Optimal. Leaf size=638 \[ \frac{7 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 c^3 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 x^3}+\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}{3 x}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{12 \sqrt{c x-1} \sqrt{c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.61533, 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 5798
Rule 5740
Rule 5683
Rule 5676
Rule 5662
Rule 90
Rule 52
Rule 5727
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 38
Rule 5729
Rule 97
Rule 12
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*}
Mathematica [A] time = 3.3581, size = 803, normalized size = 1.26 \[ \frac{-12 a^2 c^6 d^3 \sqrt{\frac{c x-1}{c x+1}} x^6+6 a b c^4 d^3 \cosh \left (2 \cosh ^{-1}(c x)\right ) x^4+112 a b c^4 d^3 \log (c x) x^4-3 b^2 c^4 d^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^4-44 a^2 c^4 d^3 \sqrt{\frac{c x-1}{c x+1}} x^4-8 b^2 c^4 d^3 \sqrt{\frac{c x-1}{c x+1}} x^4+20 b^2 c^3 d^3 (c x-1) \cosh ^{-1}(c x)^3 x^3-60 a^2 c^3 d^{5/2} \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right ) x^3-6 a b c^3 d^3 \cosh \left (2 \cosh ^{-1}(c x)\right ) x^3-112 a b c^3 d^3 \log (c x) x^3-56 b^2 c^3 d^3 (c x-1) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right ) x^3+3 b^2 c^3 d^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^3+8 a b c^2 d^3 x^2+64 a^2 c^2 d^3 \sqrt{\frac{c x-1}{c x+1}} x^2+8 b^2 c^2 d^3 \sqrt{\frac{c x-1}{c x+1}} x^2-8 a b c d^3 x+2 b d^3 (c x-1) \cosh ^{-1}(c x) \left (3 b c^3 \cosh \left (2 \cosh ^{-1}(c x)\right ) x^3+56 b c^3 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right ) x^3-6 a c^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^3-56 a c^3 \sqrt{\frac{c x-1}{c x+1}} x^3-56 a c^2 \sqrt{\frac{c x-1}{c x+1}} x^2+4 b c x+8 a c \sqrt{\frac{c x-1}{c x+1}} x+8 a \sqrt{\frac{c x-1}{c x+1}}\right )-2 b d^3 (c x-1) \cosh ^{-1}(c x)^2 \left (-30 a c^3 x^3+3 b c^3 \sinh \left (2 \cosh ^{-1}(c x)\right ) x^3+4 b \left (7 c^3 \left (\sqrt{\frac{c x-1}{c x+1}}-1\right ) x^3+7 c^2 \sqrt{\frac{c x-1}{c x+1}} x^2-c \sqrt{\frac{c x-1}{c x+1}} x-\sqrt{\frac{c x-1}{c x+1}}\right )\right )-8 a^2 d^3 \sqrt{\frac{c x-1}{c x+1}}}{24 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.434, size = 3431, normalized size = 5.4 \begin{align*} \text{output 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^{4}}, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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