Integrand size = 27, antiderivative size = 278 \[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.41 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5926, 5939, 5893, 30} \[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{24 c^2}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^4}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 30
Rule 5893
Rule 5926
Rule 5939
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{24 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.71 \[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\frac {48 a c x \sqrt {d-c^2 d x^2} \left (-3-2 c^2 x^2+8 c^4 x^4\right )-144 a \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {b \sqrt {d-c^2 d x^2} \left (-72 \text {arccosh}(c x)^2+18 \cosh (2 \text {arccosh}(c x))-9 \cosh (4 \text {arccosh}(c x))-2 \cosh (6 \text {arccosh}(c x))+12 \text {arccosh}(c x) (-3 \sinh (2 \text {arccosh}(c x))+3 \sinh (4 \text {arccosh}(c x))+\sinh (6 \text {arccosh}(c x)))\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}}{2304 c^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(877\) vs. \(2(234)=468\).
Time = 0.84 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.16
method | result | size |
default | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 c^{4} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{16 c^{4}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(878\) |
parts | \(-\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8 c^{4} d}+\frac {a x \sqrt {-c^{2} d \,x^{2}+d}}{16 c^{4}}+\frac {a d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 c^{4} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-6 c x +18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \left (c x +1\right ) c^{5} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\operatorname {arccosh}\left (c x \right )\right )}{256 \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 )}{512 \left (c x +1\right ) c^{5} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{6} x^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-64 c^{5} x^{5}-18 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\operatorname {arccosh}\left (c x \right )\right )}{2304 \left (c x +1\right ) c^{5} \left (c x -1\right )}\right )\) | \(878\) |
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\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
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\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
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\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
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\[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int { \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
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