Integrand size = 20, antiderivative size = 246 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5901, 5899, 266, 74, 267} \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {8 x \text {arccosh}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \text {arccosh}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}-\frac {4 \sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]
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Rule 74
Rule 266
Rule 267
Rule 5899
Rule 5901
Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{(-1+a x)^3 (1+a x)^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{(-1+a x)^2 (1+a x)^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}} \\ & = \frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (8 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}} \\ & = \frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arccosh}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arccosh}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.47 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {4 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \text {arccosh}(a x)+\sqrt {-1+a x} \sqrt {1+a x} \left (11-8 a^2 x^2-16 \left (-1+a^2 x^2\right )^2 \log \left (1-a^2 x^2\right )\right )}{60 a c^3 \left (-1+a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}} \]
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Time = 1.37 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.70
method | result | size |
default | \(-\frac {16 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+15 a x +16 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-8 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-64 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{7} x^{7}-64 a^{8} x^{8}+248 x^{5} a^{5} \sqrt {a x -1}\, \sqrt {a x +1}+280 a^{6} x^{6}+160 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )-340 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-456 a^{4} x^{4}-380 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )+165 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +328 a^{2} x^{2}+256 \,\operatorname {arccosh}\left (a x \right )-88\right )}{60 \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) c^{4} a}+\frac {8 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}-1\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}\) | \(419\) |
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\[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.78 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{60} \, a {\left (\frac {16 \, \sqrt {-\frac {1}{a^{4} c}} \log \left (x^{2} - \frac {1}{a^{2}}\right )}{c^{3}} + \frac {3}{{\left (a^{6} c^{3} x^{4} \sqrt {-\frac {1}{c}} - 2 \, a^{4} c^{3} x^{2} \sqrt {-\frac {1}{c}} + a^{2} c^{3} \sqrt {-\frac {1}{c}}\right )} c} - \frac {8}{{\left (a^{4} c^{2} x^{2} \sqrt {-\frac {1}{c}} - a^{2} c^{2} \sqrt {-\frac {1}{c}}\right )} c^{2}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arcosh}\left (a x\right ) \]
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Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.58 \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {1}{60} \, \sqrt {-c} {\left (\frac {16 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{a c^{4}} - \frac {24 \, a^{4} x^{4} - 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} - 1\right )}^{2} a c^{4}}\right )} - \frac {\sqrt {-a^{2} c x^{2} + c} {\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} - \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{15 \, {\left (a^{2} c x^{2} - c\right )}^{3}} \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]
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