Integrand size = 25, antiderivative size = 249 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}-\frac {b (-1+c x)^{3/2}}{12 c^5 d^3 (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c^5 d^3}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3} \]
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Time = 0.18 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5934, 5903, 4267, 2317, 2438, 75, 96, 91, 21, 37} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{4 c^5 d^3}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3}+\frac {3 b}{8 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b (c x-1)^{3/2}}{12 c^5 d^3 (c x+1)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {c x-1} (c x+1)^{3/2}}+\frac {b x^3}{12 c^2 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rule 21
Rule 37
Rule 75
Rule 91
Rule 96
Rule 2317
Rule 2438
Rule 4267
Rule 5903
Rule 5934
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x^3}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 c d^3}-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx}{4 c^2 d} \\ & = \frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {(3 b) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 c^3 d^3}-\frac {b \int \frac {x^2}{(-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{4 c^2 d^3}+\frac {3 \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx}{8 c^4 d^2} \\ & = \frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{8 c^5 d^3}-\frac {b \int \frac {-c+c^2 x}{\sqrt {-1+c x} (1+c x)^{5/2}} \, dx}{4 c^5 d^3} \\ & = \frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c^5 d^3}+\frac {(3 b) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{8 c^5 d^3}-\frac {(3 b) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{8 c^5 d^3}-\frac {b \int \frac {\sqrt {-1+c x}}{(1+c x)^{5/2}} \, dx}{4 c^4 d^3} \\ & = \frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}-\frac {b (-1+c x)^{3/2}}{12 c^5 d^3 (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c^5 d^3}+\frac {(3 b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3}-\frac {(3 b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3} \\ & = \frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}-\frac {b (-1+c x)^{3/2}}{12 c^5 d^3 (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x (a+b \text {arccosh}(c x))}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c^5 d^3}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c^5 d^3} \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.15 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {b (-2+c x) \sqrt {1+c x}}{(-1+c x)^{3/2}}+\frac {b \sqrt {-1+c x} (2+c x)}{(1+c x)^{3/2}}+\frac {12 a c x}{\left (-1+c^2 x^2\right )^2}+\frac {30 a c x}{-1+c^2 x^2}+\frac {3 b \text {arccosh}(c x)}{(-1+c x)^2}-\frac {3 b \text {arccosh}(c x)}{(1+c x)^2}-15 b \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-15 b \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+\frac {9}{2} b \text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-\frac {9}{2} b \text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-9 a \log (1-c x)+9 a \log (1+c x)+18 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-18 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{48 c^5 d^3} \]
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Time = 0.72 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {5}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {15 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+15 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-9 c x \,\operatorname {arccosh}\left (c x \right )-13 \sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c^{5}}\) | \(256\) |
default | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {5}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {15 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+15 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-9 c x \,\operatorname {arccosh}\left (c x \right )-13 \sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c^{5}}\) | \(256\) |
parts | \(-\frac {a \left (\frac {1}{16 c^{5} \left (c x +1\right )^{2}}-\frac {5}{16 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c^{5}}-\frac {1}{16 c^{5} \left (c x -1\right )^{2}}-\frac {5}{16 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c^{5}}\right )}{d^{3}}-\frac {b \left (-\frac {15 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+15 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-9 c x \,\operatorname {arccosh}\left (c x \right )-13 \sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3} c^{5}}\) | \(273\) |
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a x^{4}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b x^{4} \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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