Integrand size = 27, antiderivative size = 338 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 \sqrt {d-c^2 d x^2}}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.20 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {277, 198, 197, 5922, 12, 1813, 1634} \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3 \sqrt {d-c^2 d x^2}}{6 d^3 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )}+\frac {8 b c^3 \log (x) \sqrt {d-c^2 d x^2}}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 12
Rule 197
Rule 198
Rule 277
Rule 1634
Rule 1813
Rule 5922
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-1-6 c^2 x^2+24 c^4 x^4-16 c^6 x^6}{3 d^3 x^3 \left (1-c^2 x^2\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-1-6 c^2 x^2+24 c^4 x^4-16 c^6 x^6}{x^3 \left (1-c^2 x^2\right )^2} \, dx}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-1-6 c^2 x+24 c^4 x^2-16 c^6 x^3}{x^2 \left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{x^2}-\frac {8 c^2}{x}+\frac {c^4}{\left (-1+c^2 x\right )^2}-\frac {8 c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 \sqrt {d-c^2 d x^2}}{6 d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}-\frac {a+b \text {arccosh}(c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.70 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {2 a+12 a c^2 x^2-48 a c^4 x^4+32 a c^6 x^6-b c x \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \text {arccosh}(c x)-16 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \log (x)+8 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )-8 b c^5 x^5 \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{6 d^2 x^3 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
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Time = 1.15 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.19
method | result | size |
default | \(a \left (-\frac {1}{3 d \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+2 c^{2} \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (32 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{6} x^{6}+32 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{7} c^{7}-48 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-64 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+32 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{5} c^{5}+12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}-c^{3} x^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x^{3}}\) | \(403\) |
parts | \(a \left (-\frac {1}{3 d \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+2 c^{2} \left (-\frac {1}{d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+4 c^{2} \left (\frac {x}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}\right )\right )\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (32 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{6} x^{6}+32 \,\operatorname {arccosh}\left (c x \right ) c^{7} x^{7}-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{7} c^{7}-48 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-64 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+32 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{5} c^{5}+12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+32 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-16 \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{4}-1\right ) x^{3} c^{3}-c^{3} x^{3}+2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x \right )}{6 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) x^{3}}\) | \(403\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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none
Time = 0.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, b c {\left (\frac {8 \, c^{2} \sqrt {-d} \log \left (c x + 1\right )}{d^{3}} + \frac {8 \, c^{2} \sqrt {-d} \log \left (c x - 1\right )}{d^{3}} + \frac {16 \, c^{2} \sqrt {-d} \log \left (x\right )}{d^{3}} + \frac {\sqrt {-d}}{c^{2} d^{3} x^{4} - d^{3} x^{2}}\right )} + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} b \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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