Integrand size = 27, antiderivative size = 360 \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{256 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.52 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5930, 5926, 5939, 5893, 30, 74, 14} \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{64 c^2}-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{256 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^4}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 14
Rule 30
Rule 74
Rule 5893
Rule 5926
Rule 5930
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+\frac {1}{8} (3 d) \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 (-1+c x) (1+c x) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-x^5+c^2 x^7\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{256 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 3.47 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.94 \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d \left (-576 a c x \sqrt {d-c^2 d x^2} \left (3+2 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )-1728 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {32 b \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {b \sqrt {d-c^2 d x^2} \left (1440 \text {arccosh}(c x)^2-576 \cosh (2 \text {arccosh}(c x))+144 \cosh (4 \text {arccosh}(c x))+64 \cosh (6 \text {arccosh}(c x))+9 \cosh (8 \text {arccosh}(c x))-24 \text {arccosh}(c x) (-48 \sinh (2 \text {arccosh}(c x))+24 \sinh (4 \text {arccosh}(c x))+16 \sinh (6 \text {arccosh}(c x))+3 \sinh (8 \text {arccosh}(c x)))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )}{73728 c^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(782\) vs. \(2(304)=608\).
Time = 0.75 (sec) , antiderivative size = 783, normalized size of antiderivative = 2.18
method | result | size |
default | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{16 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{64 c^{4}}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{4}}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (128 c^{9} x^{9}-320 c^{7} x^{7}+128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+272 c^{5} x^{5}-256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-88 c^{3} x^{3}+160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c x -32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\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 ) d}{1024 \left (c x +1\right ) c^{5} \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 ) d}{1024 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+128 c^{9} x^{9}+256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-320 c^{7} x^{7}-160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+272 c^{5} x^{5}+32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-88 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+8 c x \right ) \left (1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(783\) |
parts | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{16 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{64 c^{4}}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{4}}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} d}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (128 c^{9} x^{9}-320 c^{7} x^{7}+128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+272 c^{5} x^{5}-256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-88 c^{3} x^{3}+160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+8 c x -32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\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 ) d}{1024 \left (c x +1\right ) c^{5} \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 ) d}{1024 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{8} x^{8}+128 c^{9} x^{9}+256 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}-320 c^{7} x^{7}-160 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+272 c^{5} x^{5}+32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-88 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+8 c x \right ) \left (1+8 \,\operatorname {arccosh}\left (c x \right )\right ) d}{16384 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(783\) |
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\[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
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\[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
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\[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
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