Integrand size = 29, antiderivative size = 556 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}} \]
[Out]
Time = 0.62 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5934, 5938, 5914, 5879, 75, 5883, 102, 12, 5912, 5903, 4267, 2317, 2438} \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}-\frac {2 b x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}+\frac {2 b x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (c x+1)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (c x+1)}{27 c^4 d \sqrt {d-c^2 d x^2}} \]
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Rule 12
Rule 75
Rule 102
Rule 2317
Rule 2438
Rule 4267
Rule 5879
Rule 5883
Rule 5903
Rule 5912
Rule 5914
Rule 5934
Rule 5938
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{c^2 d}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 (a+b \text {arccosh}(c x))}{(-1+c x) (1+c x)} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}-\frac {8 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^4 d}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int x^2 (a+b \text {arccosh}(c x)) \, dx}{3 c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 (a+b \text {arccosh}(c x))}{-1+c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}+\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int (a+b \text {arccosh}(c x)) \, dx}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{-1+c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^2 d \sqrt {d-c^2 d x^2}} \\ & = \frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{-1+c^2 x^2} \, dx}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \text {arccosh}(c x) \, dx}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^4 d \sqrt {d-c^2 d x^2}} \\ & = \frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 (1-c x) (1+c x)}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d \sqrt {d-c^2 d x^2}} \\ & = \frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{c^6 d \sqrt {d-c^2 d x^2}} \\ & = \frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}} \\ & = \frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d^2}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 3.00 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.71 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-36 a^2 \left (-8+4 c^2 x^2+c^4 x^4\right )+3 a b \left (135 \text {arccosh}(c x)-60 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))+72 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-72 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+62 \sinh (2 \text {arccosh}(c x))+\sinh (4 \text {arccosh}(c x))\right )-b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (378 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-378 c x \text {arccosh}(c x)+189 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2-6 \text {arccosh}(c x) \cosh (3 \text {arccosh}(c x))-54 \text {arccosh}(c x)^2 \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )+216 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )-216 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+216 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-216 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+2 \sinh (3 \text {arccosh}(c x))+9 \text {arccosh}(c x)^2 \sinh (3 \text {arccosh}(c x))+54 \text {arccosh}(c x)^2 \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{108 c^6 d \sqrt {d-c^2 d x^2}} \]
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Time = 1.46 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.40
method | result | size |
default | \(a^{2} \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-6 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+36 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-84 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+92 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-72 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+90 c x \,\operatorname {arccosh}\left (c x \right )-94 \sqrt {c x -1}\, \sqrt {c x +1}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{27 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-c^{5} x^{5}+12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-14 c^{3} x^{3}-9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+15 c x +9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{9 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}\) | \(781\) |
parts | \(a^{2} \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-6 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}+36 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-84 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+92 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-72 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}+90 c x \,\operatorname {arccosh}\left (c x \right )-94 \sqrt {c x -1}\, \sqrt {c x +1}+54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+54 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-54 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{27 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (3 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-c^{5} x^{5}+12 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-14 c^{3} x^{3}-9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+15 c x +9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{9 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}\) | \(781\) |
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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