Optimal. Leaf size=638 \[ -\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}{3 x}+\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {7 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{12 \sqrt {c x-1} \sqrt {c x+1}} \]
[Out]
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Rubi [A] time = 1.62, antiderivative size = 669, normalized size of antiderivative = 1.05, number of steps used = 29, number of rules used = 17, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {5798, 5740, 5683, 5676, 5662, 90, 52, 5727, 5660, 3718, 2190, 2279, 2391, 38, 5729, 97, 12} \[ -\frac {7 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt {c x-1} \sqrt {c x+1}}+\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {b c d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{12 \sqrt {c x-1} \sqrt {c x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 38
Rule 52
Rule 90
Rule 97
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5662
Rule 5676
Rule 5683
Rule 5727
Rule 5729
Rule 5740
Rule 5798
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^4} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^4} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}+\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 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^2} \, dx}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{x^2} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (10 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int 3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (10 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {7}{6} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (10 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^4 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 (5 b^2 c^6 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}+\frac {7 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (8 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (20 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^4 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 (5 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b^2 c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (10 b^2 c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{3 x}+\frac {23 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 b c^5 d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \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} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {7 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {14 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7 b^2 c^3 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 3.17, size = 803, normalized size = 1.26 \[ \frac {-12 a^2 c^6 d^3 \sqrt {\frac {c x-1}{c x+1}} x^6+6 a b c^4 d^3 \cosh \left (2 \cosh ^{-1}(c x)\right ) x^4+112 a b c^4 d^3 \log (c x) x^4-3 b^2 c^4 d^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^4-44 a^2 c^4 d^3 \sqrt {\frac {c x-1}{c x+1}} x^4-8 b^2 c^4 d^3 \sqrt {\frac {c x-1}{c x+1}} x^4+20 b^2 c^3 d^3 (c x-1) \cosh ^{-1}(c x)^3 x^3-60 a^2 c^3 d^{5/2} \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right ) x^3-6 a b c^3 d^3 \cosh \left (2 \cosh ^{-1}(c x)\right ) x^3-112 a b c^3 d^3 \log (c x) x^3-56 b^2 c^3 d^3 (c x-1) \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right ) x^3+3 b^2 c^3 d^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^3+8 a b c^2 d^3 x^2+64 a^2 c^2 d^3 \sqrt {\frac {c x-1}{c x+1}} x^2+8 b^2 c^2 d^3 \sqrt {\frac {c x-1}{c x+1}} x^2-8 a b c d^3 x+2 b d^3 (c x-1) \cosh ^{-1}(c x) \left (3 b c^3 \cosh \left (2 \cosh ^{-1}(c x)\right ) x^3+56 b c^3 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right ) x^3-6 a c^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^3-56 a c^3 \sqrt {\frac {c x-1}{c x+1}} x^3-56 a c^2 \sqrt {\frac {c x-1}{c x+1}} x^2+4 b c x+8 a c \sqrt {\frac {c x-1}{c x+1}} x+8 a \sqrt {\frac {c x-1}{c x+1}}\right )-2 b d^3 (c x-1) \cosh ^{-1}(c x)^2 \left (-30 a c^3 x^3+3 b c^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^3+4 b \left (7 c^3 \left (\sqrt {\frac {c x-1}{c x+1}}-1\right ) x^3+7 c^2 \sqrt {\frac {c x-1}{c x+1}} x^2-c \sqrt {\frac {c x-1}{c x+1}} x-\sqrt {\frac {c x-1}{c x+1}}\right )\right )-8 a^2 d^3 \sqrt {\frac {c x-1}{c x+1}}}{24 x^3 \sqrt {\frac {c x-1}{c x+1}} \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.03, size = 3431, normalized size = 5.38 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, {\left (10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d x + 15 \, \sqrt {-c^{2} d x^{2} + d} c^{4} d^{2} x + 15 \, c^{3} d^{\frac {5}{2}} \arcsin \left (c x\right ) + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{x} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{3}}\right )} a^{2} + \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{x^{4}} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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