Optimal. Leaf size=562 \[ -\frac{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 c^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.82139, antiderivative size = 607, normalized size of antiderivative = 1.08, number of steps used = 34, number of rules used = 18, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.621, Rules used = {5798, 5748, 5691, 5688, 5715, 3716, 2190, 2279, 2391, 5716, 39, 5754, 5721, 5461, 4182, 5746, 103, 12} \[ -\frac{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{16 c^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5748
Rule 5691
Rule 5688
Rule 5715
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 5716
Rule 39
Rule 5754
Rule 5721
Rule 5461
Rule 4182
Rule 5746
Rule 103
Rule 12
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (2 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{8 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2 c^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (16 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (16 b c^5 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{8 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (32 b c^5 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{16 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (64 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{16 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (32 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{16 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (16 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{16 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 3.68602, size = 534, normalized size = 0.95 \[ \frac{b^2 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (8 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+8 \text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )+\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{c^3 x^3}+\frac{\cosh ^{-1}(c x)}{1-c^2 x^2}+\frac{\cosh ^{-1}(c x)}{c^2 x^2}-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+\frac{c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{8 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{c x}+\frac{8 c x \cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{c x \cosh ^{-1}(c x)^2}{\left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3}-16 \cosh ^{-1}(c x)^2-16 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-16 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+\frac{a^2 \left (16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right )}{x^3 \left (c^2 x^2-1\right )}+a b c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (\frac{1}{1-c^2 x^2}+\frac{1}{c^2 x^2}+\frac{2 \left (16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right ) \left (\frac{c x-1}{c x+1}\right )^{3/2} \cosh ^{-1}(c x)}{c^3 x^3 (c x-1)^3}-16 \log (c x)-16 \log \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )\right )}{3 d^2 \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.392, size = 5251, normalized size = 9.3 \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{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{10} - 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} - d^{3} 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 (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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