Integrand size = 29, antiderivative size = 568 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
Time = 0.84 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5934, 5914, 5879, 75, 5912, 5938, 5903, 4267, 2317, 2438, 100, 21} \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {22 b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}+\frac {11 b x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \]
[In]
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Rule 21
Rule 75
Rule 100
Rule 2317
Rule 2438
Rule 4267
Rule 5879
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}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 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)^2 (1+c x)^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^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^2}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 (a+b \text {arccosh}(c x))}{(-1+c x) (1+c x)} \, dx}{3 c^3 d^2 \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))}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \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^3}-\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^2 \sqrt {d-c^2 d x^2}}+\frac {\left (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^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 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}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \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^3}+\frac {\left (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^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{-1+c^2 x^2} \, dx}{3 c^5 d^2 \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^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (-2-2 c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (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^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 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^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \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^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \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^3}+\frac {\left (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^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 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))}{3 c^6 d^2 \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}{3 c^4 d^2 \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^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \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^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \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^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (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^2 \sqrt {d-c^2 d x^2}}+\frac {\left (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^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 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 )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 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 )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \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^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \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^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (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^2 \sqrt {d-c^2 d x^2}}+\frac {\left (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^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 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 )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 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 )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \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^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \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^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 4.39 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.86 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {8 a^2 \left (8-12 c^2 x^2+3 c^4 x^4\right )+2 a b \left (25 \text {arccosh}(c x)-36 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))+3 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))+33 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-33 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+4 \sinh (2 \text {arccosh}(c x))-11 \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right ) \sinh (3 \text {arccosh}(c x))+11 \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right ) \sinh (3 \text {arccosh}(c x))-3 \sinh (4 \text {arccosh}(c x))\right )+b^2 \left (22+25 \text {arccosh}(c x)^2-4 \left (7+9 \text {arccosh}(c x)^2\right ) \cosh (2 \text {arccosh}(c x))+3 \left (2+\text {arccosh}(c x)^2\right ) \cosh (4 \text {arccosh}(c x))-66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )+66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+8 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))+22 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-22 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-6 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{24 c^6 d \left (d-c^2 d x^2\right )^{3/2}} \]
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[Out]
Time = 1.46 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.62
method | result | size |
default | \(a^{2} \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 )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-5 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )-\frac {a 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 )}{3 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(922\) |
parts | \(a^{2} \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 )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-5 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )-\frac {a 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 )}{3 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(922\) |
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
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