Optimal. Leaf size=607 \[ -\frac{b^2 c d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{15 b c^3 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b \sqrt{c x-1} \sqrt{c x+1}}+\frac{c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{31}{64} b^2 c^2 d^2 x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 c^2 d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}-\frac{89 b^2 c d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 1.29001, antiderivative size = 638, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 16, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.552, Rules used = {5798, 5740, 5685, 5683, 5676, 5662, 90, 52, 5716, 38, 5727, 5660, 3718, 2190, 2279, 2391} \[ \frac{b^2 c d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{15 b c^3 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{4} c^2 d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{31}{64} b^2 c^2 d^2 x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 c^2 d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}-\frac{89 b^2 c d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 \sqrt{c x-1} \sqrt{c x+1}} \]
Warning: Unable to verify antiderivative.
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Rule 5798
Rule 5740
Rule 5685
Rule 5683
Rule 5676
Rule 5662
Rule 90
Rule 52
Rule 5716
Rule 38
Rule 5727
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
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^2} \, 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^2} \, 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}{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 )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\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 )}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\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}{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 ) \left (a+b \cosh ^{-1}(c x)\right )}{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 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{4 \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 ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \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{b c d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\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}{x}+\frac{\left (2 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(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{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \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 c^3 d^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{11}{16} 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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\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}{x}+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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 ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(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^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{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{11 b^2 c d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{15 b c^3 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d^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{5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\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}{x}+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b c 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 )}{\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{\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{89 b^2 c d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d^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{5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\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}{x}+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c 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 )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c 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 )}{\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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d^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{5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\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}{x}+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c 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 )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c 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 )}{\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} \cosh ^{-1}(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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d^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{5}{4} c^2 d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\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}{x}+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c 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 )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{b^2 c d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 5.71888, size = 554, normalized size = 0.91 \[ \frac{d^2 \left (-256 b^2 \sqrt{d-c^2 d x^2} \left (3 c x \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+\cosh ^{-1}(c x) \left (3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+3\right )+6 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )\right )\right )+96 a^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (2 c^4 x^4-9 c^2 x^2-8\right ) \sqrt{d-c^2 d x^2}+1440 a^2 c \sqrt{d} x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-768 a b \sqrt{d-c^2 d x^2} \left (2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left (2 \log (c x)+\cosh ^{-1}(c x)^2\right )\right )+384 a b c x \sqrt{d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )-12 a b c x \sqrt{d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+64 b^2 c x \sqrt{d-c^2 d x^2} \left (4 \cosh ^{-1}(c x)^3+6 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (2 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )-b^2 c x \sqrt{d-c^2 d x^2} \left (32 \cosh ^{-1}(c x)^3+12 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (8 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )\right )}{768 x \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.411, size = 1227, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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