Optimal. Leaf size=890 \[ -\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^5}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{27} b^2 d^2 x^2 \sqrt{d-c^2 d x^2} c^4+\frac{5 b^2 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^3}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 a b d^2 x \sqrt{d-c^2 d x^2} c^3}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{6} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2-\frac{5}{2} d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 d^2 \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right ) c^2}{(1-c x) (c x+1)}-\frac{5 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{5 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{5 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{170}{27} b^2 d^2 \sqrt{d-c^2 d x^2} c^2+\frac{b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} c^2}{9 (1-c x) (c x+1)}+\frac{5 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} c^2}{3 (1-c x) (c x+1)}-\frac{b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2} \]
[Out]
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Rubi [A] time = 1.98538, antiderivative size = 921, normalized size of antiderivative = 1.03, number of steps used = 27, number of rules used = 21, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.724, Rules used = {5798, 5740, 5745, 5743, 5761, 4180, 2531, 2282, 6589, 5654, 74, 5680, 12, 460, 270, 5731, 520, 1251, 897, 1153, 205} \[ -\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^5}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{27} b^2 d^2 x^2 \sqrt{d-c^2 d x^2} c^4+\frac{5 b^2 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^3}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 a b d^2 x \sqrt{d-c^2 d x^2} c^3}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{2} d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2-\frac{5}{6} d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 d^2 \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right ) c^2}{(1-c x) (c x+1)}-\frac{5 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{5 i b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{5 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{170}{27} b^2 d^2 \sqrt{d-c^2 d x^2} c^2+\frac{b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2} c^2}{9 (1-c x) (c x+1)}+\frac{5 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} c^2}{3 (1-c x) (c x+1)}-\frac{b d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5740
Rule 5745
Rule 5743
Rule 5761
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5654
Rule 74
Rule 5680
Rule 12
Rule 460
Rule 270
Rule 5731
Rule 520
Rule 1251
Rule 897
Rule 1153
Rule 205
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^3} \, 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^3} \, 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}{2 x^2}+\frac{\left (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^2} \, 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 \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^3 d^2 x \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^5 d^2 x^3 \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{5}{6} 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-\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}{2 x^2}-\frac{\left (5 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{2 \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{-3-6 c^2 x^2+c^4 x^4}{3 x \sqrt{-1+c x} \sqrt{1+c 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 \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 x \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{2 b c^5 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{6} 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-\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}{2 x^2}+\frac{\left (5 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \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{-3-6 c^2 x^2+c^4 x^4}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \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 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\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{x \left (-3+c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{5 a b c^3 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 x \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{2 b c^5 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{6} 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-\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}{2 x^2}+\frac{\left (5 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \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 ) \int \cosh ^{-1}(c x) \, dx}{\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{x \left (-3+c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{-3-6 c^2 x^2+c^4 x^4}{x \sqrt{-1+c^2 x^2}} \, dx}{3 (-1+c x) (1+c x)}\\ &=\frac{5}{27} b^2 c^4 d^2 x^2 \sqrt{d-c^2 d x^2}+\frac{5 a b c^3 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b^2 c^3 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 x \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{2 b c^5 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{6} 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-\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}{2 x^2}+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 i b c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 i b c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (35 b^2 c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 \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{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{-3-6 c^2 x+c^4 x^2}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{6 (-1+c x) (1+c x)}\\ &=-\frac{170}{27} b^2 c^2 d^2 \sqrt{d-c^2 d x^2}+\frac{5}{27} b^2 c^4 d^2 x^2 \sqrt{d-c^2 d x^2}+\frac{5 a b c^3 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b^2 c^3 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 x \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{2 b c^5 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{6} 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-\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}{2 x^2}+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{-8-4 x^2+x^4}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 (-1+c x) (1+c x)}\\ &=-\frac{170}{27} b^2 c^2 d^2 \sqrt{d-c^2 d x^2}+\frac{5}{27} b^2 c^4 d^2 x^2 \sqrt{d-c^2 d x^2}+\frac{5 a b c^3 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b^2 c^3 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 x \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{2 b c^5 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{6} 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-\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}{2 x^2}+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-5 c^2+c^2 x^2-\frac{3}{\frac{1}{c^2}+\frac{x^2}{c^2}}\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{3 (-1+c x) (1+c x)}\\ &=-\frac{170}{27} b^2 c^2 d^2 \sqrt{d-c^2 d x^2}+\frac{5}{27} b^2 c^4 d^2 x^2 \sqrt{d-c^2 d x^2}+\frac{5 a b c^3 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac{5 b^2 c^3 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 x \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{2 b c^5 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{6} 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-\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}{2 x^2}+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{(-1+c x) (1+c x)}\\ &=-\frac{170}{27} b^2 c^2 d^2 \sqrt{d-c^2 d x^2}+\frac{5}{27} b^2 c^4 d^2 x^2 \sqrt{d-c^2 d x^2}+\frac{5 a b c^3 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac{b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac{5 b^2 c^3 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 x \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{2 b c^5 d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{5}{6} 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-\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}{2 x^2}+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{b^2 c^2 d^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{-1+c^2 x^2}\right )}{(1-c x) (1+c x)}-\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 i b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 95.217, size = 1384, normalized size = 1.56 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.444, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{{x}^{3}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}\, dx \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^{3}}, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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