rule 272
Rule 5922
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \, dx}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {8 b x \sqrt {d-c^2 d x^2}}{105 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b x^3 \sqrt {d-c^2 d x^2}}{315 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^5 \sqrt {d-c^2 d x^2}}{175 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^6 d^3} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.62 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {\sqrt {d-c^2 d x^2} \left (b c x \left (840+140 c^2 x^2+63 c^4 x^4-225 c^6 x^6\right )+105 a \sqrt {-1+c x} \sqrt {1+c x} \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right )+105 b \sqrt {-1+c x} \sqrt {1+c x} \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \text {arccosh}(c x)\right )}{11025 c^6 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(228)=456\).
Time = 0.69 (sec) , antiderivative size = 988, normalized size of antiderivative = 3.63
method | result | size |
default | \(a \left (-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{7 c^{2} d}+\frac {-\frac {4 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{35 c^{2} d}-\frac {8 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{105 d \,c^{4}}}{c^{2}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+104 c^{4} x^{4}-112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-25 c^{2} x^{2}+56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -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 )}{128 \left (c x +1\right ) c^{6} \left (c x -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 )}{128 \left (c x +1\right ) c^{6} \left (c x -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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+64 c^{8} x^{8}+112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-144 c^{6} x^{6}-56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}\right )\) | \(988\) |
parts | \(a \left (-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{7 c^{2} d}+\frac {-\frac {4 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{35 c^{2} d}-\frac {8 \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{105 d \,c^{4}}}{c^{2}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+104 c^{4} x^{4}-112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-25 c^{2} x^{2}+56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -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 )}{128 \left (c x +1\right ) c^{6} \left (c x -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 )}{128 \left (c x +1\right ) c^{6} \left (c x -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 )}{1152 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {3 \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 )}{3200 \left (c x +1\right ) c^{6} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+64 c^{8} x^{8}+112 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}-144 c^{6} x^{6}-56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right )}{6272 \left (c x +1\right ) c^{6} \left (c x -1\right )}\right )\) | \(988\) |
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Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.75 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {105 \, {\left (15 \, b c^{8} x^{8} - 18 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{7} x^{7} - 63 \, b c^{5} x^{5} - 140 \, b c^{3} x^{3} - 840 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 105 \, {\left (15 \, a c^{8} x^{8} - 18 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{11025 \, {\left (c^{8} x^{2} - c^{6}\right )}} \]
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\[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^{5} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
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Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.75 \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{105} \, {\left (\frac {15 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}{c^{2} d} + \frac {12 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{6} d}\right )} a - \frac {{\left (225 \, c^{6} \sqrt {-d} x^{7} - 63 \, c^{4} \sqrt {-d} x^{5} - 140 \, c^{2} \sqrt {-d} x^{3} - 840 \, \sqrt {-d} x\right )} b}{11025 \, c^{5}} \]
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Exception generated. \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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