Integrand size = 27, antiderivative size = 243 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}+\frac {11 b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{6 c^6 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.15 (sec) , antiderivative size = 243, 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 = {272, 45, 5922, 12, 1171, 396, 212} \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {11 b \text {arctanh}(c x) \sqrt {d-c^2 d x^2}}{6 c^6 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )} \]
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Rule 12
Rule 45
Rule 212
Rule 272
Rule 396
Rule 1171
Rule 5922
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arccosh}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-8+12 c^2 x^2-3 c^4 x^4}{3 c^6 d^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 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}-\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {-8+12 c^2 x^2-3 c^4 x^4}{\left (1-c^2 x^2\right )^2} \, dx}{3 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 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 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {17-6 c^2 x^2}{1-c^2 x^2} \, dx}{6 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 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 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}+\frac {\left (11 b \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{6 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 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 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arccosh}(c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^3}+\frac {11 b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{6 c^6 d^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {16 a-24 a c^2 x^2+6 a c^4 x^4+5 b c x \sqrt {-1+c x} \sqrt {1+c x}-6 b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+2 b \left (8-12 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)-11 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \text {arctanh}(c x)}{6 c^6 d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
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Time = 1.30 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.64
method | result | size |
default | \(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{6 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(398\) |
parts | \(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{6 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(398\) |
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Time = 0.32 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.18 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\left [-\frac {8 \, {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 8 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}, \frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) - 4 \, {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}\right ] \]
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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