Integrand size = 23, antiderivative size = 166 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {35 b d^3 x \sqrt {-1+c x} \sqrt {1+c x}}{1024 c}+\frac {35 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{1536 c}-\frac {7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac {b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}+\frac {35 b d^3 \text {arccosh}(c x)}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2} \]
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Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5914, 38, 54} \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2}+\frac {35 b d^3 \text {arccosh}(c x)}{1024 c^2}+\frac {b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{64 c}-\frac {7 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{384 c}+\frac {35 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{1536 c}-\frac {35 b d^3 x \sqrt {c x-1} \sqrt {c x+1}}{1024 c} \]
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Rule 38
Rule 54
Rule 5914
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2}+\frac {\left (b d^3\right ) \int (-1+c x)^{7/2} (1+c x)^{7/2} \, dx}{8 c} \\ & = \frac {b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2}-\frac {\left (7 b d^3\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{64 c} \\ & = -\frac {7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac {b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2}+\frac {\left (35 b d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{384 c} \\ & = \frac {35 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{1536 c}-\frac {7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac {b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2}-\frac {\left (35 b d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{512 c} \\ & = -\frac {35 b d^3 x \sqrt {-1+c x} \sqrt {1+c x}}{1024 c}+\frac {35 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{1536 c}-\frac {7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac {b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2}+\frac {\left (35 b d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1024 c} \\ & = -\frac {35 b d^3 x \sqrt {-1+c x} \sqrt {1+c x}}{1024 c}+\frac {35 b d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}{1536 c}-\frac {7 b d^3 x (-1+c x)^{5/2} (1+c x)^{5/2}}{384 c}+\frac {b d^3 x (-1+c x)^{7/2} (1+c x)^{7/2}}{64 c}+\frac {35 b d^3 \text {arccosh}(c x)}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {arccosh}(c x))}{8 c^2} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.90 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {d^3 \left (c x \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (279-326 c^2 x^2+200 c^4 x^4-48 c^6 x^6\right )+384 a c x \left (-4+6 c^2 x^2-4 c^4 x^4+c^6 x^6\right )\right )+384 b c^2 x^2 \left (-4+6 c^2 x^2-4 c^4 x^4+c^6 x^6\right ) \text {arccosh}(c x)+558 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{3072 c^2} \]
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Time = 0.54 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {-\frac {d^{3} a \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (48 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-200 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+326 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-279 c x \sqrt {c^{2} x^{2}-1}+105 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\) | \(202\) |
default | \(\frac {-\frac {d^{3} a \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (48 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-200 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+326 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-279 c x \sqrt {c^{2} x^{2}-1}+105 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\) | \(202\) |
parts | \(-\frac {d^{3} a \left (c^{2} x^{2}-1\right )^{4}}{8 c^{2}}-\frac {d^{3} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )}{4}-\frac {c^{2} x^{2} \operatorname {arccosh}\left (c x \right )}{2}+\frac {\operatorname {arccosh}\left (c x \right )}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (48 c^{7} x^{7} \sqrt {c^{2} x^{2}-1}-200 c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+326 \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}-279 c x \sqrt {c^{2} x^{2}-1}+105 \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\) | \(204\) |
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Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.11 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {384 \, a c^{8} d^{3} x^{8} - 1536 \, a c^{6} d^{3} x^{6} + 2304 \, a c^{4} d^{3} x^{4} - 1536 \, a c^{2} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} d^{3} x^{8} - 512 \, b c^{6} d^{3} x^{6} + 768 \, b c^{4} d^{3} x^{4} - 512 \, b c^{2} d^{3} x^{2} + 93 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (48 \, b c^{7} d^{3} x^{7} - 200 \, b c^{5} d^{3} x^{5} + 326 \, b c^{3} d^{3} x^{3} - 279 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{3072 \, c^{2}} \]
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\[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=- d^{3} \left (\int \left (- a x\right )\, dx + \int 3 a c^{2} x^{3}\, dx + \int \left (- 3 a c^{4} x^{5}\right )\, dx + \int a c^{6} x^{7}\, dx + \int \left (- b x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 3 b c^{2} x^{3} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{5} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{7} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (137) = 274\).
Time = 0.23 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.55 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{8} \, a c^{6} d^{3} x^{8} + \frac {1}{2} \, a c^{4} d^{3} x^{6} - \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b c^{6} d^{3} - \frac {3}{4} \, a c^{2} d^{3} x^{4} + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b c^{4} d^{3} - \frac {3}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{3} \]
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Exception generated. \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]
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