Integrand size = 29, antiderivative size = 607 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
Time = 0.65 (sec) , antiderivative size = 607, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {5928, 5897, 5895, 5893, 5883, 92, 54, 5912, 5914, 38, 5919, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {c x-1} \sqrt {c x+1}}+\frac {c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \log \left (e^{-2 \text {arccosh}(c x)}+1\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 c d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {89 b^2 c d^2 \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}}-\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \]
[In]
[Out]
Rule 38
Rule 54
Rule 92
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5883
Rule 5893
Rule 5895
Rule 5897
Rule 5912
Rule 5914
Rule 5919
Rule 5928
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-\left (5 c^2 d\right ) \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 \, dx+\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x))}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-\frac {1}{4} \left (15 c^2 d^2\right ) \int \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \, dx+\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x (a+b \text {arccosh}(c x)) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{8} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {1}{4} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {13 b^2 c d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {31}{64} b^2 c^2 d^2 x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 c^2 d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{64 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b c^3 d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 5.26 (sec) , antiderivative size = 554, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\frac {d^2 \left (96 a^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (-8-9 c^2 x^2+2 c^4 x^4\right )+1440 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-768 a b \sqrt {d-c^2 d x^2} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (\text {arccosh}(c x)^2+2 \log (c x)\right )\right )-256 b^2 \sqrt {d-c^2 d x^2} \left (\text {arccosh}(c x) \left (3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)-c x \left (\text {arccosh}(c x) (3+\text {arccosh}(c x))+6 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )\right )+3 c x \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+384 a b c x \sqrt {d-c^2 d x^2} (\cosh (2 \text {arccosh}(c x))+2 \text {arccosh}(c x) (\text {arccosh}(c x)-\sinh (2 \text {arccosh}(c x))))+64 b^2 c x \sqrt {d-c^2 d x^2} \left (4 \text {arccosh}(c x)^3+6 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-3 \left (1+2 \text {arccosh}(c x)^2\right ) \sinh (2 \text {arccosh}(c x))\right )-12 a b c x \sqrt {d-c^2 d x^2} \left (8 \text {arccosh}(c x)^2+\cosh (4 \text {arccosh}(c x))-4 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )-b^2 c x \sqrt {d-c^2 d x^2} \left (32 \text {arccosh}(c x)^3+12 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))-3 \left (1+8 \text {arccosh}(c x)^2\right ) \sinh (4 \text {arccosh}(c x))\right )\right )}{768 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
[In]
[Out]
Time = 1.24 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a^{2} c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a^{2} c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a^{2} c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-72 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+72 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-33 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+40 \operatorname {arccosh}\left (c x \right )^{3} x c -64 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}-64 \operatorname {arccosh}\left (c x \right )^{2} x c +128 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \,\operatorname {arccosh}\left (c x \right )+64 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-8 c^{5} x^{5}-144 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+72 c^{3} x^{3}+120 \operatorname {arccosh}\left (c x \right )^{2} x c -128 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-128 c x \,\operatorname {arccosh}\left (c x \right )+128 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\) | \(589\) |
parts | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a^{2} c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a^{2} c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a^{2} c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 \sqrt {c x +1}\, \sqrt {c x -1}\, \operatorname {arccosh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}+2 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{4} x^{4}-72 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+72 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )-33 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+40 \operatorname {arccosh}\left (c x \right )^{3} x c -64 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}-64 \operatorname {arccosh}\left (c x \right )^{2} x c +128 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \,\operatorname {arccosh}\left (c x \right )+64 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-8 c^{5} x^{5}-144 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+72 c^{3} x^{3}+120 \operatorname {arccosh}\left (c x \right )^{2} x c -128 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-128 c x \,\operatorname {arccosh}\left (c x \right )+128 \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) x c -33 c x \right ) d^{2}}{64 \sqrt {c x -1}\, \sqrt {c x +1}\, x}\) | \(589\) |
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\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^2} \,d x \]
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